Welcome to Alan Thompson's 2 Unit Mathematics Quiz Sheets Links: Matching Items in Printable Format.

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  Question/Answer single topic sheet links below


  

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   Question Sheets collated with Answer Sheets, all 59 topics.


  

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   Question Sheets only, all 59 topics.


  

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The following links open Printable Single Topic pages.

 These pages are composed of a page of matching items and a page with the answers. Using CTRL-P or right-click-mouse (then select print from pop-up menu), will lead to a 2 page print out. Questions on the first page. Answers on the second page.


    Worksheet   1    Basic Arithmetic and Algebra 1

    Worksheet   2    Basic Arithmetic and Algebra 2

    Worksheet   3    Basic Arithmetic and Algebra 3

    Worksheet   4    Basic Arithmetic and Algebra 4

    Worksheet   5    Basic Arithmetic and Algebra 5

    Worksheet   6    Basic Arithmetic and Algebra 6

    Worksheet   7    Basic Arithmetic and Algebra 7

    Worksheet   8    Plane Geometry 1

    Worksheet   9    Plane Geometry 2

    Worksheet  10   Plane Geometry 3

    Worksheet  11   Plane Geometry 4

    Worksheet  12   Plane Geometry 5

    Worksheet  13   Plane Geometry 6

    Worksheet  14   Functions and Graphs 1

    Worksheet  15   Functions and Graphs 2

    Worksheet  16   Functions and Graphs 3

    Worksheet  17   Functions and Graphs 4

    Worksheet  18   Functions and Graphs 5

    Worksheet  19   Trigonometry 1

    Worksheet  20   Trigonometry 2

    Worksheet  21   Trigonometry 3

    Worksheet  22   Trigonometry 4

    Worksheet  23   Trigonometry 5

    Worksheet  24   Trigonometry 6

    Worksheet  25   Straight Line Graphs 1

    Worksheet  26   Straight Line Graphs 2

    Worksheet  27   Straight Line Graphs 3

    Worksheet  28   Straight Line Graphs 4

    Worksheet  29   Introduction to Calculus 1

    Worksheet  30   Introduction to Calculus 2

    Worksheet  31   Quadratic Functions 1

    Worksheet  32   Quadratic Functions 2

    Worksheet  33   Quadratic Functions 3

    Worksheet  34   Locus Parabola Circle 1

    Worksheet  35   Locus Parabola Circle 2

    Worksheet  36   Locus Parabola Circle 3

    Worksheet  37   Series 1 1

    Worksheet  38   Series 1 2

    Worksheet  39   Series 1 3

    Worksheet  40   Geometrical Applications of Calculus 1

    Worksheet  41   Geometrical Applications of Calculus 2

    Worksheet  42   Geometrical Applications of Calculus 3

    Worksheet  43   Integration 1

    Worksheet  44   Integration 2

    Worksheet  45   Integration 3

    Worksheet  46   Integration 4

    Worksheet  47   Exponential and Logarithmic Functions 1

    Worksheet  48   Exponential and Logarithmic Functions 2

    Worksheet  49   Exponential and Logarithmic Functions 3

    Worksheet  50   Exponential and Logarithmic Functions 4

    Worksheet  51   Trigonometric Functions 1

    Worksheet  52   Trigonometric Functions 2

    Worksheet  53   Trigonometric Functions 3

    Worksheet  54   Applications of Calculus to the Real World 1

    Worksheet  55   Applications of Calculus to the Real World 2

    Worksheet  56   Series 2 1

    Worksheet  57   Series 2 2

    Worksheet  58   Probability 2 1

    Worksheet  59   Probability 2 2



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Questions: Sheet

1

Basic Arithmetic and Algebra 1

  1   rational number any real number that can be expressed in the form a/b (but b is not 0), includes integers, fractions, percentages, definite decimals, repeating decimals, surds with integer roots (e.g. √4), and can be negative.
  2   4x2 +12x + 9 = (2x+3)2 an example of an algebraic perfect square: the coefficient of x equals twice the product of the square root of the coefficient of x2 and the square root of the third expression (a positive integer)
  3   substitution evaluation of expressions involving the four operations, and powers and roots, by replacing pronumerals with integers, fractions, decimals or surds. Also used with common formulae (e.g. F = ma)
  4   grouping in pairs a factorisation process used to determine whether or not other factorisation processes can be used eg common factor, difference of squares, trinomials
  5   hollow points absolute value terms like |x - y| are represented on the number line by these at the extremities of the range
  6   square root and absolute value √x2 = |x|: square root of a number is an absolute value, i.e. it can be expressed as either + or -
  7   rationalising converting the denominator of a fraction containing surds into a rational number by multiplying top and bottom by the surd, or using the quadratic difference of squares principle
  8   |x - y| the distance between x and y on the number line, hollow points indicate the extremities of the range
  9   a1/n = n√a fraction indice to root rule
10   absolute value arithmetic size of a number, disregarding its signage (+ or -)
11   am×an = am+n adding indices when multiplying a base rule
12   am/n = n√am positive fraction indice to root rule
13   significant figures the digits in a number that begin with the first non-zero digit, and which correspond to the accuracy of data, or which have been requested
14   subtract indices process used when dividing a number (root) that has been expressed with indices
15   repeating decimal to fraction let the repeating decimal equal n, multiply by a suitable factor of 10 (e.g. 10, or 100), then subtract n, and hence 9n = a whole number, or 99n = a whole number etc. divide by the 9 or 99 and cancel down to the simplest fraction
16   n decimal places this is the number of digits required after a decimal point if rounding off correct to n decimal places



  Basic Arithmetic and Algebra 1Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

2

Basic Arithmetic and Algebra 2

  1   order of operation when expanding algebraic expression: brackets, indices, division and multiplication (from left to right), addition and subtraction (from left to right) BIDMAS
  2   100n - 10n = 1a.aaaaa - 1.aaaaa repeating decimal to fraction method for converting 0.1aaaaa to a fraction, e.g. 0.13333. = 12/90
  3   adding fractions process that involves getting a common denominator by multiplying denominators, and getting a numerator by multiplying each numerator by the other (fraction's original) denominator and then adding the numerators derived
  4   [a/b + c/d] = (da + bc)/bd adding fractions
  5   perfect squares trinomial expressions used in factorisation where the result is the sum or difference of two numbers squared eg a2 ± 2ab + b2
  6   irrational number any real number that cannot be expressed in the form a/b, surds that do not have integer roots, π, natural logarithm base: e, indefinite decimals (can be negative)
  7   simplification the removal of grouping symbols and the collecting of like terms involving numbers and algebraic expressions
  8   |x| a point pair of points on the number line, locating how far x is from the origin, blackened points
  9   (x > -y) = (-x < y) inequality and negative number multiply or divide resulting in reversal of the inequality symbol
10   sum of two cubes factorisation pattern for two cubed numbers if they are added: a3 + b3 = (a + b)(a2 - ab + b2)
11   root index (n√x) the denominator of an index expressed as a fraction e.g. x1/n
12   √(x2) = |x| square root of a number is an absolute value, ie it can be expressed as either + or -
13   add indices process used with multiplying a number (root) that has been expressed with indices
14   test solution a method used to determine if points on a number line solve an absolute value inequality
15   rounding off writing a number with a prescribed number of digits
16   two possible answers these are written out for solutions of absolute value calculations involving pronumerals, to take into account both the positive and the negative outcomes e.g. evaluate |2x-1| = 13



  Basic Arithmetic and Algebra 2Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

3

Basic Arithmetic and Algebra 3

  1   evaluate convert a complicated number to its simplest form
  2   a-m/n = 1/(n√(nam) negative fraction indice to root rule
  3   transcendental a real irrational number that is not the root of any polynomial equation with rational roots, or a surd(e.g. include π, base of natural logarithms e, log23)
  4   am/an = am-n rule for subtracting indices and the division of a base
  5   non-significant zeros zero digits prior to non-zero digits after a decimal place, eg 0.002345235
  6   quadratic surds real numbers in the form of integers mixed with surds, allowing factorisation or rationalisation of denominators using the rules of order of operations, fractions and algebra (difference of squares)
  7   fraction from recurring decimal multiply by a power of ten and subtract to eliminate the recurring part
  8   |a| this absolute value equals +a when a is greater than or equal to zero, or equals -a when a is less than zero
  9   multiplying fractions process: multiply numerators and also multiply denominators.
10   blackened points |x| is represented by these on the number line, locating how far x is from the origin
11   real number any rational number (can be expressed as a/b) or irrational number (indefinite non-repeating decimals)
12   reverses this happens to an inequality sign if the inequality is multiplied or divided by a negative numbers
13   a-n = 1/(an) negative index to positive index rule
14   parentheses brackets, grouping symbols
15   a3 + b3 = (a + b)(a2 - ab + b2) factorising sum of two cubes
16   rounded down this is done if the digit beyond significant figures is less than 5 when rounding off to a prescribed number of significant figures



  Basic Arithmetic and Algebra 3Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

4

Basic Arithmetic and Algebra 4

  1   common factor factorisation process involving the grouping of numbers that have a common factor in parentheses
  2   denominator number on the bottom of a fraction
  3   triangle inequality |a+b| ≤ |a| + |b|: the absolute value of a sum of two numbers is less than or equal to the sum of the absolute values of the two numbers
  4   fraction a real rational number that takes the actual form of a/b
  5   |x| = +x or -x absolute numbers have two values
  6   index the power to which a root is raised to obtain a particular number e.g. 34 = 81 (the 4)
  7   multiply indices process used with a number (the root) that has been expressed with an index to the power of another index in order to provide a single indice
  8   significant zeros zero digits amongst non-zero digits, and zero digits that occur after non-zero numbers after a decimal place, eg 2.0235100 has 3 of these
  9   x2 - 2x - 15 = (x+3)(x-5) quadratic factorisation
10   exact answers these are obtained when irrational numbers are expressed in surd (√2) or symbol form (π); or certain fractions e.g. 1/7
11   number line a simple graphing axis that can be used to illustrate the location of values from absolute value calculations
12   percentage (amount divided by total) times 100
13   (x > y ) = (ax > ay) inequality and positive number multiply or divide, same to both sides, inequality symbol retains direction
14   (a/b) x (c/d) = (ac)/(bd) multiplying fractions
15   reciprocal to convert a negative index into a positive index, the number is written as the denominator, with a positive index, eg x-2
16   imaginary number any number that involves the square root of a negative number



  Basic Arithmetic and Algebra 4Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

5

Basic Arithmetic and Algebra 5

  1   at least one number of extra digits that must be used at each step of a computation to find a final answer before rounding off to requested significant figures
  2   rounded up this is done if the digit beyond significant figures is 5 or higher when rounding off to a prescribed number of significant figures
  3   numerator number on the top of a fraction
  4   difference of two cubes factorisation pattern for two cubed numbers if they are subtracted: a3 - b3 = (a - b)(a2 + ab + b2)
  5   surd an irrational number expressed with a root symbol (square root, cube root), and which does not have integer roots
  6   difference of two squares factorisation process that can be done with pronumerals and numbers that have integer square roots e.g. 4x2 - 1
  7   |a+b| ≤ |a| + |b| triangle inequality, the absolute value of a sum of two numbers is less than or equal to the sum of the absolute values of the two numbers
  8   conjugate of a surd the number that will result in a rational number when multiplied by the surd e.g. √a x √a or (√a + √b)(√a - √b)
  9   root a number with is multiplied a given number of times (its index) to obtain a particular number e.g. for 34 = 81 it’s the 3
10   quadratics trinomial expressions used in factorisation where the result is different groups of numbers multiplied together, eg, 2x2 + x - 6
11   (am)n = amn multiplying indices when an indexed number is raised to the power of a further index
12   > and < and absolute value |a|: this absolute value equals +a when a is greater than or equal to zero, or equals -a when a is less than zero
13   one (1) the value of any number raised to the power of index 0
14   remains the same this happens to inequality signs if a number is added to or subtracted from both sides of an inequality or if the inequality is multiplied or divided by a positive number
15   inexact answers these are obtained when irrational numbers (surds, π) and certain fractions (2/7), are expressed in decimal form
16   a0 = 1 any number to the power of zero



  Basic Arithmetic and Algebra 5Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

6

Basic Arithmetic and Algebra 6

  1   %= amount x 100/total percentage formula
  2   removing the first period technique that is used to convert repeating decimals to fractions, and add it on after, e.g. 5.2636363 = 5 + 0.2636363 = 5 29/110
  3   dividing by fraction process: invert and multiply
  4   scientific notation also called standard notation: expressing a number as a number between 1 and 10 multiplied with a power of 10
  5   complex number the sum of a real number and an imaginary number (ie a mixture of numbers that includes those that can be rational (a/b) or integers, or irrational numbers (indefinite non-repeating decimals) and the square root of a negative number)
  6   [a/b ÷ c/d] = a/b x d/c = ad/(bc) dividing by a fraction
  7   integer any positive or negative whole number, including 0
  8   a3 - b3 = (a - b)(a2 + ab + b2) factorising the difference between two cubes
  9   (x2 - 1) = (x + 1)(x - 1) difference of squares
10   factorise and cancel process for simplifying algebraic fractions
11   product and absolute value |ab| = |a|×|b|: absolute value of a product of two numbers equals the value of the product of the absolute values of the two numbers
12   trinomials quadratic expressions and perfect squares used in factorisation process involving three numbers, at least one of which is squared
13   √a x √a conjugate of a surd, simple multiply
14   |ab| = |a|×|b| absolute value of a product of two numbers equals the value of the product of the absolute values of the two numbers
15   inequalities equations expressed with < and > symbols with or without = symbol (i.e. ≤ and ≥)
16   (√a - √b)(√a + √b) conjugate of a surd, difference of squares method



  Basic Arithmetic and Algebra 6Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

