DEPARTMENT OF MATHEMATICS

Transcription

1 DEPARTMENT OF MATHEMATICS A2 level Mathematics Core 3 course workbook Name:

2 Welcome to Core 3 (C3) Mathematics. We hope that you will use this workbook to give you an organised set of notes for revision and that it will help you enjoy and benefit from the course more. ORGANISATION Keep this workbook, worksheets, homework and assessments in a file + BOOK SCRUTINY You will be asked to hand this in/bring it along to a progress interview with your teacher or Mr Gower during the year REVISION BOOKLET TRIAL EXAM We will produce revision a revision programme once you have completed all or most of the Core 3 course. As last year there will be an expectation that you do one chapter a week/fortnight to prepare for trial and actual exams. There will be one opportunity to resit the trial exam You may also find the following useful: Free Electronic copy of C3 textbook: (linked from the department website) Dropbox.com For resources and links to video tutorials; exam papers and solutions, formula booklet: Use the Mathematics department website: ccwmaths.wordpress.com And CCW Maths on For exam practice questions with solutions or exam past papers use (linked from the department website) Examsolutions.net Physicsandmathstutor.com Mathspapers.co.uk Mr J Gower Head of KS5 Mathematics

3 Exponential and natural logarithm functions: In general: e x = y and so: x = ln y FORMULA SHEET The usual laws of logarithms apply, so: a ln a + ln b = ln ab ln a ln b = ln b ln a k = k ln a Trigonometric Identities You need to know the trig identities from C2: sin 2 2 tan and sin cos 1 cos You can easily use the second of these to derive these C3 identities: sec 2 2 x 1 tan x and cosec 2 2 x 1 cot x There are a few other identities which you need to be able to use (but the next few are given in the formula book): sin(a ± B) = sin A cos B ± cos A sin B cos(a ± B) = cos A cos B sin A sin B tan A tan B tan( A B) 1 tan A tan B By making A = B in the above identities, you can easily derive these double angle formulae (these are not given in the formula book): sin 2A = 2sin A cos A cos 2A = cos 2 A sin 2 A = 2 cos 2 x 1 = 1 2 sin 2 x tan 2 2 tan A 1 tan A A 2 Reciprocal Trig functions cosec x = 1, sec x = 1, cot x = 1, sin x cos x tan x Inverse Trig functions arcsin x = sin 1 x, arccos x = cos 1 x, arctan x = tan 1 x,

4 Differentiation: The Chain Rule (for differentiating a function of a function) Very simply: dy dx dy du (where y is a function of u, and u is a function of x) du dx The Product Rule (for differentiating products of functions) If u and v are functions of x, and y = uv, then: dy dx dv du u v (this is not given in the formula book, so learn it!) dx dx The Quotient Rule (for differentiating quotients of functions) du dv v u u dy If u and v are functions of x, and y =, then: dx dx v 2 dx v (this is in the formula book, but in a different form) Trig differentiation f(x) sin kx cos kx tan kx cosec x sec x cot x f (x) k cos kx k sin kx k sec 2 kx cosec x cot x sec x tan x cosec 2 x

5 Contents 1. Algebra and Functions 2. Exponential and logarithm functions 3. Trigonometry 4. Differentiation 5. Numerical methods 6. Revision

6 C3WB: Algebra and Functions Algebra Notes BAT find equivalent expressions by dividing by the denominator BAT add, subtract, multiply and divide rational expressions BAT simplify rational expressions

7 C3WB: Algebra and Functions WB1 Express x 6 (x+3) (x 1) as a single fraction in its simplest form. WB2 Common factor in the denominator Express x 4 x 2 as a single fraction in its simplest form

8 C3WB: Algebra and Functions WB3 Exam Q Express Given that 4x (x 4) 2(x 4)(2x 5) f(x) = as a single fraction in its simplest form 4x 4 2(x 4) 36 2(x 4)(2x 5) 2 b) Show that f(x) = 12 (2x 5)

9 C3WB: Algebra and Functions WB4 Simplify: 12x 96 x(x 8)(7x+7) 3 (7x+7) 2

10 C3WB: Algebra and Functions Top heavy fractions Notes BAT Manipulate algebra fluently and efficiently BAT Simplify top heavy algebraic fractions

11 C3WB: Algebra and Functions WB5 Top heavy x 2x fraction Simplify x 1 2 WB6 Top heavy fraction (evil) 3 2 2x 14x 25x 3 C Ax B 2 Show that x 7x 12 x 4

12 C3WB: Algebra and Functions Functions Introduction notes BAT understand definition of a function as a one to one mapping BAT recognise odd and even functions BAT state the domain and range of functions

13 C3WB: Algebra and Functions WB7 Sketch the graph of y = f(x) where f(x) = x 2 + 1, x -2 What is the domain of f? What is the range of f?

