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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 |

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

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## Questions: Sheet | ## 2 | ## Basic Arithmetic and Algebra 2 |

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

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## Questions: Sheet | ## 3 | ## Basic Arithmetic and Algebra 3 |

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

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## Questions: Sheet | ## 4 | ## Basic Arithmetic and Algebra 4 |

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

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## Questions: Sheet | ## 5 | ## Basic Arithmetic and Algebra 5 |

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

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## Questions: Sheet | ## 6 | ## Basic Arithmetic and Algebra 6 |

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

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## Questions: Sheet | ## 7 | ## Basic Arithmetic and Algebra 7 |

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

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## Questions: Sheet | ## 8 | ## Plane Geometry 1 |

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

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## Questions: Sheet | ## 9 | ## Plane Geometry 2 |

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

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## Questions: Sheet | ## 10 | ## Plane Geometry 3 |

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## Questions: Sheet | ## 11 | ## Plane Geometry 4 |

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

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## Questions: Sheet | ## 12 | ## Plane Geometry 5 |

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

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## Questions: Sheet | ## 13 | ## Plane Geometry 6 |

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

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## Questions: Sheet | ## 14 | ## Functions and Graphs 1 |

Functions and Graphs 1 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 15 | ## Functions and Graphs 2 |

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

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## Questions: Sheet | ## 16 | ## Functions and Graphs 3 |

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

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## Questions: Sheet | ## 17 | ## Functions and Graphs 4 |

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

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## Questions: Sheet | ## 18 | ## Functions and Graphs 5 |

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

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## Questions: Sheet | ## 19 | ## Trigonometry 1 |

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

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## Questions: Sheet | ## 20 | ## Trigonometry 2 |

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

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## Questions: Sheet | ## 21 | ## Trigonometry 3 |

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

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## Questions: Sheet | ## 22 | ## Trigonometry 4 |

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

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## Questions: Sheet | ## 23 | ## Trigonometry 5 |

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

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## Questions: Sheet | ## 24 | ## Trigonometry 6 |

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

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## Questions: Sheet | ## 25 | ## Straight Line Graphs 1 |

Straight Line Graphs 1 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 26 | ## Straight Line Graphs 2 |

Straight Line Graphs 2 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 27 | ## Straight Line Graphs 3 |

Straight Line Graphs 3 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 28 | ## Straight Line Graphs 4 |

Straight Line Graphs 4 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 29 | ## Introduction to Calculus 1 |

Introduction to Calculus 1 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 30 | ## Introduction to Calculus 2 |

Introduction to Calculus 2 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 31 | ## Quadratic Functions 1 |

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

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## Questions: Sheet | ## 32 | ## Quadratic Functions 2 |

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

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## Questions: Sheet | ## 33 | ## Quadratic Functions 3 |

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

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## Questions: Sheet | ## 34 | ## Locus, Parabola, Circle 1 |

Locus, Parabola, Circle 1 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 35 | ## Locus, Parabola, Circle 2 |

Locus, Parabola, Circle 2 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 36 | ## Locus, Parabola, Circle 3 |

Locus, Parabola, Circle 3 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 37 | ## Series 1 1 |

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

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## Questions: Sheet | ## 38 | ## Series 1 2 |

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## Questions: Sheet | ## 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 + ar^{2} + ar^{3} + ....... + ar^{(n-1)}, first term a, common ratio r, number of terms n |

2 | S_{n} = a(r^{n} - 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 S_{n} = a(1 - r^{n})/(1 - r) where the absolute value of r is less than 1, first term is a and the number of terms is n |

4 | T_{n} = 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 = T_{2} - T_{1} = T_{3} - T_{2} |

6 | nth term | for a geometric series, is found by T_{n} = ar^{n-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. S_{n} |

9 | sequence | the terms of a series, i.e. a_{1}, a_{2}, a_{3}, a_{4}, ..... a_{n}; or a_{1}, a_{2}, a_{3}, ...... |