7

Basic Arithmetic and Algebra 7

  1   simultaneous equations two equations with two unknowns that are satisfied by the same solution, solved by substitution or elimination
  2   substitution method of solving simultaneous equations in which an expression is formed for one of the unknowns from one equation and then used to replace the same unknown in the other equation
  3   elimination method of solving simultaneous equations in which equations are added or subtracted after ensuring that one of the unknowns has equal coefficients in both equations
  4   a3 - b3 = (a - b)(a2 + ab + b2) factorisation of a difference of two cubes
  5   check by substitution this process should be carried out to ensure that the correct solution for two unknowns have been found for simultaneous equations
  6   quadratic formula a formula that is used to find the possible values of an unknown in a quadratic equation (unknown to the power of 2): x = (- b ± √(b2-4ac))/(2a)
  7   factorising process of simplifying a quadratic expression by converting it to bracketed expressions
  8   x = (- b ± √(b2-4ac))/(2a) quadratic equation solution formula
  9   zero a rule used to find the possible values of an unknown in a quadratic equation which can be factorised: if the product of two numbers is zero then at least one number must be ........
10   quadratic inequations these are solved by quadratic equation solution, plotting answers on a number line, and then testing by substituting values either outside or inside to determined the range
11   linear equations simple equations with one pronumeral, with an assumed indice of 1
12   use ab and a+b for factorising simple quadratic equations (coefficient of x2 is 1): example: x2 + (a+b)x + ab
13   y = 3x + 7 example of a linear equation (the assumed indice is 1)
14   equation solving isolate the unknown by performing the same operation to both sides of the equation
15   x2 + (a+b)x + ab expansion of the quadratic expression: (x + a)(x + b)
16   sum of two cubes factorisation pattern for two cubed numbers if they are added: a3 + b3 = (a + b)(a2 - ab + b2)



  Basic Arithmetic and Algebra 7Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

8

Plane Geometry 1

  1   collinear points a series of points through which a straight line passes
  2   specific mention Syllabus: any results may be used by students in the proofs of subsequent exercises as long as ........ is made of it in the solution of the exercise (the working out method counts)
  3   common side term describing the situation where two triangles have been drawn in such a way that one side is a part of each triangle
  4   foot of perpendicular a point on a line where another line meets the line at right angles
  5   square area side length squared, l2
  6   parallelogram tests a quadrilateral is this if: opposite sides are equal; opposite angles are equal; one pair of opposite sides are parallel and equal OR diagonals bisect each other
  7   rhombus a parallelogram with adjacent sides equal (hence all sides are equal), diagonals bisecting at right angles and bisecting the angles they pass through
  8   l2 area of a square formula
  9   corresponding angles similarity test for triangles: 3 pairs of these are equal; (actually it need only be two pairs because the third angle of the triangles must then be equal)
10   square a rectangle with a pair of adjacent sides equal, diagonals are perpendicular (and equal), and make angles of 45o with the sides (bisect the angles they pass through)
11   ∵ symbol for because
12   corresponding sides similarity test for triangles: three pairs of these are in proportion
13   equilateral triangle a triangle with 3 equal sides, and 3 equal angles (60o each)
14   sketch diagram a quickly drawn diagram, roughly to scale, used to indicate size of angles, length of lines, parallel lines, in pencil, with straight lines; drawings of data from written expressions
15   parallel lines lines that never intersect
16   interior angles the three angles of a triangle, which sum up to 180o



  Plane Geometry 1Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

9

Plane Geometry 2

  1   intersecting lines lines that cross
  2   acute angle an angle between 0o and 90o
  3   regular polygon a polygon with all sides equal and all angles equal
  4   ½h(a+b) trapezium area half perpendicular distance between parallel sides time the sum of the parallel sides
  5   lb rectangle area, length times bredth
  6   allied angles angles that form when a transversal crosses parallel lines, located between the parallel lines, and on one side of the transversal, they are supplementary, also called cointerior angles.
  7   cointerior angles angles that form when a transversal crosses parallel lines, located between the parallel lines, and on one side of the transversal, have a sum of 180o
  8   diagonals lines joining opposite corners of a quadrilateral, they bisect each other if in a parallelogram, and are of equal length if the parallelogram is a rectangle or square, and are perpendicular for square and rhombus
  9   cyclic order sequence in which the angles of shapes (triangle, parallelogram quadrilateral, polygon) should be given: ABCD etc
10   similar triangles two triangles that have the same shape but not necessarily the same size, satisfying one of 3 tests concerning corresponding angles, sides ratio or a combination of these
11   rhombus tests a quadrilateral is this if: all sides are equal OR diagonals bisect each other at right angles
12   ∴ symbol for "is congruent to", ie, is exactly the same shape and size, corresponding sides and angles are equal
13   triangle area half the base times the perpendicular height: ½bh
14   isosceles triangle a triangle that has two equal sides as well as the angles opposite these sides (the base angles) are equal in size
15   trapezium a quadrilateral with one pair of sides parallel
16   vertically opposite term describing angles between two intersecting lines which are equal and directly opposite each other



  Plane Geometry 2Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

10

Plane Geometry 3

  1   sides, angles similarity test for triangles: two pairs of ....... are in proportion and their included ....... are equal
  2   ∠ABC standard notation for angles, three letter, with vertex of angle at the centre letter
  3   exterior angle an angle on the outside of a triangle with one side extended, and which is equal to the sum of the 2 opposite interior angles
  4   transversal a line that crosses (intersects with) two parallel lines
  5   square a rectangle with a pair of adjacent sides equal.
  6   right angle an angle of 90o
  7   theoretical exercises these involve proving congruency, similarity, shape, or finding angle or side comparisons, given only that sides have the same length, bisect, are at right angles, or are parallel
  8   obtuse angle an angle between 90o and 180o
  9   rectangle area length times breadth, lb
10   rectangle a parallelogram (i.e. a shape with the same properties as a parallelogram) but additionally has diagonals of equal length
11   RHS rule triangle congruency test: both triangles have a right angle, their hypotenuses are equal and one other pair of corresponding sides are equal
12   parallel two lines are this, if they have both been shown to be parallel to a third line
13   ∴ symbol for therefore
14   extension part of a line beyond the given amount, eg X is a point on the line AB extropolation
15   right triangle a triangle that has a right angle
16   rectangle tests a quadrilateral is this if: diagonals are equal
17   angle of revolution an angle of 360o, also termed an angle at a point



  Plane Geometry 3Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

11

Plane Geometry 4

  1   ½bh triangle area
  2   kite a quadrilateral with two pairs of adjacent sides equal
  3   Pythagoras' theorem the square on the hypotenuse in any right-angled triangle is equal to the sum of the squares on the other two sides, c2 = a2 + b2
  4   radius when triangles are drawn in a circle with an angle at the centre and the base cutting out a secant of the circle, then two sides are equal because they are both a ..........
  5   Δ ABC standard notation for triangle ABC
  6   rhombus a parallelogram with a pair of adjacent sides equal
  7   arrow heads used to indicate that two lines are parallel
  8   numerical exercises in plane geometry mathematics, these involve proving congruency, similarity, shape, or finding the size of angles or side length, given numerical values for angles and or sides
  9   alternate angles angles that form when a transversal crosses parallel lines, and are located between the parallel lines and on opposite sides of the transversal, they are equal in size
10   trapezium area half the perpendicular height times the sum of the length of the parallel sides: ½h(a+b)
11   corresponding angles angles that form when a transversal crosses parallel lines, located on the same side of the transversal, and each parallel line, they are equal in size
12   bisect to cut into equal halves, the diagonals of a parallelogram do this, producing a pair of congruent triangles
13   complementary angles two angles whose sum is 90o
14   AAS congruency test: two pairs of angles and one pair of corresponding sides are equal
15   ⊥ symbol for is perpendicular to, ie, meets at right angles
16   scalene triangle a triangle that has no sides equal, all sides of different lengths



  Plane Geometry 4Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

12

Plane Geometry 5

  1   proportion a line parallel to one side of a triangle divides the other two sides in the same …...
  2   reflex angle an angle between 180o and 360o
  3   lengths in triangle congruency proof statements, these are being referred to when stating e.g. AB = CD
  4   trapezium a quadrilateral is this if: two sides are parallel
  5   parallelogram area base times perpendicular height: bh
  6   argument sequence of steps in proving a congruency, similarity, shape
  7   polygons geometrical figures in which an area is bounded by any number of straight sides (3 or more), external angles total 360o and interior angles total (2n - 4) x 90o, where n is the number of sides
  8   rectangle a parallelogram with one angle a right angle
  9   c2 = a2 + b2 Pythagoras theorem
10   similar triangle properties corresponding angles are equal and corresponding sides are in proportion
11   congruent triangles triangles that are exactly the same shape and size, satisfying one of four tests: SSS, SAS, AAS, or RHS
12   ½xy rhombus area
13   cross strokes used on geometrical diagrams to indicate that two lines are equal in length
14   regular polygons polygons with all sides equal and all interior angles equal
15   ∥ symbol for is parallel to, ie two lines that never meet
16   SAS congruency test: two pairs of corresponding sides and their included angles are equal
17   cointerior angles angles that form when a transversal crosses parallel lines, located between the parallel lines, and on one side of the transversal, they are supplementary, also called allied angles



  Plane Geometry 5Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

13

Plane Geometry 6

  1   ⦀ symbol for is similar to, ie, corresponding angles are equal, and the corresponding sides are in the same proportions (ratios) as other corresponding sides
  2   straight angle an angle of 180o
  3   intercepts distance along transversals cut off by parallel lines. For a group of parallel lines these have the same ratio
  4   supplementary angles two angles whose sum is 180o
  5   midpoints a line joining the ........ of two sides of a triangle is parallel to the third side and half its length
  6   lines or intervals these are being referred to when stating AB∥CE
  7   converse the opposite, a statement turned the other way around, eg p = q has the converse q = p Note, that which holds true in the forward direction may be untrue in the reverse direction eg honest but poor has different meaning to poor but honest
  8   kite a quadrilateral is this if: two pairs of adjacent sides are equal
  9   parallelogram a quadrilateral with both pairs of opposite sides are parallel
10   bh parallelogram area
11   square a quadrilateral is this if: diagonals are equal AND adjacent sides are equal
12   (2n - 4) x 90o total of internal angles for geometrical figures bounded by 3 or more straight sides
13   rhombus area half the product of the lengths of the diagonals: ½xy where x and y are the lengths of the diagonals
14   quadrilateral a polygon with 4 sides, interior angles add up to 360o
15   360o total of external angles for geometrical figures bounded by 3 or more straight lines
16   SSS congruency test: all 3 pairs of corresponding sides of two triangles are equal in length
17   parallelogram properties quadrilateral with both pairs of opposite sides equal, both pairs of opposite sides parallel, both pairs of opposite angles equal, one pair of opposite sides are equal and parallel, and/or diagonals bisect each other



  Plane Geometry 6Two page printable: Student Answer Sheet followed by the Answers


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14

Functions and Graphs 1

  1   y = ax2 + bx +c general parabola equation
  2   Cartesian coordinate x and y values for plotting on a grid with appropriate axes at right angles
  3   y = -a directrix equation, for the parabola x2 = 4ay
  4   (x - a)2 + (y - b)2 = r2 general circle equation
  5   y-axis even functions are symmetrical about this when they have been graphed (ie, when f(x) = f(-x))
  6   directrix a line that is equidistant from the locus of a parabola with the distance from the focus to the locus of the parabola, equation: y = -&frac14; coefficient of x2
  7   locus a set of points obeying a given condition
  8   boundary term describing the location of a region on a graph representing a system of inequalities where a line forming an outline of the area is being referred to
  9   simultaneous inequations used to find regions on a graph grid which satisfies all the relations indicated
10   f(x) = -f(-x) odd function test result
11   axis a line through the focus of a parabola and perpendicular to the directrix of the parabola
12   range if y = f(x), this is all the set of all possible values of the dependent variable y
13   focus point on a parabola that is equidistant from the locus of the parabola with the perpendicular distance from the directrix to the locus, (&frac14; coefficient of x2)
14   pair of lines the locus of points that satisfies the equation x2 = y2 is this because the square root of a number is either positive or negative
15   f(x) the value function f at x



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15

Functions and Graphs 2

  1   relation a rule uniting two quantities where one quantity (the dependent variable) depends on the value of the other quantity (the independent variable) eg s = u + at, s relates to t, or y = 2x - 3, relates x and y
  2   straight line ax + by + c = 0, when graphed provides this shape
  3   parabola formula x2 = 4ay, concave up, y axis symmetry, focus (0,a), directrix y = -a, axis equation x = 0, vertex (0,0)
  4   logarithmic function y = logax, when graphed, provides this shape
  5   independent variable x is called this because it may be chosen freely within the domain of function f
  6   table of values technique used to locate key points for sketching graphs for relations (including functions), eg y vs x or f(x) vs x, includes 0 for axes intercepts if applicable
  7   bounded by term describing the location of a region on a graph representing a system of inequalities where the equation of the line or curve is being referred to
  8   broken the line of a graph of an inequation is drawn like this if the equation is not included in the region, together with appropriate shading, eg y > x2
  9   asymptote a straight line that continually approaches a curve but does not meet it within a finite distance, e.g. for the f(x) = 1/x, it's the x and y axes
10   vertex the point where the axis of a parabola cuts the parabola
11   odd function a function is this if it is changed (in sign) by substitution of a -x for each x, i.e. f(x) = -f(-x), e.g. f(x) = x3
12   semicircle y = ± √(r2 - x2), when graphed, provides this shape
13   origin odd functions are symmetrical about this, when they have been graphed (i.e., when f(x) = -f(-x))
14   arc term describing the location of a region on a graph representing a system of inequalities where portion of a curve between two points on the curve are being referred to
15   discontinuous function a relation that has sudden jumps in value, or takes widely differing values over a short range, e.g. hyperbola, y = 1/x, can't have x = 0 (divide by zero), thus there is a gap in the curve



  Functions and Graphs 2Two page printable: Student Answer Sheet followed by the Answers