14 C3WB: Algebra and Functions WB8 The function h(x) is defined by h(x) = 1 + 2, xεr x 0 x Solve these equations h(x) = 3 h(x) = 4 h(x) = 1 Explain why the equation h(x) = 2 has no solution Sketch a graph of h(x)

15 C3WB: Algebra and Functions WB9 Draw a sketch of the function defined by: 2x + 1, 3 < x < 4 f: x ( ) 13 x, 4 x < 10 and state the range of f(x)

16 C3WB: Algebra and Functions WB10 Draw a sketch of the function defined by g(t) = 3 t + 2, and state it s domain and range

17 C3WB: Algebra and Functions WB11 Draw a sketch of the function defined by f(x) = 6 x+1, x 1 State it s domain and range Is f(x) an odd function? Give a reason for your answer

18 C3WB: Algebra and Functions Inverse Functions notes BAT Find the inverse of functions BAT Find the domain and range of inverse functions BAT understand the inverse of a function graphically

19 C3WB: Algebra and Functions WB12 Find the Inverse function: 2 x f ( x) 5, 3 x WB13 Find the Inverse function: f(x) = 6 3x 2, x 2 3

20 C3WB: Algebra and Functions WB14 Find the Inverse function: f(x) = 3x+2 2x 5 WB15 Find the Inverse function: sin 2x 1 f ( x) 5, 3 x

21 C3WB: Algebra and Functions WB16 Inverse function graphically Sketch on the same axes f(x) and its inverse Sketch on the same axes g(x) and its inverse

22 C3WB: Algebra and Functions Composite Functions notes BAT Find composite functions BAT Find the domain and range of composite functions

23 C3WB: Algebra and Functions WB 17 A gas meter indicates the amount of gas in cubic feet used by a consumer. The number of therms of heat from x cubic feet of gas is given by the function f where f(x) = 1.034x, x > 0 A particular gas company s charge in for t therms is given by the function g where g(t) = t (i) (ii) Find the cost of using 100 cubic feet Find a single rule for working out the cost given the number of cubic feet of gas used WB 18 2 f : x x, x The functions f and g are defined by: g : x 2x 1 Find i) fg(x) ii) fg(0) iii) gf(x) iv) gf(1) v) ff(x) vi) ff(-2) vii) gg(x) viii) gg(-7)

24 C3WB: Algebra and Functions WB 19 The function f has domain [, ] and is defined by f(x) = 4e x The function g has domain [1, ] and is defined by g(x) = 3 ln x a) Explain why gf( 3) does not exist b) Find in its simplest form an expression for fg(x) stating its domain and range

25 C3WB: Algebra and Functions Modulus Functions and transformations notes BAT Understand and draw graphs of modulus functions BAT Find intersections of graphs, including modulus graphs BAT Solve inequalities three methods BAT apply transformations to graphs

26 C3WB: Algebra and Functions Review. Transformations of graphs (i) Shifts f ( x A) is a shift in the x direction (x, y) (x + A, y) +A Learn these and practice them f(x) = x 3 sketch f(3x) sketch f(x+2) sketch 4 f(x) (x, y + A) +A (x, y) f ( x) A is a shift in the y direction f(x) = 2x sketch f(3x) sketch f(x+2) sketch 4 f(x) f(x) = 2x 3 + 4x sketch f(2x) sketch f(x 1) sketch 3f(x) (ii) Stretches f(x) = Sin x sketch f(2x) sketch f(x + 90) sketch 3 f(x) f (Ax) is a stretch by scale factor 1 in the x direction A ( x, y) (x, y) f(x) = Cos x sketch f(x) sketch f(x 30) sketch 2 f(x) (x, Ay) f(x) = Tan x sketch f(2x) sketch f(x + 180) sketch f ( x) (x, y) Af (x) is a stretch by scale A factor A in the y direction (iii) Reflections f ( x) is a reflection of the graph in the y axis (-x, y) (x, y) (x, y) f (x) is a reflection of the graph in the x axis (x, y)