10 | T_{2} = ½(T_{1} + T_{3}) | formula for the arithmetic mean for two terms of an arithmetic series |

11 | sum arithmetic series | the formula for this is S_{n} = ½n[2a + (n-1)d] if the first term (a) is known, and the common difference (d) |

12 | r = T_{2}/T_{1} = T_{3}/T_{2} | 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: T_{2} = ± √(T_{1}.T_{3}) |

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 |

Series 1 3 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 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 = x^{2} + 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 | d^{2}y/dx^{2}, f"(x), y", d^{2}/dx^{2} (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 = x^{n}, then y = x^{n+1}/(n+1) + C, where C is a constant and can be determined from the coordinates of known points or the y axis intercept |

Geometrical Applications of Calculus 1 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 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 = x^{n+1}/(n+1) + C | formula for finding the primitive function given the first derivative, e.g. dy/dx = x^{n}, 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, d^{2}y/dx^{2}, f"(x), y", d^{2}/dx^{2} (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 |

Geometrical Applications of Calculus 2 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 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 = πr^{2} |

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 |

Geometrical Applications of Calculus 3 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 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: ∫^{b}_{a} f(x).dx ≑ ½h[(y_{o} + y_{n}) + 2(y_{1} + y_{2} + y_{3} + ......... + 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 | ∫^{b}_{a} 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 | ∫^{b}_{a} f(x).dx ≑ ½h[(y_{o} + y_{n}) + 2(y_{rest} 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 | ∫ x^{n}.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 ∫^{a}_{0} f(x).dx |

11 | π∫^{b}_{a} y^{2}.dx | formula for finding the volume of space obtained when a curve is rotated about x-axis; first step is to make y^{2} 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 x^{2} the subject, then apply: V = π∫^{b}_{a} x^{2}.dy, then substitute for x^{2} so that V = π∫^{b}_{a} 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[(y_{0}+y_{n}) + 4(y_{1}+y_{3}+y_{5} +...) + 2(y_{2}+y_{4}+y_{6}+....)], 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 |

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

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## Questions: Sheet | ## 44 | ## Integration 2 |

1 | volume about x-axis | to find this, square both sides of the equation and make y^{2} the subject, then apply: V = π∫^{b}_{a} y^{2}.dx, then substitute for y^{2} so that V = π∫^{b}_{a} 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 × x^{0} which integrates to 1 × x^{0 + 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 = ∫^{b}_{a} 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 | ∫^{b}_{a} 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: ∫ x^{n}.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: ∫^{b}_{a} f(x).dx ≑((b-a)/6)[f(a) + 4f((a + b)/2) + f(b)] ≑ (b-a)/6[y_{first} + 4y_{mid} + y_{last}] |

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 x_{2} and x_{1} in f(x), i.e. f(x_{2}) - f(x_{1}) |

15 | f(-x) = f(x), and ∫^{a}_{-a} f(x).dx = 2∫^{a}_{0} f(x).dx | even functions rules, when f(x) is symmetrical about the y-axis |

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

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## Questions: Sheet | ## 45 | ## Integration 3 |

1 | ∫ kx^{n}.dx = k∫ x^{n}.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 | ∫^{b}_{a} f(x).dx ≑ ⅓h[(y_{0} + y_{n}) + 4(y_{sum odds}) + 2(y_{sum 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(x^{2}) or f(y^{2}), (found by squaring both sides of the equation), so that the volume can be calculated: V = π∫^{b}_{a} y^{2}.dx for x axis |

5 | y-axis areas | these are found by finding f(y) and integrating it as follows: A = ∫^{b}_{a} f(y).dy using the same rules as for x |

6 | area estimation errors | for trapezoidal rule this is approximately 1/n^{2}, while for Simpson's rule it is approximately 1/n^{4} where n is the number of subdivisions being used to find the area under a graph |