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16

Functions and Graphs 3

  1   f(x) = +f(-x) and -f(-x) neither odd nor even function test result, eg for x2 + 2x
  2   parabola the locus of a point whose distance from a given fixed point (its focus) equals its distance from a given fixed line (its directrix)
  3   (0,a) focus coordinates for the parabola x2 = 4ay
  4   y = ax exponential equation
  5   x2 = y2 equation that results in two lines because the square root of a number is either positive or negative
  6   domain of f the set of real numbers on which function f is defined (values of x)
  7   function special type of relation where for every value of the independent variable (x) there is only one value of the dependent variable (y); checked with the vertical line test (a circle is not a function)
  8   ax + by + c = 0 straight line equation
  9   focus the point in a parabola shaped reflector where light from a source will reflect off the parabola to form a parallel beam of light
10   exterior to term describing the location of a region on a graph representing a system of inequalities where the area is totally outside
11   unbroken the line of a graph of an inequation is drawn like this if the equation is included in the region, together with appropriate shading, eg y ≥ x2
12   f(x) = f(-x) even function test result
13   function properties look for: is it positive, negative, zero?, where is it increasing, decreasing?, is there symmetry properties?, are there gaps? sharp corners? is there an asymptote?
14   (0,0) vertex coordinates for the parabola x2 = 4ay
15   sector term describing the location of a region on a graph representing a system of inequalities where the area is bounded by a curve and a straight line



  Functions and Graphs 3Two page printable: Student Answer Sheet followed by the Answers


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17

Functions and Graphs 4

  1   y = ± √(r2 - x2) semi circle equation
  2   domain if y = f(x), this is all the set of all possible values of the independent variable x
  3   continuous function a relation that does not suddenly jump in value or take widely differing values over a short range or domain, no gaps in the curve/line
  4   neither odd nor even a function is this, if it is only partially changed by the substitution of -x for each x, e.g. f(x) = x2 + 2x
  5   describing regions using terms such as interior to, inside, exterior to, outside, bounded by, boundary, sector, common to, arc (in relation to a graphed system of inequalities)
  6   exponential functions y = ax, when graphed, provides this shape
  7   focal length distance from the focus of a parabola to the vertex (along the axis), &frac14; coefficient of x2
  8   circle the locus of points about a particular point is this
  9   range or image the set of values of f(x) obtained as x varies over the domain of the f (function)
10   x2 = 4ay formula for a parabola which is concave up, y axis symmetry, focus (0,a), directrix equation y = -a, axis equation x=0, vertex (0,0)
11   vertical line test a graph of a relation is a function if this results in just one value/intersection with the graph
12   parabola y = ax2 + bx +c, when graphed provides this shape
13   y = logax logarithmic function equation
14   function notation method of writing a function, or the values that are to be substituted into a function: y = f(x), f(x) = 2x - 6, f(3) allows 3 to be substituted for x in a given function, f(x) = 3 means find x if the expression = 3
15   region this is used to represent an inequality in x and y on a graph grid, they are tested with substituted values from inside and outside, and then shaded



  Functions and Graphs 4Two page printable: Student Answer Sheet followed by the Answers


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18

Functions and Graphs 5

  1   dependent variable y = f(x) is called this because its value depends on the value chosen for x
  2   even function a function is this if it is not changed by the substitution of a -x for each x, i.e. f(x) = f(-x), e.g. f(x) = x4
  3   circle (x - a)2 + (y - b)2 = r2, when graphed provides this shape, with centre at (a, b)
  4   interior to term describing the location of a region on a graph representing a system of inequalities where the area is totally inside, eg a circle
  5   inequalities these can be used in pairs of functions to indicate that below a certain value y = one f(x), while above that value y = another f(x), resulting in a composite graph
  6   curve sketching symmetry of odd and even functions (f(x) = -f(-x), and f(x) = f(-x) respectively) is useful for this purpose
  7   hyperbola xy = a or y = a/x, when graphed, provides this shape
  8   find the locus this means find a simple algebraic or geometric description of the set of all points which satisfy a function
  9   x = 0 axis of symmetry equation for the parabola x2 = 4ay
10   common to term describing the location of a region on a graph representing a system of inequalities where the area is found in both eg, at the intersection of two circles
11   xy = a or y = a/x hyperbola equation
12   limits these are values for x where f(x) approaches zero, and because y = f(a)/f(x), domain is expressed as: all real x, x ≠ b, x ≠ c etc
13   essential features sketches drawn of functions must show these, eg, shape, interceptions with the axes, arrows to indicate the line/curve continues
14   a focal length for the parabola x2 = 4ay
15   straight line the locus of points equidistant from two given points A and B, found by circle A = circle B (=r2)



  Functions and Graphs 5Two page printable: Student Answer Sheet followed by the Answers


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19

Trigonometry 1

  1   OPP/ADJ tangent ratio for a right angle triangle
  2   sin 30o, cos 60o, tan 45o these give exact ratios of 1 for right angle triangles
  3   y = tan x graph of tan curve shape, a repeating pattern that has gaps because the graph is asymptotic at 90o and 270o, starts at zero, a maximum at 90o limit, recommence with a minimum after 90o limit
  4   sin 0o, cos 90o, tan 0o these give exact ratios of zero (0) for right angle triangles
  5   ASTC (all stations to central) acronym for positive trigonometric ratios by quadrant up to 360o, All positive, then Sine (and cosec) positive, then Tangent (and cotangent) positive, then Cosine (and secant) positive, moving anticlockwise around (0,0)
  6   √3/2 exact value of either cos 30o, sin 60o
  7   anticlockwise for angles of any magnitude, the angles are positive no matter what the magnitude if measured in this direction from the positive x axis
  8   cosine rule a trigonometric formula used in non-right-angled triangles instead of Pythagoras rule to calculate the length of a side opposite a known angle: a2 = b2 + c2 -2bcCos A
  9   1/√2 exact value of sin 45o, cos45o
10   cos(90o - q) = sin q complementary angle identity from cosine to sine
11   largest angle in any triangle, this is opposite the longest side (useful in solving cosine rule questions, sine rule, and area of triangle)
12   cosine ratio the ratio of the adjacent side and hypotenuse in a right-angled triangle
13   Pythagorean identities trigonometric ratio relationships found using the Pythagoras theorem, eg sin2 q + cos2 q = 1, 1 + tan2 q = sec2 q, and 1 + cot2 q = cosec2 q
14   bearings these are angles measured clockwise from north



  Trigonometry 1Two page printable: Student Answer Sheet followed by the Answers


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20

Trigonometry 2

  1   square root don't forget to take this when finding the length of a side using the cosine rule
  2   sec(90o - q) = cosec q complementary angle identity from secant to cosecant
  3   OPP/HYP sine ratio for a right angled triangle
  4   triangle shape for which the following formula is used to calculate area: A = ½abSin C
  5   tan cot relationship the following complementary angle identities apply tan(90o - q) = cot q, cot(90o - q) = tan q
  6   tan2 q + 1 = sec2 q Pythagorean identity for trigonometric ratio relationships found by dividing sin2 q + cos2 q = 1 by cos2 q
  7   1/√3 exact value of either tan 30o, cot 60o
  8   tan(90o - q) = cot q complementary angle identity from tangent to cotangent
  9   divide by sin2 q converting the basic Pythagorean identity to the form 1 + cot2 q = cosec2 q
10   cosec reciprocal of the sine of an angle, i.e., the ratio of the hypotenuse and the opposite side in a right angled triangle
11   ½ exact value of either sin 30o or cos 60o
12   sine rule a trigonometric formula used on non-right-angled triangles allowing length of a side to be found if two angles and a side are known or an angle if two sides and an angle are known
13   trigonometric identities rules relating trigonometric ratios together, eg complementary angles: 90o - q, ratios: sin ÷ cos = tan, Pythagorean: sin2 + cos2 = 1
14   obtuse angles angles between 90o and 180o which can create difficulties in interpretation of results for sine rule, cos rule, and area of triangle calculations when finding the angle



  Trigonometry 2Two page printable: Student Answer Sheet followed by the Answers


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21

Trigonometry 3

  1   sine rule for a non-right angled triangle: a/sin a = b/sin b = c/sin c
  2   cosine x ÷ sine x this ratio results in the cotangent trigonometric identity
  3   1:1:√2 relative lengths of the sides of a 45o and right angled triangle permitting exact trigonometric ratios
  4   negative angles these are measured when dealing with angles of any magnitude by proceeding clockwise around the quadrants, ie -80o = + 280o
  5   cosine rule this allows us to calculate the length of the third side of a triangle when the length of two sides and the included angles are known: a2 = b2 + c2 - 2bcCos A
  6   ADJ/OPP cotangent ratio for a right angle triangle, the inverse of tangent
  7   sin 90o, cos 0o, tan 45o these give exact ratios of 1 for right angle triangles
  8   sin(90o - q) = cos q complementary angle identity from sine to cosine
  9   ADJ/HYP cosine ratio for a right angled triangle
10   sin2 q + cos2 q = 1 Pythagorean identity for trigonometric ratio relationships, the others can be found by dividing by either sin2 q or cos2 q
11   angle of elevation the angle through which you must look upwards from the horizontal to sight an object
12   quadrant angles moving in a clockwise direction: positive x axis: 0o, negative y axis: -90o, negative x axis: -180o, positive y axis: -270o, positive x axis: -360o
13   unit circle a circle with centre at origin and radius 1, and equation x2 + y2 = 1, used to determine trigonometric ratios, as points on the circumference have coordinates that can be used directly in the ratios, together with the radius (1 unit)
14   isoceles triangle type of right angled triangle with other angles 45o, resulting in a tan or cot ratio of 1



  Trigonometry 3Two page printable: Student Answer Sheet followed by the Answers


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22

Trigonometry 4

  1   1:2:√3 relative lengths of the sides of a 30o, 60o and right angled triangle permitting exact trigonometric ratios
  2   positive angles for angles measured in an anticlockwise direction about origin and from the x axis: x axis = 0o, y axis = 90o, negative x axis = 180o, negative y axis = 270o, positive x axis = 360o
  3   tan 90o, cot 90o these give ratios of infinity for right angle triangles, ie, given as an error in the calculator, due to divide by zero error
  4   cot(90o - q) = tan q complementary angle identity from cotangent to tangent
  5   1 + cot2 q = cosec2 q Pythagorean identity for trigonometric ratio relationships found by dividing sin2 q + cos2 q = 1 by sin2 q
  6   sine rule the ratio of each side of a triangle to the sine of its opposite angle will be equal, that is a/sin A = b/ sin B = c/sin C
  7   HYP/OPP cosecant ratio for a right angle triangle, the inverse of sine
  8   cos 30o, sin 60o these give exact ratios of √3/2 for right angle triangles
  9   zero (0) exact values of any of: sin 0o, cos 90o, tan 0o
10   π alternative angle notation to degrees, represents 180o
11   cot this is the result of cos x/sin x ratio, one of the trigonometric identities
12   cot reciprocal of the tangent of an angle, i.e., the ratio of the adjacent side and the opposite side in a right angled triangle
13   y = sin x this function provides a graph of sine curve shape, over 360o between 1 and -1 on the y axis, starting at zero (sine 0 = zero)
14   a2 = b2 + c2 - 2bcCos A cosine rule, used to calculate the length of a side of any triangle when two sides and the included angle are known



  Trigonometry 4Two page printable: Student Answer Sheet followed by the Answers


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23

Trigonometry 5

  1   longest side in any triangle, this is opposite the largest angle (useful in solving cosine rule, sine rule, and area of triangle questions)
  2   one (1) exact value of any of: sin 90o, cos 0o, tan 45o
  3   sec cosec relationship the following complementary angle identities apply sec(90o - q) = cosec q, cosec(90o - q) = sec q
  4   basic Pythagorean identity sin2 q + cos2 q = 1: a relationship between trigonometric ratios obtained from Pythagoras' theorem, the others can be obtained by dividing by either sin2 q or cos2 q
  5   sine ratio the ratio of the opposite side and the hypotenuse in a right angled triangle
  6   angle of depression the angle through which you must look downwards from the horizontal to sight an object
  7   quadrant angles moving in an anticlockwise direction: positive x axis: 0o, positive y axis: 90o, negative x axis: 180o, negative y axis: 270o, positive x axis: 360o
  8   tan 30o, cot 60o these give exact ratios of 1/√3 for right angle triangles
  9   three-figure notation method used to denote compass bearings clockwise from the north, e.g. 023o
10   tan this is the result of sin x/cos x ratio, one of the trigonometric identities
11   alternative angle notation π/2 = 90o, π = 180o, 3π/2 = 270o, 2π = 360o (radians instead of degrees)
12   secant reciprocal of the cosine of an angle, ie, the ratio of the hypotenuse and the adjacent side in a right angled triangle
13   sine x/cosine x this ratio results in the tangent trigonometric identity
14   negative sign of trigonometric ratios by quadrant, going anticlockwise from positive x axis: none, then cosine (secant) and tangent (cotangent), then sine (cosecant) and cosine (secant), then sine (cosecant) and tangent (cotangent)



  Trigonometry 5Two page printable: Student Answer Sheet followed by the Answers


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24

Trigonometry 6

  1   a/sin a = b/sin b = c/sin c formula for the sine rule, used to find unknown angles and side lengths, knowing an angle and its opposite side length, along with either another angle or side length
  2   HYP/ADJ secant ratio for a right angle triangle, the inverse of cosine
  3   sin 45o, cos45o these give exact ratios of 1/√2 for right angle triangles
  4   sin cos relationship the following complementary angle identities apply sin(90o - q) = cos q, cos(90o - q;) = sin q;
  5   y = cos x this function provides a graph of the sine curve shape, over 360o between 1 and -1 on the y axis, starting at 1 (cos 0 = one)
  6   tan 60o, cot 30o these give exact ratios of √3 for right angle triangles
  7   from x axis the numerical value of trigonometric ratios is the same as that of the angle measured from this axis, e.g., sin 150o = sin 30o, or sin 210o = -sin 30o, or sin 330o = - sin 30o
  8   A = ½ ab Sin C formula for finding the area of a triangle knowing the length of two sides and the included angle
  9   √3 exact value of either tan 60o, cot 30o
10   cosec(90o - q) = sec q complementary angle identity from cosecant to secant
11   divide by cos2 q converting the basic Pythagorean identity to the form tan2 q + 1 = sec2 q
12   tan ratio the ratio of the opposite side and the adjacent side in a right angled triangle
13   exact ratios term describing the use of surds to express trigonometric values instead of decimal numbers, the surds concerned to not have integer roots
14   positive x axis moving either clockwise or anticlockwise from this provides the same absolute value of trigonometric ratios e.g. |sin -30o| = |sin 330o| = sin 30o = sin -330o