27 C3WB: Algebra and Functions WB 20 The functions f and g are defined by: f(x) = x + 3 g(x) = 4x + 6 Solve the inequality fg(x) > 12

28 C3WB: Algebra and Functions WB21 Solve the Inequality x 12 > 3x

29 C3WB: Algebra and Functions WB22 Given that f(x) = x and g(x) = x + 3 Sketch the graphs of the composite functions fg(x) and gf(x) Indicating clearly which is which

30 C3WB: Algebra and Functions WB23 Solve the inequality 3x 1 < 6x 1

31 C3WB: Algebra and Functions WB24 a) Solve the equation 9x 2 61 = 60 b) Hence, or otherwise, solve the inequality 9x

32 C3WB: Algebra and Functions WB25 f(x) = x 4 4x 240 a) Show that there is a root of f(x) = 0 in the interval [-4, -3] b) Find the coordinates of the turning point on graph of y = f(x) Given that f(x) = (x 4)(x 3 + ax 2 + bx + c) find the values of a,b and c d) Sketch the graph of y = f(x) e) Hence sketch the graph of y = f(x)

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34 C3 WB: Exponential & natural Logarithm Exponential and Natural logarithm Graphs - Notes BAT Understand and sketch transformations of exponential and natural logarithm graphs

35 C3 WB: Exponential & natural Logarithm WB1 Sketch the graph of: y = 10e -x WB2 Sketch the graph of: y = 3 + 4e 0.5x

36 C3 WB: Exponential & natural Logarithm WB3 Sketch the graph of: y = 3 + ln(2x) WB4 Sketch the graph of: y = ln(3 - x)

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38 C3 WB: Exponential & natural Logarithm Exp and ln - Solving equations - Notes BAT solve equations and manipulate algebra using rules of indices and logarithms

39 C3 WB: Exponential & natural Logarithm WB5 Solve i) e x = 3 ii) e x+2 = 7 iii) ln x = 4 iv) ln(3x 2) = 3 WB6 Make x the subject of: i) lny lnx = 4t ii) y = 5e 2x

40 C3 WB: Exponential & natural Logarithm WB7 Solve i) lnx 16 lnx ii) e 2x 8e x + 12 = 0 WB8 Exam Q Find the exact solutions to these equations a) ln(3x 8) = 2 b) 3 x e 8x+3 = 18

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42 C3 WB: Exponential & natural Logarithm Growth Functions - Notes BAT Solve real life problems involving growth functions of the form y = Ae bx+c

43 C3 WB: Exponential & natural Logarithm WB9 The Price of a used car is given by the formula: P = 1600e t 10 a) Calculate the value of the car when it is new b) Calculate the value after 5 years c) What is the implied value of the car in the long run (ie what value does it tend towards?) d) Sketch the Graph of P against t

44 C3 WB: Exponential & natural Logarithm WB10 The number of elephants in a herd can be represented by the equation: N = e 1 40 Where n is the number of elephants and t is the time in years after a) Calculate the number of elephants in the herd in 2003 b) Calculate the number of elephants in the herd in 2007 c) Calculate the year when the population will first exceed 100 elephants d) What is the implied maximum number in the herd?

45 C3WB: Trigonometry Trig Reciprocal Functions - Notes BAT understand sec, cosec and cot and their graphs BAT solve equations involving sec, cosec and cot

46 C3WB: Trigonometry WB1 a) Will cosec 200 be positive or negative? b) Find the value of sec 280 to 2 decimal places WB2 Solve these equations: ( 0 < 2π ) a) sec θ = 10 b) 5 cosec θ = 25 c) cot 3θ + 7 = 6. 5

47 C3WB: Trigonometry Graphs of Reciprocal functions - sec x, cosec x, cot x

48 C3WB: Trigonometry WB3 Sketch, in the interval 0 θ 360, the graph of y = 1 + sec 2θ

49 C3WB: Trigonometry Trig Identities and Equations - Notes BAT prove identities using reciprocal functions BAT know and use new trig identities