7 | ∫^{b}_{a} 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∫^{b}_{a} x^{2}.dy | formula for finding the volume of space obtained when a curve is rotated about y-axis; first step is to make x^{2} 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 | ∫^{b}_{a} 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)]^{b}_{a} | 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) |

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

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## Questions: Sheet | ## 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 y^{2} the subject, then apply: V = π ∫^{b}_{a} y^{2}.dx = π∫^{b}_{a} 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)]^{b}_{a} |

5 | ∫^{b}_{a} 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[y_{first} + 4y_{mid} + y_{last}] |

6 | integration and constant factor | if f(x) contains a factorizable constant: ∫ kx^{n}.dx = k∫x^{n}.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∫^{b}_{a} x^{2}.dy = π∫^{b}_{a} 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 x^{2} 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 × x^{0} which integrates to 1 × x^{0 + 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: &int^{b}_{a} f(x).dx ≑ ½(b-a)[f(a) + f(b)] |

13 | definite integral | ∫^{b}_{a} 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 |

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

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## Questions: Sheet | ## 47 | ## Exponential and Logarithmic Functions 1 |

1 | base | the number to which an exponential is applied to produce another defined number: in y = a^{x}, x is the exponential applied to the a to produce y |

2 | y = a^{x} | an expression showing base a with exponent x |

3 | differentiating e^{f(x)} | if the exponential function y = e^{f(x)} then the gradient is given by dy/dx = f'(x).e^{f(x)}, i.e. the product of the differentiation of the exponent, and the original function |

4 | log_{a}x + log_{a}y = log_{a}(xy) | multiplication is carried out in logarithms by adding logarithms in the same base |

5 | integrating e^{x} | if the area under the graph of an exponential function is being found: ∫ e^{x}.dx = e^{x} + C, ie the exponent function is not changed in the process |

6 | log_{a}x - log_{a}y = log_{a}(x/y) | division is carried out in logarithms by subtracting logarithms in the same base |

7 | differentiating log_{e}f(x) | with the logarithmic function y = log_{e}f(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 = log_{e}f(x), dy/dx = f'(x)/f(x) | the gradient of y = log_{e}f(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: ∫^{e}_{1} (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/x^{2} |

12 | logarithmic function to a base a | log_{a}; defined as the inverse of the exponent function a^{x}, thus if y = a^{x}, then log_{a}y = x (after logging both sides of y = a^{x} to base a and then using indice rule and apply log _{a}a = 1 ) |

13 | log definition | in general, if logarithm of y to base a equals x, then a to the indice x = y, i.e.: log_{a}y = x then a^{x} = y |

14 | log_{a}y = x | log definition: in general, if logarithm of y to base a equals x, then a to the indice x = y, i.e.: a^{x} = y |

15 | zero | the logarithm of one (1) in any base is zero (any number to the indice of zero = 1 (x^{0} = 1), and the logarithm is actually the indice of a base) |

Exponential and Logarithmic Functions 1 Two page printable: Student Answer Sheet followed by the Answers

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## Questions: Sheet | ## 48 | ## Exponential and Logarithmic Functions 2 |

1 | ∫ f'(x).dx/f(x) = log_{e}f(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 e^{ln x} = x, i.e., e^{logex} = x |

3 | a^{3/4} = ∜a^{3} | example of an indice involving fraction: if a > 0 and r = p/q > 0, then a^{r} is the qth root of a^{p} |

4 | ln | abbreviation for log_{e}, also called natural logarithm |

5 | dy/dx = e^{x} | differentiating y = e^{x}, this results in the same expression, i.e. e^{x} 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 = e^{x}, y = n^{x}, passes through 1 on the y axis (e^{0} = 1) |

7 | exponent | index, the number of times a number is multiplied by itself, e.g. a × a × a × a × a = a^{5}, and the 5 is the index |

8 | differentiating log_{e}x | if the logarithmic function y = log_{e}x, then dy/dx = 1/x, i.e. log x to base e differentiates to be the inverse of x |