  Trigonometry 6Two page printable: Student Answer Sheet followed by the Answers


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25

Straight Line Graphs 1

  1   geometrical problems straight line equations intersection point, perpendicular distances, gradients can be used for proving: vertices of an isosceles triangle, corners of a square, or parallelogram
  2   m = (y2 - y1)/(x2 - x1) gradient formula for a line joining two points (x1,y1) and (x2,y2) or rise over run
  3   y = b equation of a straight line that is parallel to the x axis
  4   m = tan q alternate gradient formula where the angle a line makes with the x axis is known (q)
  5   y - y1 = m(x - x1) point gradient formula for finding the equation of a straight line through a point (x1,y1)
  6   intersection this point is found when solving simultaneous equations for two lines
  7   concurrent several lines are this if they pass through the one point (solve two simultaneously, then substitute the point derived to test the rest)
  8   parallel for the two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0; if a1/a2 = b1/b2 ≠ c1/c2 then they are this ….....
  9   coincident for the two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0; if a1/a2 = b1/b2 = c1/c2 then they are this....
10   distance formula for two points, (x1,y1) and (x2,y2), (potentially on a straight line), this formula applies: d = √((x2-x1)2 + (y2-y1)2)
11   y = mx + b gradient/intercept form of the equation for a straight line, used where the gradient and y intercept are known to find the equation
12   intercept form equation of a straight line of the form x/a + y/b = 1, where a = x-intercept and b = y-intercept
13   y-intercept for a straight line equation, this is found by substituting x = 0 into the equation



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26

Straight Line Graphs 2

  1   perpendicular distance formula for the point (x1,y1), and the line ax + by + c = 0: d =|(ax1 + by1 + c)/(√(a2 + b2))|
  2   parallel two lines are this if their gradients (in equations) are equal, ie, m1 = m2
  3   points location test if ax1 + by1 + c, from the perpendicular distance formula: d =|(ax1 + by1 + c)/(√(a2 + b2))|, yields the same sign for 2 points they lie on the same side, different signs, opposite sides, while 0 means on the line
  4   unbroken line for an inequality, this shows that the line is included in the region being shown on a number plane: eg y ≥ x +2
  5   x-intercept for a straight line equation, this is found by substituting y = 0 into the equation
  6   gradient formula for two points, (x1,y1) and (x2,y2), (potentially on a straight line), this formula applies m = (y2-y1)/(x2-x1)
  7   ax1 + by1 + c test for location of two points: same sign means same side of a the line, different signs, opposite sides, 0 means on the line. Derived from the perpendicular formula: d =|(ax1 + by1 + c)/(√(a2 + b2))|
  8   perpendicular two lines are this if the product of their gradients is -1; i.e. m1m2 = -1 or m1 = -1/m2
  9   gradient m = tan q where q is the angle the line makes with the x-axis in the positive direction (anticlockwise)
10   point gradient formula for the point, (x1,y1) the equation of the straight line passing through it is y - y1 = m(x - x1) where m is the gradient
11   if a1/a2 ≠ b1/b2 intersecting lines test for the two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
12   two-point formula for two points, (x1,y1) and (x2,y2), the equation of the straight line through them is given by (y - y1)/(x - x1) = (y2 - y1)/(x2 - x1)
13   ax + by + c = 0 general form of the equation of a straight line



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27

Straight Line Graphs 3

  1   if a1/a2 = b1/b2 ≠ c1/c2 parallel lines test for the two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
  2   intersection line formula for the two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, this applies: (a1x + b1y + c1) + k(a2x + b2y + c2) = 0, where k is a constant found by substituting a given point
  3   y = 0 formula for finding the x-axis intercept from the equation of a straight line
  4   d = √((x2-x1)2 + (y2-y1)2) distance formula, for the distance between (x1,y1) and (x2,y2)
  5   two regions a straight line divides the number plane (graph) into this, each of which can be defined by a linear inequality: e.g. y≥ x + 2
  6   x = 0 formula for finding the y-axis intercept from the equation of a straight line
  7   d =|(ax1 + by1 + c)/(√(a2 + b2))| perpendicular distance formula, from the point (x1,y1), to the line ax + by + c = 0:
  8   parallel to axes the straight lines x = a and y = b, are this respect to y-axis and x-axis respectively
  9   x = a equation of a straight line that is parallel to the y axis
10   m1 = m2 test for two lines to determine whether or not they are parallel, also formula for substituting into the gradient form of the equation of straight line for two parallel lines
11   non-zero for ax + by + c to represent a straight line, then at least one of a and b must be this
12   m1m2 = -1 or m1 = -1/m2 test for two lines to determine if they are perpendicular, also formula for substituting into gradient equation form to obtain two straight lines that are perpendicular



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28

Straight Line Graphs 4

  1   intersect for the two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0; if a1/a2 ≠ b1/b2 then they do this....
  2   general form equation of a straight line of the form ax + by + c = 0
  3   (y - y1)/(x - x1) = (y2 - y1)/(x2 - x1) two-point formula for a straight line passing through the points (x1,y1) and (x2,y2)
  4   line through intersection find the point of intersection then use it with another point (given) or gradient to obtain the equation
  5   gradient form equation of a straight line of the form y = mx + b, where m = gradient and b = y-intercept
  6   x-intercept for a straight line equation, this is found by substituting y = 0 into the equation
  7   (a1x + b1y + c1) + k(a2x + b2y + c2) = 0 intersection line formula for the two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where k in the formula is a constant found by substituting a given point
  8   if a1/a2 = b1/b2 = c1/c2 coincident lines test for the two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
  9   midpoint formula for two points, (x1,y1) and (x2,y2), (potentially on a straight line), this formula applies P = ((x1+x2)/2, (y1+y2)/2)
10   x/a + y/b = 1 intercept form of the equation of a straight line, used where the x and y axis intercepts (a and b respectively) are known
11   broken line for an inequality, this shows that the line is not included in the region being shown on a number plane and is merely the boundary of the region: eg y > x +2
12   P = ((x1+x2)/2, (y1+y2)/2) mid point formula for locating a point midway between two points (x1,y1) and (x2,y2)



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29

Introduction to Calculus 1

  1   calculus branch of mathematics that is useful for finding the properties of curves from their equations: maximums, minimums, points of inflexion, gradients at a point
  2   gradient this measure the rate of change in y as x varies
  3   dy/dx = nxn-1 differentiation formula for y = xn
  4   automatically zero gradient if an equation includes only constants, i.e. y = k, then dy/dx = 0
  5   dy/dx = n[f(x)]n-1f'x chain rule or the derivative of a function-of-a-function rule: applies if y =[f(x)]n e.g. (3x2 + 7)5 differentiates to: 5(3x2 + 7)46x
  6   product rule if y = uv where u and v are separate functions of x (in brackets) then dy/dx = u.dv/dx + v.du/dx = uv' + vu'
  7   differentiation the process of finding the derivative of a function, i.e., finding the gradient of the curve for all values of x: dy/dx, or f'(x)
  8   curve sketching a use of the derivative (result of differentiation or curve gradient), once gradient is known at a series of points the general curve shape can be shown
  9   dy/dx = f'(x) ± g'(x) differentiation that applies if y = two functions of x being added or subtracted ie: y = f(x) ± g(x)
10   common constant and differentiation if function x can be factorised to place a constant outside brackets, then it can be differentiated as follows: y = f(kx) = kf(x), so dy/dx = kf'(x)
11   dy/dx = (v.du/dx - u.dv/dx)/v2 = (vu' - uv')/v2 quotient rule, applies where y = u/v where u and v are separate functions of x (in brackets)
12   normal this is the line perpendicular to the tangent, and so the product of the gradients of the tangent and this line = -1: m1m2 = -1, enabling its equation to be found
13   first principles differentiation process using a graph: draw a series of secants through the point in question and another point, until both points correspond, then find gradient: ie dy/dx = lim as δx -> 0 (δy/δx)
14   differentiation by calculation substitute (x + h) for x in the function (f(x)), expand, then subtract the original f(x) prior to placing in the equation: f'(x) = lim as h->0 (f(x+h) - f(x))/h, the h will then cancel after expansion and collecting like terms
15   dy/dx = ½x = ½√x for differentiation of surd forms, convert the root forms to indice: if y = √x, then y = x½
16   sharp bend the graph of y = |x| has this



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30

Introduction to Calculus 2

  1   differentiation inverse forms convert to indice form over 1 (i.e. invert), eg if y = 1/x5, then y = x-5, so dy/dx = -5x-6 = -5/x6
  2   differentiation formula if y = xn, then dy/dx = nxn-1
  3   dy/dx = -5x-6 = -5/x6 for differentiation of inverse forms, first convert to indice over 1 (i.e. invert), eg if y = 1/x5, then y = x-5
  4   chain rule also called derivative of a function-of-a-function rule: y =[f(x)]n e.g. (3x2 + 7)5 differentiates to: 5(3x2 + 7)46x
  5   tangent a line that just touches a curve and which has a gradient the same as that of the curve at the point it touches the curve
  6   derivative the result of the mathematical process of differentiation, for finding curve gradient as x approaches 0, expressed in the form: dy/dx, or f'(x)
  7   dy/dx = 0 for the equation y = k, the gradient is zero (due to y = kx0 which results in a multiply by zero calculation)
  8   added/subtracted differentials if y = two functions of x being added or subtracted, then: for y = f(x) ± g(x), dy/dx = f'(x) ± g'(x)
  9   dy/dx = u.dv/dx + v.du/dx = uv' + vu' product rule applies where y = uv where u and v are separate functions of x (in brackets)
10   quotient rule if y = u/v where u and v are separate functions of x (in brackets) then dy/dx = (v.du/dx - u.dv/dx)/v2 = (vu' - uv')/v2
11   gradient function the result of differentiation, can be a straight line (for y = x2), or a curve (for y = x3), even not a line (for y = c)
12   dy/dx = kf'(x) differentiation that applies if function x can be factorised to place a constant outside brackets, ie: y = f(kx) = kf(x)
13   differentiation surd forms convert root forms to indice: if y = √x, then y = x½, so dy/dx = ½x = ½/√x = 1/(2√x)
14   f'(x) = lim as h-->0 (f(x+h) - f(x))/h in differentiation from first principles using calculations, substitute (x + h) for x in the function (f(x)), expand, then subtract the original (f(x)) prior to placing in the equation: f'(x) = lim as h-->0 (f(x+h) - f(x))/h, the h will then cancel
15   tangent when the two points through which a secant passes that are located at the same place then the line is this
16   differential calculus basis if the gradient f'(x)exists at (x,f(x)), the slope of the tangent at the point is defined to be f'(x)



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31

Quadratic Functions 1

  1   concave down general parabola shape if the coefficient (a) of x2 in the general quadratic equation ax2 + bx + c is less than zero (a<0)
  2   Δ = b2 - 4ac must be ≥ 0 for real roots the discriminant formula: the part of the quadratic formula that allows you to decide if a quadratic function has real roots (crosses the x axis)
  3   quadratic formula a formula that is used to find the possible values of an unknown in a quadratic equation (unknown to the power of 2): x = (- b ± (√(b2-4ac)))/2a
  4   (b2 -4ac) < zero & a < 0 a quadratic expression, ax2 + bx + c = 0, is negative definite if the discriminate and a, the coefficient of x2, have these values, ALSO it has no real roots, only imaginary, doeSn't cross the x-axis
  5   sum of roots if α and β are two roots of ax2 + bx + c = 0, [ie, (x - α)(x - β) = 0, and x2 - (α + β)x + αβ = 0], then α + β = -b/a
  6   quadratic function a function where the independent variable is raised to the power of 2, eg y = ax2 or y = ax2 + bx + c
  7   y = ax2 or y = ax2 + bx + c examples of equations that are quadratic functions (independent variable raised to the power of 2)
  8   (b2 -4ac) = zero two equal real roots (α = β) are obtained for a quadratic equation when the discriminate yields this result
  9   two unequal real roots a quadratic equation has this/these if the discriminate (b2 -4ac) > zero. They are rational if Δ is a perfect square, or irrational if Δ is not a perfect square
10   αβ = c/a product of roots for a quadratic function ax2 + bx + c = 0, [ie, (x - α)(x - β) = 0, and x2 - (α + β)x + αβ = 0] where α and β are the roots
11   convert to quadratics this can be done to allow the use of quadratic formula to find roots with equations such as: 22x - 2x - 12 =0 (let t = 2x) or x - 1/x = 3 (multiply throughout by x)
12   x = -b/2a the formula for the axis of symmetry for the quadratic function, y = ax2 + bx + c