9 | dy/dx = f'(x).e^{f(x)}) | gradient of the exponential function y = e^{f(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: log_{a}x - log_{a}y = log_{a}(x/y) |

11 | logarithm | the exponential to which a base is raised to produce a particular number: in y = a^{x}, log_{a}y = 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 | x^{0} = 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: log_{a}x^{n} = nlog_{a}x |

15 | integrating e^{f(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 |

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

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## Questions: Sheet | ## 49 | ## Exponential and Logarithmic Functions 3 |

1 | x^{1} = x | rule for raising any real number to the power of 1 |

2 | ∫^{e}_{1} (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 = log_{e}x + 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 | log_{a}a = 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): a^{1} = a |

5 | logarithmic function ln | log_{e}; defined by ln x = log_{e}x, where x>0, and ∫(1/x)dx = ln x, where x > 0 |

6 | qth root | if a > 0 and r = p/q> 0 then a^{r} is this of a^{p}, e.g. a^{2/3} is the cube root of a^{2} = ∛(a^{2}) |

7 | log_{a}1 = 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 a^{r} is the qth root of a^{p}, e.g. a^{3/4} means the 4th root of a^{3} = ∜(a^{3}) |

9 | log_{a}x = log_{b}x/log_{b}a | 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: log_{a}x + log_{a}y = log_{a}(xy) |

11 | e^{ln x} = x or e^{logex} = x | a base raised to the power of a logarithm of a number in the same base equals that original number |

12 | y = e^{x} | 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 = a^{5} | rule for converting the repeated multiplication into exponent or index format |

14 | exponential function | the function f such that f(x) = e^{x} or exp x, for all values of x in real numbers, has properties of other indices, d (exp x)/dx = exp x |

15 | d(log_{e}x)/dx = 1/x | if the logarithmic function is y = log_{e}x, it differentiates to be the inverse of x |

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

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## Questions: Sheet | ## 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 | log_{a}y = x | expressing y = a^{x} 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; log_{a}a = 1 |

4 | ∫ e^{x}.dx = e^{x} + C | formula for integrating e^{x}: where the area under the graph of an exponential function is being found; note: the exponent function is not changed in the process |

5 | log_{a}x^{n} = nlog_{a}x | 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 = log_{e}x + 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 e^{f(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: log_{a}a = 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) = log_{e}f(x) + C |

10 | x^{-a} = 1/x^{a} | rule for converting a negative index into a positive index by inversion |

11 | ∫(1/x)dx = ln x = log_{e}x | where x > 0, the integration of 1/x, results in the logarithmic function ln, where ln x = log_{e}x |

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: log_{a}x = log_{b}x/log_{b}a |

13 | x^{p/q} = ^{q}√(x^{p}) | rule for indices composed of fractions, the denominator of the exponent is called a root factor, eg qth root |

14 | differentiating e^{x} | if the exponential function is y = e^{x}, then dy/dx = e^{x}, i.e. it is not changed by the process |

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

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## Questions: Sheet | ## 51 | ## Trigonometric Functions 1 |

1 | π radians = 180^{o} | 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 360^{o} (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 | sec^{2}x integrals | ∫ sec^{2}x dx = tan x + C or ∫ sec^{2}(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 360^{o} and π radians to 180^{o} |

10 | circumference | product of π and the diameter or of 2π and the radius: C = 2πr = πd |

11 | ∫ sec^{2}x dx = tan x + C | sec^{2}x integral for simple sec^{2}x 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 = sec^{2}x | tax x derivative for simple tan x function, where x is in radians |

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

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## Questions: Sheet | ## 52 | ## Trigonometric Functions 2 |

1 | angle notation | an angle in degrees is written x^{o} while an angle in radians is written x |

2 | d(tan(ax + b))/dx = a sec^{2}(ax + b) | tax x derivative for complex tan x function, where x is in radians [or alternatively tan f(x) = f'(x).sec^{2} 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 r^{2} θ |