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32

Quadratic Functions 2

  1   minimum or maximum value concave up or down, whichever is applicable to a quadratic function, is found by substituting f(-b/2a) into the quadratic equation, i.e., the y value obtained when x = -b/2a (the axis of the parabola)
  2   (b2 -4ac) ≥ zero a quadratic expression, ax2 + bx + c = 0, is indefinite if the discriminate has this value, and it has real roots
  3   positive definite a quadratic expression, ax2 + bx + c = 0, is this if the discriminate < 0 and a, and the coefficient of x2, > 0. It has no real roots, only imaginary. Concave up above the x axis
  4   secant the straight line passing through two given points on a curve
  5   minimum value concave up parabolas have this, i.e., a position on the curve below which there are no values for the dependent variable y, this occurs when a > 0
  6   chord a straight line joining any two given points on a curve
  7   negative definite a quadratic expression, ax2 + bx + c = 0, is this if the discriminate < 0 and a, the coefficient of x2, < 0 It has no real roots, only imaginary, doeSn't cross the x axis
  8   coefficient (a) of x < 0 concave down is the result for a general parabola shape for the equation y = ax2 + bx + c
  9   maximum value concave up parabolas have this, i.e., a position on the curve above which there are no values for the dependent variable y, when a < 0
10   x = (- b ± (√(b2-4ac)))/2a the quadratic formula; a formula that is used to find the possible values of an unknown in a quadratic equation (unknown to the power of 2). Locates where the parabola crosses the x axis
11   two equal real roots a quadratic equation has this/these if the discriminate (b2 -4ac) = zero
12   α + β = -b/a sum of roots for a quadratic function ax2 + bx + c = 0, [ie, (x - α)(x - β) = 0, and x2 - (α + β)x + ab = 0] where α and β are the roots



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33

Quadratic Functions 3

  1   product of roots if α and β are two roots of ax2 + bx + c = 0, [i.e., (x - α)(x - β) = 0, and x2 - (α + β)x + αβ = 0] then αβ = c/a
  2   x-axis location of the solutions to a quadratic equation, i.e., when y = 0
  3   axis of symmetry for the quadratic function, y = ax2 + bx + c, this line is found by: x = -b/2a
  4   (b2 -4ac) > zero two unequal real roots are obtained for a quadratic equation when the discriminate (&Delta) yields this result
  5   indefinite a quadratic expression is this if the discriminate ≥ 0, and may be positive or negative
  6   quadratic identities if a1x2 + b1x c1 is equivalent to a2x2 + b2x + c2, then a1 = a2, b1 = b2, and c1 = c2
  7   concave up general parabola shape if the coefficient (a) of x2 in the general quadratic equation y = ax2 + bx + c is greater than zero (a > 0)
  8   complete the square a method of solving quadratic equations by completing the square, eg x2 - 10x + 3 = 0, as x2 - 10 + 25 = 22, so (x - 5)2 = 22, and x - 5 = ± √22
  9   coefficient (a) of x > 0 concave up is the result for a general parabola shape for the equation y = ax2 + bx + c
10   x = -b/2a the minimum or maximum value, whichever of these is applicable to a quadratic function, is found by substituting f(-b/2a), ie, the y coordinate of the lowest or highest value is found
11   discriminant the part of the quadratic formula that allows you to decide if a quadratic function has real roots Δ = b2 - 4ac must be ≥ 0 for real roots
12   (b2 -4ac) < zero & a > 0 a quadratic expression, ax2 + bx + c = 0, is positive definite if the discriminate (Δ) and a, the coefficient of x2, have these values, also it has no real roots, only imaginary, doeSn't cross the x axis



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34

Locus, Parabola, Circle 1

  1   focal length distance from the focus of a parabola to the vertex (along its axis of symmetry), 1/4 coefficient of x2
  2   x2 + y2 = r2 equation of the locus of a point that moves so that it is r units from the origin, a circle radius r, centre (0, 0)
  3   general circle equation (x - a)2 + (y - b)2 = r2 which is the equation of the locus of a point that moves so that it is r units from the point (a, b), a circle radius r, centre (a, b)
  4   focal chord a line drawn between two points on a parabola and passing through the focus
  5   locus a path of a point that moves according to certain conditions
  6   chord a line drawn between two points on a parabola, also called line segment. Cuts off an arc
  7   parabola the locus of a point moving so that it is equidistant from a fixed point and a fixed line (called the focus and directrix respectively)
  8   simple parabola equation x2 = -4ay with focus (0,-a), directrix y = a, focal length = a, axis x = 0, vertex (0,0), concave down
  9   x2 = -4ay focus (0,-a), directrix y = a, focal length = a, axis x = 0, vertex (0,0), concave down
10   x = -b/a axis of symmetry equation for the parabola y = ax2 + bx + c
11   greatest or least value of y for the parabola y = ax2 + bx + c, the value of y at the vertex, this property can be found with (4ac - b2)/4a, or - Δ/4a
12   focus the point in a parabola shaped reflector where light from a source placed there will reflect off the parabola to form a parallel beam of light



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35

Locus, Parabola, Circle 2

  1   directrix a line that is equidistant from the locus of a parabola with the distance from the focus to the locus of the parabola, equation: y = - 1/4 coefficient of x2
  2   general parabola equation: (x - h)2 = 4a(y - k), vertex (h, k), focal length a, axis parallel to the y axis at x = h, focus ((h+a), k), directrix y = k - a, concave up
  3   (x - h)2 = -4a(y - k) parabola vertex (h, k), focal length a, axis parallel to the y axis at x = h, focus (h , (k - a)), directrix y = k + a, concave down
  4   negative definite if a quadratic function is negative for all values of x then it has this property, Δ < 0, a < 0, concave down and completely below the x axis
  5   indefinite if a quadratic function is sometimes positive and sometimes negative, then it has this property, Δ ≥ 0, could be concave up or concave down, and either touches or crosses the x axis
  6   focus point in a parabola that is equidistant from the locus and the perpendicular distance from the directrix to the locus, has coordinates (x, 1/4 coefficient of x2)
  7   find parabola equation this can be found from two or more of the following: focus, vertex, focal length, axis or directrix, but may be indefinite as the concavity sign may be either + or - eg axis and vertex given
  8   simple parabola equation x2 = 4ay with focus (0,a), directrix y = -a, focal length = a, axis x = 0, vertex (0,0), concave up
  9   x2 = 4ay equation of a parabola with the following properties: focus (0,a), directrix y = -a, focal length = a, axis x = 0, vertex (0,0), concave up
10   (x - a)2 + (y - b)2 = r2 equation of the locus of a point that moves so that it is r units from the point (a, b), a circle radius r, centre (a, b)
11   axis of symmetry for the parabola y = ax2 + bx + c, this property has the equation: x = -b/a
12   arc curved line between two points on a parabola or circle or other curved locus



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36

Locus, Parabola, Circle 3

  1   straight line the locus of a point moving so that it is equidistant from two fixed points
  2   circle the locus of a point moving so that it is equidistant from one fixed point
  3   sideways parabola y2 = 4ax which is the locus of a point that is equidistant from (a,0) (focus) and the fixed line x = -a (directrix), focal length a, axis y = 0, vertex (0,0), concave right
  4   y2 = 4ax the locus of a point that is equidistant from (a,0) (focus) and the fixed line x = -a (directrix), focal length a, axis y = 0, vertex (0,0), concave right
  5   (4ac - b2)/4a greatest or least value of y for the parabola y = ax2 + bx + c, is obtained with this formula ( -Δ/4a )
  6   positive definite if a quadratic function is positive for all values of x then it has this property, discriminate (Δ) < 0, a > 0, concave up and lies above the x axis
  7   vertex the point where the axis of a parabola cuts the parabola
  8   axis a line through the focus of a parabola and perpendicular to the directrix of the parabola
  9   (y - k)2 = -4a(x - h) general equation of a parabola with: vertex (h, k), focal length a, axis parallel to the x axis at y = k, focus ((h - a), k), directrix x = h + a, concave left
10   simple circle equation x2 + y2 = r2: which is the equation of the locus of a point that moves so that it is r units from the origin, a circle radius r, centre (0, 0)
11   complete the square this can be done to convert a quadratic equation from the y = ax2 + bx + c form into the (x-h)2 = 4a(y - k) form to find the focus, directrix, axis, vertex
12   completing square process from the equation y = ax2 + bx + c, isolate ax2 + bx, then divide throughout the equation by the coefficient of x2 --> x2 + (b/a)x, then add (b/a/2)2 to both sides of the equation --> x2 + (b/a)x + (b/a/2)2, and finally factorise --> (x + b/a/2)2, with the final result being: y/a - c/a + (b/a/2)2 = (x + b/a/2)2



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37

Series 1 1

  1   a1 + a2 + a3 + a4 + ......... + an a series is the sum of a sequence of numbers
  2   sum arithmetic series the formula for this is Sn = ½n(a + l) if the first and last (or nth) term are known
  3   T2 = ± √(T1.T3) formula for the geometric mean for two terms of a geometric series
  4   sum geometric series the formula for this is Sn = a(rn - 1)/(r - 1) where the absolute value of r is greater than 1
  5   no limiting value for geometric series in which the absolute value of r is greater or equal to 1 this principle is applied to the sum to infinity
  6   a1, a2, a3, a4, ..... infinite sequence, contains an indefinite number of terms
  7   nth term for an arithmetic series this is found by the formula: Tn = a + (n-1)d
  8   a + ar + ar2 + ar3 + ....... + ar(n-1) geometric series, the terms in this type of series are formed by multiplying the preceding term by a constant: the common ratio (r), or any particular term (n) can be found from the first term (a) multiplied by the common ratio raised to the power of (n - 1)
  9   common ratio the number used in a geometric series to find subsequent terms (r)
10   Sn = a(1 - rn)/(1 - r) formula for the sum of a geometric series, where the absolute value of the common ratio (r) is less than 1
11   sigma notation or summation notation: a symbol indicating either an arithmetic or geometric series, showing the first and final value for n or r and a function for the terms
12   arithmetic series the terms of this series type are found by adding a constant to the preceding term: a + (a + d) + (a + 2d) + (a + 3d).... + [a + (n-1)d] + .... where d is the common difference, first term a
13   d = T2 - T1 = T3 - T2 common difference for an arithmetic series, this is found by subtracting succeeding terms
14   arithmetic mean for two terms of an arithmetic series, this is found with the formula: T2 = ½(T1 + T3)



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38

Series 1 2

  1   Tn = arn-1 formula for the nth term of a geometric series where a is the first term, and r is the common ratio
  2   common ratio for a geometric series, this is found by dividing terms into subsequent terms: r = T2/T1 = T3/T2
  3   limiting sum or sum to infinity (terms) for a geometric series, formula: S = a/(1 - r) if the absolute value of r is less than 1 (terms get progressively smaller)
  4   a1, a2, a3, a4, ..... an a sequence, the terms of a series
  5   S = a/(1 - r) formula for the limiting sum, or sum to infinity (of terms) for a geometric series; where the absolute value of r is less than 1 (ie, terms get progressively smaller)
  6  
10
(3n + 2)
n = 1
arithmetic series indicated using sigma notation, involving starting term n = 1, final term 10, and a formula to be used to obtain terms
  7   finite sequence the terms of the series a1, a2, a3, a4, ..... an where n is a positive number and is the length of the sequence
  8   Sn = ½n[2a + (n-1)d] formula for the sum of an arithmetic series, where the first term (a), common difference (d) and number of terms (n) are known
  9   series a sum of terms of a sequence of numbers: a1 + a2 + a3 + a4 + ......... + an
10  
15
2n
n = 4
geometric series indicated using sigma notation, involving starting term n = 4, final term 15, and a formula to be used to obtain terms
11   a1, a2, a3, a4, ..... an a finite sequence, has a fixed number of terms (n is a positive number and is the length of the sequence
12   geometric series applications many problems can be reduced to finding the nth term or sum of n terms, e.g., a plant originally 420 mm, grows 10cm in a week and then a percentage (r%) of that in the next week: a trend that continues in the following weeks, find height after 8 weeks
13   infinite sequence the terms of the series a1, a2, a3, …......., or an indefinite number of terms
14   Sn = ½n(a + l) formula for the sum of an arithmetic series, where the first (a) and last (l) terms and number (n) of terms are known



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39

Series 1 3

  1   geometric series the terms in this type of series are formed by multiplying the preceding term by a constant: a + ar + ar2 + ar3 + ....... + ar(n-1), first term a, common ratio r, number of terms n
  2   Sn = a(rn - 1)/(r - 1) formula for the sum of a geometric series, where the absolute value of r (the common ratio) is greater than 1
  3   sum geometric series the formula for this is Sn = a(1 - rn)/(1 - r) where the absolute value of r is less than 1, first term is a and the number of terms is n
  4   Tn = a + (n-1)d formula for the nth term of arithmetic series, where the knowns are first term (a), common difference (d) and number of terms (n)
  5   common difference for an arithmetic series, this is found by subtracting succeeding terms: d = T2 - T1 = T3 - T2
  6   nth term for a geometric series, is found by Tn = arn-1, where a is the first term, r common ratio
  7   a + (a + d) + (a + 2d) + (a + 3d).... + [a + (n-1)d] + ... an arithmetic series, in which the terms of the series are found by adding a constant (the common difference d) to the preceding term, first term a
  8   sigma (Σ) notation problems for arithmetic and geometric series: write down a few terms, and the last term and use these to find common difference or ratio, then apply other series formulae, e.g. Sn
  9   sequence the terms of a series, i.e. a1, a2, a3, a4, ..... an; or a1, a2, a3, ......
10   T2 = ½(T1 + T3) formula for the arithmetic mean for two terms of an arithmetic series
11   sum arithmetic series the formula for this is Sn = ½n[2a + (n-1)d] if the first term (a) is known, and the common difference (d)
12   r = T2/T1 = T3/T2 formula for the common ratio of a geometric series, found by dividing terms into subsequent terms
13   geometric mean for two terms of a geometric series, this is found with the formula: T2 = ± √(T1.T3)
14   common difference the number used in arithmetic series to find subsequent terms (d)
15   arithmetic series applications many problems can be reduced to finding the nth term or sum of n terms, eg, pay increases from $1600 by $600 per month, what will be earned in the 12th month, what is the annual income