5 | A = ½ r^{2}θ | 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 r^{2}(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 = πr^{2}, 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 = πr^{2} | 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 = 180^{o} (i.e. divide both sides by 180, and then multiply both sides by the angle (in degrees) that needs converting) |

Trigonometric Functions 2 Two 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 360^{o} (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 0^{o}, max at 90^{o} (π/2) limit, recommence with min at 90^{o} 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 | 360^{o} = 2π rad | relationship between degrees and radians |

5 | tan x derivatives | d(tanx)/dx = sec^{2}x or d(tan(ax + b))/dx = a sec^{2}(ax + b) [or tan f(x) = f'(x).sec^{2} f(x)] {where x is in radians} |

6 | ∫ sec^{2}(ax + b) = (1/a)tan(ax + b) + C | sec^{2}x integral for complex sec x function, {where x is in radians} |

7 | one revolution | this amount of a circle is 360^{o} 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 0^{o} limit, reach min at π (180^{o}), recommence with max at same limit, symmetrical at multiples of π |

11 | periodicity | repeating, as in the graphs of trigonometric functions, every 2π (360^{o}) |

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 5.4^{o} 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 = 180^{o} (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: ½r^{2}(θ - sin θ), where θ is in radians |

Trigonometric Functions 3 Two 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 = ẍ = dv/dt = d(dx/dt)/dt = d^{2}x/dt^{2} | 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 = Q_{0}e^{kt}, where Q_{0} is the initial quantity |

5 | instantaneous | this is the type of measure of rate of change determined by using dN/dt = kN = N_{t} = N_{0}e^{kt}, 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 = N_{t} = N_{0}e^{kt} |

7 | velocity = 0 | this occurs where displacement is a maximum or a minimum, the first derivative dx/dt = 0, or ẋ = 0 |

8 | second derivative | this gives the maximum or minimum velocity when it equals zero: ẍ or dv/dt = 0, i.e. the acceleration = zero |

9 | dQ/dt = -kQ | rate of decay of a quantity, found by differentiating Q = Q_{0}e^{-kt}, where Q_{0} 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 = P_{0}e^{kt} or N_{t} = N_{0}e^{kt}, where P_{0} or N_{0} 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 = ẋ = dx/dt; can be positive or negative according to the direction the object is travelling |

13 | velocity | if an acceleration (ẍ) equation is given, and this quantity is required, then the expression is found by integration: = ∫ ẋ dt |

14 | v = ∫ a dt or ẋ = ẍ 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 Q̇ (first derivative) as functions of t if possible |

Applications of Calculus to the Real World 1 Two 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 = Q_{0}e^{kt}, this process is performed to simplify the result: Q̇ = kQ_{0}e^{kt} = 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 = V̇ dt |

4 | decimal or fraction | a rate given as a percentage needs to be converted to this before using it in y = Ae^{kt} equations for growth (k>0) or decay (k<0) |

5 | acceleration = 0 | this is the case when velocity is constant, second derivative d^{2}x/dt^{2} = 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 = Q_{0}e^{kt}, where Q_{0} is the initial quantity, and k is the growth constant |

8 | decay | situation where dQ/dt = -kQ because Q = Q_{0}e^{-kt}, where Q_{0} 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 = P_{0}e^{kt} or N_{t} = N_{0}e^{kt} | formula for population size, which can be differentiated to find the rate of population growth: dP/dt = kP. Note Po and N_{0} are the original populations |

13 | volume | the rate of change of this quantity for water is shown by dV/dt |

14 | v = ẋ = 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 = ẍ = dv/dt = d(dx/dt)/dt = d^{2}x/dt^{2} |

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

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Num. | Answer | Question/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 S_{n} = a(r^{n}-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 S_{n} = a(r^{n} - 1)/(r - 1), which is then divided into P(1 + r)^{n}, such that M = P(1 + r)^{n}/(1(r^{n} - 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 1 Two 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 | S_{n} = 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 2 Two 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 1 Two 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 2 Two page printable: Student Answer Sheet followed by the Answers

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