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40

Geometrical Applications of Calculus 1

  1   differential calculus the part of Mathematics that develops from the definition of the derivative of a function, or the gradient of a graph, the subject concerned with the rate of change of one quantity with respect to another
  2   f'(x) = 0 and f"(x) = 0 test that shows a horizontal point of inflexion has occurred in a graph, i.e. there is a change in concavity, which is tested by substitution
  3   family of curves this is the result when given f'(x)and asked to find f(x) i.e., the primitive; the original equation has an unknown constant, e.g.if dy/dx = 2x, then y = x2 + c
  4   first derivative this shows us whether a curve is increasing (positive gradient), decreasing (negative gradient) or stationary (horizontal), dy/dx, f'(x)
  5   higher derivatives these are found by differentiating previous differentiations or derivatives: f(x) differentiated to provide f'(x), differentiated to provide f"(x), differentiated to provide f'"(x) or the third derivative
  6   second derivative signs concave upwards if f"(x) > 0, change in concavity or point of inflexion if f"(x) = 0, or concave down if f"(x) < 0
  7   absolute maximum the greatest y value of a curve in a given domain (set of values of x)
  8   absolute minimum the least y value of a curve in the given domain (set of values of x)
  9   d2y/dx2, f"(x), y", d2/dx2 (y) notation for the second derivative which shows us the concavity of a curve
10   first derivative sign this shows us if the curve is increasing (f'(x) > 0), stationary (f'(x) = 0) or decreasing (f'(x) < 0)
11   stationary points points on a graph where the gradient is zero, that is, the curve is neither increasing or decreasing, it is horizontal
12   f"(x) < 0 second derivative sign which indicates the curve is concave down
13   testing concavity this is done by substituting values from either side of an inflexion, if y" > 0 concave up and y" < 0 concave down
14   zero to find horizontal points on a curve, let f'(x) equal this value, because that means the graph is horizontal at that value of x
15   primitive function formula if dy/dx = xn, then y = xn+1/(n+1) + C, where C is a constant and can be determined from the coordinates of known points or the y axis intercept



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41

Geometrical Applications of Calculus 2

  1   differential rate of change, the result of a differentiation, the gradient of the tangent to a curve
  2   y = xn+1/(n+1) + C formula for finding the primitive function given the first derivative, e.g. dy/dx = xn, where C is a constant that is determined from points or the y axis intercept
  3   (f'(x) < 0) first derivative sign which indicates that the curve is decreasing, found by substitution of x values
  4   possible stationary points minimum, maximum, horizontal inflexion points, occur when f'(x) = 0, the second derivative is needed to determine which one is occurring
  5   f'(x) = 0 and f"(x) < 0 test that shows the point where a maximum has occurred in a graph, i.e. concave down
  6   > 0 means the result is positive
  7   point of inflexion this occurs in a curve if it changes concavity e.g. from concavity up to concavity down, or concavity left to concavity right, along the locus of the curve
  8   concavity change this indicates a point of inflexion in a curve
  9   curve sketching a use of information found by using first and second derivatives to determine stationary points, concavities, maxima, minima, inflexions; in conjunction with x and y axis intercepts
10   maxima these occur with functions that when graphed provide a concave down locus
11   relative maxima and minima these are stationary points, or turning points on a curve where concavity tests give the same result on both sides of the stationary point, i.e. not points of inflexion
12   dy/dx or f'(x) notation for the first derivative, which shows us whether a curve is increasing, decreasing or stationary
13   second derivative this shows us the concavity of a curve, d2y/dx2, f"(x), y", d2/dx2 (y)
14   f"(x) > 0 second derivative sign which indicates the curve is concave up
15   f"(x) = 0 second derivative sign which indicates the curve is changing concavity ie, a point of inflexion



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42

Geometrical Applications of Calculus 3

  1   rotate clockwise shorthand way of determining if f"(x) is concave up or down and has a maxima: f"(x) > 0 is concave up, and f"(x) < 0 is concave down and has a maxima
  2   maxima and minima problems determine equation using area or volume formulae, find first and second derivatives, determine maxima/minima/inflexions and concavities
  3   dA/dr or f'(A) example of the appearance of a first derivative found for a maxima/minima problem involving circle area and radius: A = πr2
  4   < 0 means the result is negative
  5   primitive function the original equation used to obtain a given gradient by dy/dx (differentiation)
  6   f'(x) > 0 first derivative sign which indicates that the curve is increasing, found by substitution
  7   f'(x) = 0 first derivative sign which indicates that the curve is at a stationary point, found by substitution, could be maximum, minimum or point of inflexion
  8   types of stationary points minimum if f'(x) = 0 and f"(x) > 0, maximum if f'(x) = 0 and f"(x) < 0, horizontal inflexion if f'(x) = 0 and f"(x) = 0 and concavity has been found to change (by testing)
  9   f'(x) = 0 and f"(x) > 0 test that shows a minimum has occurred in a graph, ie concave up
10   two variables when solving maximum and minimum problems use information given about physical measurements (length, time, volume, area etc) to derive equations and then substitute to get to the point of being able to differentiate only one variable against another
11   horizontal not all points of inflexion are this, they are merely the place on a curve where concavity changes
12   stationary points these are turning points, maxima or minima or inflexions that are horizontal, parallel to the x axis, gradient = 0
13   minima these occur with functions that when graphed provide a concave up locus
14   geometric formulae maxima and minima problems utilise these relations because they have squared length measurements (area) or cubed length (volume) e.g. maximum area a given length of fence will allow



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43

Integration 1

  1   integration mathematical technique to measure the exact area under a graph
  2   trapezoidal rule (many) formula for estimation of area between two curves for the situation where the curve requires many trapezia: ∫ba f(x).dx ≑ ½h[(yo + yn) + 2(y1 + y2 + y3 + ......... + y(n-1))] where h = (b - a)/n
  3   trapezium geometrical shape used in trapezoidal rule which reduces the accuracy if the area between or under smooth curves is being calculated because it is based on straight line sides
  4   ∫ba f(x)dx = F(x) + C indefinite integral, where F(x) is a primitive function of f(x), and C is a constant related to the possible y-axis intercept
  5   ∫ba f(x).dx ≑ ½h[(yo + yn) + 2(yrest total)] trapezoidal rule for the situation where the curve requires many trapezia
  6   positive integrals (∫ f(x).dx) have this sign if the area they are being used to calculate is above the x axis
  7   ∫ xn.dx = x(n+1)/(n+1) + C general integration formula, which is found in Standard Integral tables; note: the following conditions apply: n ≠ -1, and x ≠ 0 if n < 0 (also the C is left off in integral tables)
  8   A = ∫[f(x) - g(x)].dx or ∫f(x).dx - ∫g(x).dx formula to find areas between curves y = f(x) and y = g(x)
  9   integration this mathematical procedure is performed to find the exact area between a curve and the x axis over a given domain (x values)
10   even functions f(x) is this when it is symmetrical about the y axis, and the following rule applies: f(-x) = f(x), and ∫a-a f(x).dx = 2 ∫a0 f(x).dx
11   π∫ba y2.dx formula for finding the volume of space obtained when a curve is rotated about x-axis; first step is to make y2 the subject of the equation, (after squaring both sides)
12   volume about y-axis to find this, square both sides of the equation and make x2 the subject, then apply: V = π∫ba x2.dy, then substitute for x2 so that V = π∫ba f(y).dy
13   sum of limits the area under a graph is defined as this quantity, and is generated by dividing the area into smaller and smaller rectangles, until the calculated area approaches the actual area: sum of y values × x values: ∫ f(x).dx, with suitable limit notation
14   Simpson's rule (many) when a region under a curve is divided into n strips and parabolas, with (n+1) function values: ∫ f(x).dx ≑ ⅓h[(y0+yn) + 4(y1+y3+y5 +...) + 2(y2+y4+y6+....)], where h = (b - a)/n, i.e. ∫ f(x).dx ≑ ⅓h[(first and last) + 4 × odds sum + 2 × evens sum]
15   original area estimation between a curve and an axis, this was based on the sum of areas of rectangles, S ≑ f(x).D x, replaced later by trapezoidal (trapezium areas) and Simpson's (parabola areas) rules



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44

Integration 2

  1   volume about x-axis to find this, square both sides of the equation and make y2 the subject, then apply: V = π∫ba y2.dx, then substitute for y2 so that V = π∫ba f(x).dx
  2   parabola for this shape, i.e., the graph of a quadratic function, Simpson's rule is exact
  3   ∫ 1.dx = x + C integration of numerals formula for example where f(x) = f(1) because 1 = 1 × x0 which integrates to 1 × x0 + 1/(0 + 1)
  4   integrating combined functions if two functions of x are being integrated then they can be treated as separate integrations: ∫ [f(x) ± g(x)].dx = ∫ f(x).dx ± ∫ g(x).dx
  5   A = ∫ba f(y).dy formula for finding the area between a curve and the y-axis using the same rules as for x. The equation first needs to be transposed to make x the subject of the equation so that x = f(y)
  6   net area this is the result of applying integration over the full range of values indicated for some functions, and may not result in the required answer because positive and negative areas cancel out
  7   integrating f(ax + b)n no need to expand the brackets, the following applies: ∫ (ax + b)n.dx = (ax + b)n+1/(a(n+1)) + C
  8   indefinite integral ba f(x).dx = F(x) + C, where F(x) is a primitive function of f(x), and C is a constant related to the possible y-axis intercept
  9   integral calculus the subject that arose from the problem of trying to find the area of a region with a curved boundary, which can only be approximated by using small rectangles; relates to differential calculus through primitive functions; finds the product of two quantities
10   volumes the capacity of an object with a curved side can be calculated by rotation of the curve about an axis, through integration
11   general integration formula this is found on Standard Integrals sheets: ∫ xn.dx = x(n+1)/(n+1) + C, n ≠ -1, and x ≠ 0 if n < 0 (note: the C is left off)
12   negative integrals (∫ f(x).dx) have this sign if the area they are being used to calculate is below the x axis
13   Simpson's rule (one) one strip divided into two subintervals under a curve, and a parabola drawn through the 3 values, where x is a or b, or (a + b)/2: ∫ba f(x).dx ≑((b-a)/6)[f(a) + 4f((a + b)/2) + f(b)] ≑ (b-a)/6[yfirst + 4ymid + ylast]
14   integral calculus this is based on the area under a curve between two values of the variable: if f'(x) is the curve, then the area is difference between x2 and x1 in f(x), i.e. f(x2) - f(x1)
15   f(-x) = f(x), and ∫a-a f(x).dx = 2∫a0 f(x).dx even functions rules, when f(x) is symmetrical about the y-axis



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45

Integration 3

  1   ∫ kxn.dx = k∫ xn.dx = k(x(n+1)/(n+1) + C) integration and a constant factor formula for when f(x) contains a factorizable constant
  2   estimating area the trapezoidal rule and Simpson's rule are used for this purpose, being the sum of approximate trapezia and parabolic areas respectively
  3   ∫ba f(x).dx ≑ ⅓h[(y0 + yn) + 4(ysum odds) + 2(ysum evens)] Simpson's rule used where under the curve can be divided into n strips and parabolas, has (n+1) function values AND where h = (b - a)/n
  4   rotating the curve process that is performed through the integration of f(x2) or f(y2), (found by squaring both sides of the equation), so that the volume can be calculated: V = π∫ba y2.dx for x axis
  5   y-axis areas these are found by finding f(y) and integrating it as follows: A = ∫ba f(y).dy using the same rules as for x
  6   area estimation errors for trapezoidal rule this is approximately 1/n2, while for Simpson's rule it is approximately 1/n4 where n is the number of subdivisions being used to find the area under a graph
  7   ∫ba f(x).dx ≑ ½(b-a)[f(a) + f(b)] trapezoidal rule for the situation where one trapezium is suitable
  8   areas between curves if two curves are y = f(x) and y = g(x), then this can be found, with the following formula: A = ∫ [f(x) - g(x)].dx or ∫ f(x).dx - ∫ g(x).dx
  9   integral the result of an integration, the quantity of which a given function is the differential or differential coefficient, an expression from which a function can be derived by differentiation
10   π∫ba x2.dy formula for finding the volume of space obtained when a curve is rotated about y-axis; first step is to make x2 the subject of the equation, (after squaring both sides if necessary)
11   trapezoidal rules there are two of these, one for simple situations where a curve can be approximated by a straight line between two points, and another where the curve needs to be divided into many trapezia
12   ∫ba f(x).dx = F(b) - F(a) definite integral formula, where F(x) is a primitive function of f(x)
13   ∫ (ax + b)n.dx = (ax + b)n+1/(a(n+1)) + C formula to use when integrating f(ax + b)n, no need to expand the brackets
14   sketch and subdivide when using integration to find areas between curves this is often better than applying an "area between curves" formula because it does not discriminate included/omitted areas, negative/positive areas
15   [F(x)]ba definite integrals notation showing the primitive into which the values of x are to be substituted to find the area bounded by x = a, the curve (f'(x)), x = b and the x-axis: Area = F(b) - F(a)



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46

Integration 4

  1   limit a number (l) that f(x) approaches as x approaches a given number (a): eg lim x->a f(x) = l, a may be zero, eg in finding the tangent of a curve at a point by making smaller secants until the secant is the tangent
  2   Simpson's rules there are two of these, one for simple situations where a curve can be approximated by a parabola through three values, and another where the curve needs to be divided into many parabolas
  3   volume about x-axis to find this, square both sides of the equation and make y2 the subject, then apply: V = π ∫ba y2.dx = π∫ba f(x).dx
  4   definite integrals the following notation is used to show that the values of x = a and x = b are to be substituted into primitives when finding the area between x = a, the curve (f'(x)), x = b, and the x-axis: [F(x)]ba
  5   ∫ba f(x).dx ≑ (b-a)/6[f(a) + 4f((a + b)/2) + f(b)] Simpson's rule used where one strip is divided off under a curve, and a parabola drawn through the 3 x values a, b, and (a + b)/2 i.e., (b-a)/6[yfirst + 4ymid + ylast]
  6   integration and constant factor if f(x) contains a factorizable constant: ∫ kxn.dx = k∫xn.dx = k(x(n+1)/(n+1) + C)
  7   odd functions f(x) is this when it is symmetrical about the origin, and the following rule applies: f(-x) = -f(x), and ∫a-a f(x).dx = 0
  8   π∫ba x2.dy = π∫ba f(y).dy formula for finding the volume of space obtained when a curve is rotated about the y axis; first step is to make x2 the subject of the equation, (after squaring both sides)
  9   integration of numerals if f(x) = f(1), then ∫x.dx = ∫ 1.dx = x + C, because 1 = 1 × x0 which integrates to 1 × x0 + 1/(0 + 1)
10   f(-x) = -f(x), and ∫a-a f(x).dx = 0 odd functions rules, when f(x) is symmetrical about the origin
11   integrating polynomials the integral of any sum is a sum of the integrals, so any polynomial may be integrated by finding the primitives of each term
12   trapezoidal rule (one) for the situation where one trapezium is suitable: &intba f(x).dx ≑ ½(b-a)[f(a) + f(b)]
13   definite integral ba f(x)dx = F(b) - F(a), where F(x) is a primitive function of f(x)
14   ∫ [f(x) ± g(x)].dx = ∫ f(x).dx ± ∫ g(x).dx integrating combined functions formula, the two functions are treated as separate integrations



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47

Exponential and Logarithmic Functions 1

  1   base the number to which an exponential is applied to produce another defined number: in y = ax, x is the exponential applied to the a to produce y
  2   y = ax an expression showing base a with exponent x
  3   differentiating ef(x) if the exponential function y = ef(x) then the gradient is given by dy/dx = f'(x).ef(x), i.e. the product of the differentiation of the exponent, and the original function
  4   logax + logay = loga(xy) multiplication is carried out in logarithms by adding logarithms in the same base
  5   integrating ex if the area under the graph of an exponential function is being found: ∫ ex.dx = ex + C, ie the exponent function is not changed in the process
  6   logax - logay = loga(x/y) division is carried out in logarithms by subtracting logarithms in the same base
  7   differentiating logef(x) with the logarithmic function y = logef(x) the gradient is given by dy/dx = f'(x)/f(x), i.e. differentiation of the function divided by the the original function
  8   y = logef(x), dy/dx = f'(x)/f(x) the gradient of y = logef(x) is the differentiation of the function divided by the original function
  9   same number this is the result of raising any real number to the index power of 1
10   natural logarithm base e; which is defined by the integration: ∫e1 (1/t)dt = 1, an irrational number with value of 2.718 281 82.....
11   inverse a negative index can be written as a positive index if this process is done: x-2 = 1/x2
12   logarithmic function to a base a loga; defined as the inverse of the exponent function ax, thus if y = ax, then logay = x (after logging both sides of y = ax to base a and then using indice rule and apply log aa = 1 )
13   log definition in general, if logarithm of y to base a equals x, then a to the indice x = y, i.e.: logay = x then ax = y
14   logay = x log definition: in general, if logarithm of y to base a equals x, then a to the indice x = y, i.e.: ax = y
15   zero the logarithm of one (1) in any base is zero (any number to the indice of zero = 1 (x0 = 1), and the logarithm is actually the indice of a base)



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48

Exponential and Logarithmic Functions 2

  1   ∫ f'(x).dx/f(x) = logef(x) + C integrating f'(x).dx/f(x): in which the quotient of a differentiated function of x and the original function is integrated to the logarithm of the original function in base e
  2   the number a base raised to the power of a logarithm of a number in the same base equals this; eg eln x = x, i.e., elogex = x
  3   a3/4 = ∜a3 example of an indice involving fraction: if a > 0 and r = p/q > 0, then ar is the qth root of ap
  4   ln abbreviation for loge, also called natural logarithm
  5   dy/dx = ex differentiating y = ex, this results in the same expression, i.e. ex is not changed when it is differentiated because the differential of x is 1.
  6   exponential graph a curve, asymptotic with the x axis, rising to infinity on the y axis, y = ex, y = nx, passes through 1 on the y axis (e0 = 1)
  7   exponent index, the number of times a number is multiplied by itself, e.g. a × a × a × a × a = a5, and the 5 is the index
  8   differentiating logex if the logarithmic function y = logex, then dy/dx = 1/x, i.e. log x to base e differentiates to be the inverse of x
  9   dy/dx = f'(x).ef(x)) gradient of the exponential function y = ef(x) is the product of the differentiation of the exponent, and the original function
10   division this mathematical process is carried out in logarithms by subtracting logarithms in the same base: logax - logay = loga(x/y)
11   logarithm the exponential to which a base is raised to produce a particular number: in y = ax, logay = x, for real numbers
12   y = 1/x hyperbola function which forms a pair of curves that are asymptotic to both x and y axis, forming a mirror image diagonally across the axes
13   x0 = 1 rule for raising any real number to the exponent (index) power of zero
14   indice and logs rule the logarithm of a number that has an indice = the product of the indice and the logarithm of that number: logaxn = nlogax
15   integrating ef(x) the product of the reciprocal of the coefficient of the x in the exponent function and the original function with an added constant: ∫ e(ax + b).dx = (1/a).e(ax + b) + C



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49

Exponential and Logarithmic Functions 3

  1   x1 = x rule for raising any real number to the power of 1
  2   ∫e1 (1/t).dt = 1 the definition of the natural logarithm base e, which is an irrational number with value of 2.718 281 82.... .
  3   ∫ dx/x = ∫ (1/x).dx = logex + C integrating 1/x: the area under the graph of a reciprocal function (the reciprocal of a number) is integrated to the log of the number in base e
  4   logaa = 1 the logarithm of a number in its own base is one because the logarithm is actually the indice of a base, (so a base raised to power of one yields the base): a1 = a
  5   logarithmic function ln loge; defined by ln x = logex, where x>0, and ∫(1/x)dx = ln x, where x > 0
  6   qth root if a > 0 and r = p/q> 0 then ar is this of ap, e.g. a2/3 is the cube root of a2 = ∛(a2)
  7   loga1 = 0 the logarithm of one (1) in any base is zero (any number to the indice of zero = 1, and the logarithm is actually the indice of a base)
  8   indice involving fraction if a > 0 and r = p/q > 0, then ar is the qth root of ap, e.g. a3/4 means the 4th root of a3 = ∜(a3)
  9   logax = logbx/logba logarithm change of base rule: this can be done by finding the quotient of the new base log of the original number to the new base log of the original base
10   multiplication this mathematical process is carried out in logarithms by adding logarithms in the same base: logax + logay = loga(xy)
11   eln x = x or elogex = x a base raised to the power of a logarithm of a number in the same base equals that original number
12   y = ex exponential function that forms a curved graph, which is asymptotic with the x axis, and rising to infinity on the y axis, after passing through 1 on the y axis
13   a × a × a × a × a = a5 rule for converting the repeated multiplication into exponent or index format
14   exponential function the function f such that f(x) = ex or exp x, for all values of x in real numbers, has properties of other indices, d (exp x)/dx = exp x
15   d(logex)/dx = 1/x if the logarithmic function is y = logex, it differentiates to be the inverse of x



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50

Exponential and Logarithmic Functions 4

  1   hyperbola graph a curve that is asymptotic to both x and y axes, and has a mirror image diagonally across the axis: y = 1/x
  2   zero index for any real number, this provides the answer 1
  3   logay = x expressing y = ax so that the exponent can be processed: log both sides with the same base that the exponent has originally permitting the use of a number logged to itself = 1; logaa = 1
  4   ∫ ex.dx = ex + C formula for integrating ex: where the area under the graph of an exponential function is being found; note: the exponent function is not changed in the process
  5   logaxn = nlogax indice and logs: the logarithm of a number that has an indice = the product of the indice and the logarithm of that number
  6   integrating 1/x if the area under the graph of a reciprocal function is being found: ∫ dx/x = ∫ (1/x).dx = logex + C, i.e. the reciprocal of a number is integrated to the log of the number in base e
  7   ∫ e(ax + b).dx = (1/a).e(ax + b) + C integrating ef(x): the product of the reciprocal of the coefficient of the x in the exponent function and the original function with an added constant
  8   one the logarithm of a number in its own base: logaa = 1 because the logarithm is actually the indice of a base (so a base raised to power of one yields the base)
  9   integrating f'(x).dx/f(x) the quotient of a differentiated function of x and the original function is integrated to the logarithm of the original function in base e: ∫ f'(x).dx/f(x) = logef(x) + C
10   x-a = 1/xa rule for converting a negative index into a positive index by inversion
11   ∫(1/x)dx = ln x = logex where x > 0, the integration of 1/x, results in the logarithmic function ln, where ln x = logex
12   change of base rule this can be done by finding the quotient of the log of the original number to the log of the original base: logax = logbx/logba
13   xp/q = q√(xp) rule for indices composed of fractions, the denominator of the exponent is called a root factor, eg qth root
14   differentiating ex if the exponential function is y = ex, then dy/dx = ex, i.e. it is not changed by the process



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51

Trigonometric Functions 1

  1   π radians = 180o to convert radians to degrees, apply the unitary method on this equation: i.e. divide both sides by π, and then multiply both sides by the angle (in radians) that needs converting
  2   cos x integrals ∫ cos x dx = sin x + C or ∫ cos(ax + b) dx = (1/a)sin(ax + b) + C {where x is in radians}
  3   don't forget after using the calculator to find a trigonometric function of a angle in radians, it is essential to return the calculator back to deg(rees) mode
  4   small angles if x in radians is this then sin x = x, tan x = x and cos x = 1 - x
  5   sine graph graph of a curve, over 360o (0 to 2π) between 1 and -1 on the y axis, starting at zero ( sin 0 = zero)
  6   cos x derivatives d(cos x)/dx = - sin x, or d(cos(ax + b) = -asin(ax + b) [or cos f(x) dx = -f'(x).sin f(x)] {where x is in radians}
  7   d(cos x)/dx = - sin x cos x derivative for a simple cos x function, where x is in radians
  8   sec2x integrals ∫ sec2x dx = tan x + C or ∫ sec2(ax + b) = (1/a)tan(ax + b) + C {where x is in radians}
  9   radian the angle that an arc of 1 unit subtends at the centre of a circle of radius 1 unit, note: 2π radians is equivalent to 360o and π radians to 180o
10   circumference product of π and the diameter or of 2π and the radius: C = 2πr = πd
11   ∫ sec2x dx = tan x + C sec2x integral for simple sec2x function, where x is in radians
12   C = 2πr = πd circle circumference, the product of π and the diameter, or of 2π and the radius
13   cosecant graph graph showing consecutive concave up, concave down curves with minima/maxima at y = 1 or y = -1 respectively at x = odd multiples of π/2, and asymptotic at x = integral multiples of π
14   secant graph graph showing consecutive concave up, concave down curves with minima/maxima at y = 1 or y = -1 respectively at x = integral multiples of π, and asymptotic at x = odd multiples of π/2
15   d(tan x)/dx = sec2x tax x derivative for simple tan x function, where x is in radians



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52

Trigonometric Functions 2

  1   angle notation an angle in degrees is written xo while an angle in radians is written x
  2   d(tan(ax + b))/dx = a sec2(ax + b) tax x derivative for complex tan x function, where x is in radians [or alternatively tan f(x) = f'(x).sec2 f(x)]
  3   preferred for higher maths natural logarithms (base e) rather than logarithms to base 10 or any other base, radians as a measure of angle to degrees
  4   sector area product of half, radius squared, and the angle in radians: ie A = ½ r2 θ
  5   A = ½ r2θ sector area: the product of half, radius squared, and the angle in radians
  6   trigonometric limits when x is small, sin x --> 0, cos x --> 1, tan x --> 0, and the limit as x --> 0 of (sinx)/x = 1 {where x is in radians}
  7   sin x derivatives d(sin x)/dx = cos x or d(sin(ax + b)/dx = acos(ax + b) [or sin f(x) dx = f'(x).cos f(x)] {where x is in radians}
  8   ∫ sin(ax + b) dx =(-1/a) cos(ax + b) + C sin x integral for complex sin x function, {where x is in radians}
  9   ½ r2(θ - sin θ) minor segment area: this is found by using the formula
10   d(sin x)/dx = cos x sin x derivative for a simple sin x function, where x is in radians
11   d sin(ax + b)/dx = acos(ax + b) sin x derivative for a complex sin x function, where x is in radians [or alternatively sin f(x) dx = f'(x).cos f(x)]
12   ∫ cos x dx = sin x + C cos x integral for simple cos x function, where x is in radians
13   circle area A = πr2, i.e. this amount is the product of π and the square of the radius
14   ∫ cos(ax + b) dx = (1/a)sin(ax + b) + C cos x integral for complex cos x function, {where x is in radians}
15   A = πr2 circle area, ie the product of π and the square of the radius
16   degrees to radians use unitary method on the equation π radians = 180o (i.e. divide both sides by 180, and then multiply both sides by the angle (in degrees) that needs converting)



  Trigonometric Functions 2Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

53

Trigonometric Functions 3

  1   cosine graph graph with a curve shape, over 360o (0 to 2π) between 1 and -1 on the y axis, starting at 1 ( cos 0 = one)
  2   tangent graph graph of a curve shape, a repeating pattern that has gaps because the graph is asymptotic at multiples of π/2, starts at 0o, max at 90o (π/2) limit, recommence with min at 90o limit, symmetrical at integral multiples of π
  3   d(cos(ax + b) = -asin(ax + b) cos x derivative for a complex cos x function, where x is in radians [or alternatively cos f(x) dx = -f'(x).sin f(x)]
  4   360o = 2π rad relationship between degrees and radians
  5   tan x derivatives d(tanx)/dx = sec2x or d(tan(ax + b))/dx = a sec2(ax + b) [or tan f(x) = f'(x).sec2 f(x)] {where x is in radians}
  6   ∫ sec2(ax + b) = (1/a)tan(ax + b) + C sec2x integral for complex sec x function, {where x is in radians}
  7   one revolution this amount of a circle is 360o or 2π radians
  8   arc length product of radius and the angle subtended at the centre of a circle in radians: l = rθ
  9   l = rθ arc length: the product of radius and the angle subtended at the centre of a circle in radians
10   cotangent graph graph of a curve shape, a repeating pattern that has gaps because the graph is asymptotic at integral multiples of π, max at 0o limit, reach min at π (180o), recommence with max at same limit, symmetrical at multiples of π
11   periodicity repeating, as in the graphs of trigonometric functions, every 2π (360o)
12   sin x integrals ∫ sin x.dx = -cos x + C, or ∫ sin(ax + b) dx =(-1/a) cos(ax + b) {where x is in radians}
13   rad(ians) mode to find sine of an angle in radians, change calculator mode to this and then proceed with the angle as radians, eg sin 5.4 is sine of an angle in radians, whereas sin 5.4o is for angle in degrees
14   ∫ sin x dx = -cos x + C sin x integral for simple sin x function, where x is in radians
15   radians to degrees use unitary method on the equation π radians = 180o (i.e. divide both sides by π, and then multiply both sides by the angle (in radians) that needs converting)
16   minor segment area this is found by using the formula: ½r2(θ - sin θ), where θ is in radians



  Trigonometric Functions 3Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

54

Applications of Calculus to the Real World 1

  1   a = = dv/dt = d(dx/dt)/dt = d2x/dt2 formula to find the rate of change of velocity with respect to time or acceleration, can be either positive or negative depending on the direction it is acting in
  2   constant velocity this occurs when acceleration = 0
  3   growth constant in exponential growth or decay, the rate of change of some quantity is in proportion to the quantity itself: dQ/dt = kQ, where k is the ..........
  4   growth situation where dQ/dt = kQ because Q = Q0ekt, where Q0 is the initial quantity
  5   instantaneous this is the type of measure of rate of change determined by using dN/dt = kN = Nt = N0ekt, rather than the average rate of change, quantity: units per unit time
  6   exponential growth uses in the study of population growth and decay, depletion of natural resources, inflation, industrial productions; dN/dt = kN = Nt = N0ekt
  7   velocity = 0 this occurs where displacement is a maximum or a minimum, the first derivative dx/dt = 0, or = 0
  8   second derivative this gives the maximum or minimum velocity when it equals zero: or dv/dt = 0, i.e. the acceleration = zero
  9   dQ/dt = -kQ rate of decay of a quantity, found by differentiating Q = Q0e-kt, where Q0 is the initial quantity, and -k is the decay constant
10   population the rate of change of this in a town is given by dP/dt = kP = P0ekt or Nt = N0ekt, where P0 or N0 is the original
11   direct distance displacement (x), either positive or negative from the origin depending on its direction from the origin
12   velocity rate of change of displacement with respect to time: v = = dx/dt; can be positive or negative according to the direction the object is travelling
13   velocity if an acceleration () equation is given, and this quantity is required, then the expression is found by integration: = ∫ dt
14   v = ∫ a dt or = dt the integration of an acceleration equation determines the velocity, i.e. the area under an acceleration graph is velocity
15   dV/dt formula applied to find the rate of change of the volume of water (i.e., the rate of flow) etc.
16   draw sketches these should be done for rate of change questions for Q and (first derivative) as functions of t if possible



  Applications of Calculus to the Real World 1Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

55

Applications of Calculus to the Real World 2

  1   substitute equation when finding rate of decay or growth by differentiation of Q = Q0ekt, this process is performed to simplify the result: = kQ0ekt = kQ
  2   x = ∫ v dt if a velocity equation is given, and the displacement is required, then it can be found by integration; displacement is area under a velocity vs time graph
  3   integrate process performed to go from rate of water flow equation to volume equation V = dt
  4   decimal or fraction a rate given as a percentage needs to be converted to this before using it in y = Aekt equations for growth (k>0) or decay (k<0)
  5   acceleration = 0 this is the case when velocity is constant, second derivative d2x/dt2 = 0
  6   first derivative this gives the maximum or minimum displacement when it equals zero: dx/dt = 0, i.e. the velocity = zero
  7   dQ/dt = kQ rate of growth of a quantity, found by differentiating Q = Q0ekt, where Q0 is the initial quantity, and k is the growth constant
  8   decay situation where dQ/dt = -kQ because Q = Q0e-kt, where Q0 is the initial quantity
  9   displacement direct distance, either positive or negative from the origin depending on its direction from the origin; when at origin, x = 0
10   dv/dt = 0 second derivative alternative: this gives the maximum or minimum velocity when it equals zero: ie the acceleration = zero
11   time in motion equations and differentiations, this quantity is never negative, but initially it equals zero
12   P = P0ekt or Nt = N0ekt formula for population size, which can be differentiated to find the rate of population growth: dP/dt = kP. Note Po and N0 are the original populations
13   volume the rate of change of this quantity for water is shown by dV/dt
14   v = = dx/dt formula to find the rate of change of displacement with respect to time or velocity; can be positive or negative according to the direction the object is travelling
15   acceleration the rate of change of velocity with respect to time, can be either positive or negative depending on the direction it is acting in, a = = dv/dt = d(dx/dt)/dt = d2x/dt2
16   displacement if a velocity equation is given, and this quantity is required, then the expression is found by integration: x = ∫ v dt



  Applications of Calculus to the Real World 2Two page printable: Student Answer Sheet followed by the Answers


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Num.AnswerQuestion/Statement



Questions: Sheet

56

Series 2 1

  1   A = P(1 + r/100)n compound interest formula: amount value (A) of an investment (P(rincipal)) after it has been invested for (n) periods of time at an interest rate of r%
  2   N = A/(1 + r)n the single sum investment (N) that would yield the same final annuity value (A) if invested at the same interest rate (r) for the same amount of time (n)
  3   superannuation regular investments at compounded interest means that the amount of money at the end of the investment is the sum of n terms of a geometric series, so that the first term (a) = 1+ interest = r, allowing use of Sn = a(rn-1)/(r-1), multiplied by instalment (P)
  4   present value of an annuity formula used to calculate this, knowing the contribution M and interest per period r and the number of periods n: N = M{((1 + r)n - 1)/(r(1 + r)n)}
  5   loan repayments these can be calculated by transposing the present value (N or loan) formula to make the periodic payments (M) the equation subject: M = N{r(1 + r)n/((1 + r)n - 1)}
  6   N =M {((1 + r)n - 1)/(r(1 + r)n)} formula used to calculate present value of an annuity, knowing the contribution (M) and interest (r) per period and the number of periods (n)
  7   M = N{r(1 + r)n/((1 + r)n - 1)} formula for calculating loan repayments, M is the repayment and N is the amount borrowed, r the interest , n the number of payment periods
  8   house repayments regular repayments with compounded interest means that the amount of money paid is in instalments so that none is owed at the end of the loan and is the sum of n terms of a geometric series, so that the first term (a) = 1, interest = r (as 1 + i), allowing the use of Sn = a(rn - 1)/(r - 1), which is then divided into P(1 + r)n, such that M = P(1 + r)n/(1(rn - 1)/(r - 1))
  9   series applications saving a constantly increasing amount per year, converting a repeating decimal to fraction, distance moved by bouncing ball, compound interest, monthly repayments on variable house loans
10   time to repay loan this can be found from the present value of an annuity formula (if given an amount of a periodic payment) through a process of trial and error by substituting various repayment periods until a similar periodic repayment is obtained: M = N{r(1 + r)n/((1 + r)n - 1)}
11   repeating decimal these can be converted to fractions using series formulae by converting the number to a series, finding the common ratio and substituting in sum to infinity formula: S = a/(1 - r)



  Series 2 1Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

57

Series 2 2

  1   M = N{r(1 + r)n/((1 + r)n - 1)} formula that can be used to calculate the time taken to repay a loan, given M repayments($), where N is the loan, r interest per period, n repayments (periods - unknown), using a process of trial and error
  2   S = a/(1 - r) geometric series formula that can be used to convert repeating decimals to fractions by converting them into a number series and finding the common ratio
  3   future value of an annuity a formula used to find out how much money will have accumulated if a periodic investment is made at the end of each year (n = number of years) and interest is compounded A = M{((1 + r)n - 1)/r}
  4   constant increase a problem involving this (e.g. an extra $10 saved per year): an arithmetic series, so the sum of n terms = (n/2)(a + l) where a and l are the first and last terms respectively
  5   A = M{((1 + r)n - 1)/r} a formula used to find out how much money will have accumulated (A) if a periodic investment (M) is made at the end of each year (n = number of years) and interest is compounded (r)
  6   Sn = ½n(a + l) arithmetic series formula that can be used to solve problems involving a constant increase(eg an extra $10 saved per year) where a and l are the first and last terms respectively
  7   future value of an annuity a formula that can be used to find how much needs to be contributed to an investment to reach a certain goal M = Ar/((1 + r)n - 1)
  8   bouncing ball distance a sum to infinity problem that involves a single sum to infinity of a geometric problem in which a number (x) is followed by two series because of the up/down factor, so total = x + 2(a/(1 - r)), a is the first upward movement r is the common ratio
  9   M = Ar/((1 + r)n - 1) a formula that can be used to find how much needs to be contributed to an investment to reach a certain goal (future value of an annuity A), r interest, n periods, M contributions per period
10   compound interest this is the amount value (A) of an investment, or principal (P) after it has been invested for (n) periods of time at an interest rate of r (given as a percentage): A = P(1 + r/100)n
11   present value of an annuity the single sum investment (N) that would yield the same final annuity value if invested at the same interest rate for the same amount of time: N = A/(1 + r)n



  Series 2 2Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

58

Probability 2 1

  1   P(E') = 1 - P(E) complementary probability formula, determining the probability that an event does NOT occur, care needs to be taken and probability trees help
  2   mutually exclusive events are this if only one can occur e.g., numbers thrown on dice, probability found with addition law: P(A ∪ B) = P(A) + P(B), i.e. P(A or B)
  3   independent two events are this if the occurrence of either does not affect the probability of the other occurring, probability can be calculated with the product rule: P(A ∩ B) = P(A).P(B), e.g. a series of coin tosses, two two die thrown at the same time
  4   product rule that is applied to find the probability of independent events occurring, e.g. a series of coin tosses, two two coins tossed at the same time: P(A ∩ B) = P(A).P(B)
  5   probability of an event P(E) = (number of ways the event can occur)/(total number of possible outcomes), found by using a sample space, e.g., table, Venn diagram
  6   mutually non-exclusive events are this if more than one thing can happen at the same time, P(A U B) = P(A) + P(B) - P(A ∩ B), use Venn diagrams to show sample space eg find probability of drawing an even number card or a card less than 8
  7   P(A &cup: B) = P(A) + P(B) - P(A ∩ B) formula for mutually non-exclusive events (more than one thing can happen at the same time), use Venn diagrams to show sample space eg find probability of drawing an even number card or a card less than 8
  8   add probabilities this is done if an outcome can be obtained in more then one way as observed with a probability tree or tree diagram between separate branches
  9   multiply probabilities this mathematical operation is performed on probabilities linked along branches of a probability tree or tree diagram
10   certain this is the situation if the probability of an event is 1
11   probability a measure of an event occurring as a result of an experiment, expressed on a scale of 0 (impossible) to 1 (certain)
12   union A ∪ B: the event that is either A or B occurs, also called the sum of A and B, eg P(A ∪ B) = P(A) + P(B) for mutually exclusive events
13   A ∪ B notation for union or the probability either A or B occurs, also called the sum of A and B, eg P(A ∪ B) = P(A) + P(B) for mutually exclusive events, A union B
14   intersection A ∩ B: the event that is when both A and B occur, shown as intersecting circles on Venn diagrams



  Probability 2 1Two page printable: Student Answer Sheet followed by the Answers


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Questions: Sheet

59

Probability 2 2

  1   A ∩ B notation for intersection or the probability both A and B occurs in the same event, A intersection B
  2   Venn diagram a method for writing the sample space for events that are mutually non-exclusive so that either A or B (A ∪ B) and A and B (A &cap B) outcomes can be determined: rectangle containing overlapping circles
  3   addition law method used to find the probability of two events occurring when they are mutually exclusive, ie only one of the events can occur at a time (numbers on a dice): P(A ∪ B) = P(A) + P(B) i.e. A or B
  4   P(A ∪ B) = P(A) + P(B) formula for the addition law of probability, can only be used for mutually exclusive events, i.e. to find the probability of event A or event B occurring
  5   P(A ∩ B) = P(A).P(B) product rule that is applied to find the probability of independent events occurring, e.g. a series of coin tosses, two prizes in a raffle (allow for ticket reduction)
  6   tree diagrams a technique used when using the product rule of probability because it shows all the possible sequences of events, probabilities multiplied along branches, and branches added if more than one branch has suitable results
  7   P(E) = favourable/possible outcomes probability of an event formula, data being found using a sample space such as a table or a Venn diagram
  8   impossible this is the situation if the probability of an event is zero
  9   intersection A ∩ B: the event that can be described as both A and B occur
10   A ∩ B notation for intersection or the event in probability that can be described as both A and B occur
11   equally likely outcomes these occur when each element of the sample space for a probability experiment is equally likely to occur, e.g. coin toss, die roll
12   changing probabilities these occur where previous experiments have altered the number of items in a sample space, eg, drawing tickets for more than one prize
13   complementary result the event or events that occurs when a particular event does not occur
14   complementary probability probability that event E does not occur = 1 - probability that event E does occur: P(E') = 1 - P(E); e.g. probability of a baby not having blue eyes = 1 - probability of having blue eyes
15   sample space all the possible outcomes of an experiment, can be expressed in table form or Venn diagram (the latter being especially for mutually non-exclusive events)



  Probability 2 2Two page printable: Student Answer Sheet followed by the Answers


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