50 C3WB: Trigonometry WB4 Simplify sinθ cotθ sec θ WB5 Simplify sin θ cos θ (sec θ + cosec θ)

51 C3WB: Trigonometry WB6 Show that cot θ cosec θ sec 2 θ+ cosec 2 θ cos3 θ WB7 Prove that cosec 4 θ cot 4 θ 1+cos2 θ 1 cos 2 θ

52 C3WB: Trigonometry WB8 Prove that sec 2 θ cos 2 θ sin 2 θ (1 + sec 2 θ) WB9 Solve the equation 4cosec θ 9 = cotθ in the interval 0 θ 360

53 C3WB: Trigonometry WB10 Solve these equations: a) 4cos 2 θ + 5 sin θ = 3 in the interval π θ π b) 2tan 2 θ + sec θ = 1 in the interval π θ π

54 C3WB: Trigonometry Inverse trig questions - Notes BAT use and understand applications and graphs of the inverse Trig functions

55 C3WB: Trigonometry Graphs of arcsin x, arcos x, arctan x

56 C3WB: Trigonometry WB11 a) Work out in degrees, the value of arcsin ( 2 2 ) b) Work out in radians, the value of cos [arcsin ( 1)] WB12 a) Work out, in radians, the value of arcsin(0.5) b) Work out, in radians, the value of arctan( 3)

57 C3WB: Trigonometry WB13 a) Work out, in radians, the value of arcsin(0.5) b) Work out, in radians, the value of arctan( 3)

58 C3WB: Trigonometry Addition Formulae - Notes BAT understand where the addition formula come from and know the Identities for sin 2x, cos 2x and tan 2x BAT use the addition formulae to solve show that problems BAT use the addition formulae to solve equations

59 C3WB: Trigonometry WB14 Show that sin 15 = WB15 Given that sin A = in the range 180 < A < 270 and cos B = where B is Obtuse. Find the value of tan(a + B)

60 C3WB: Trigonometry WB16 Rewrite the following as a single Trigonometric function: 2 sin θ cos θ cos θ 2 2 WB17 Show that 1 + cos 4θ can be written as 2cos 2 2θ

61 C3WB: Trigonometry WB18 Given that cos x = 3 4 in the range [180, 360] find the exact value of 2 sin 2x WB19 Given that x = 3 sin θ and y = 3 4 cos 2θ Eliminate and express y in terms of x

62 C3WB: Trigonometry WB20 Solve the equation 3 cos 2x cos x + 2 = 0 in the range 0 x 360

63 C3WB: Trigonometry R cos ( x + alpha) - Notes BAT write expressions of the form acosθ + bsinθ, where a and b are constants, as a sine or cosine function only BAT solve equations of the form acosθ + bsinθ, or find their max/min value

64 C3WB: Trigonometry WB21 Show that 3 sin x + 4 cos x can be expressed in the form where R > 0, 0 < α < 90 R sin(x+ ) WB22 a) Show that you can express sin x 3 cos x in the form R sin(x+ ) where R > 0, 0 < α < 90 b) Hence, sketch the graph of y = sin x 3 cos x

65 C3WB: Trigonometry WB23 a) Express 2 cos θ + 5 sin θ can be expressed in the form R cos(θ ) where R > 0, 0 < α < 90 b) Hence, solve 2 cos θ + 5 sin θ = 3 in the range 0 < θ < 360 WB24 a) Express 12 cos θ + 5 sin θ in the form R cos(θ ) where R > 0, 0 < α < 90 b) Hence, find the maximum value of 12 cos θ + 5 sin θ and the smallest positive value of at which it arises

66 C3WB: Trigonometry WB23 a) Express 2 cos θ + 5 sin θ can be expressed in the form R cos(θ ) where R > 0, 0 < α < 90 b) Hence, solve 2 cos θ + 5 sin θ = 3 in the range 0 < θ < 360 WB24 a) Express 12 cos θ + 5 sin θ in the form R cos(θ ) where R > 0, 0 < α < 90 b) Hence, find the maximum value of 12 cos θ + 5 sin θ and the smallest positive value of at which it arises

67 C3WB: Differentiation Chain rule - Notes BAT differentiate a function of a function, using the chain rule BAT apply dy dx = 1 dx dy to solve a problem

68 C3WB: Differentiation WB1 Differentiate y = (3x 4 + x) 5 WB2 Given that y = 5x Find the value of f (x) at (4,9)

69 C3WB: Differentiation WB3 Given that y = (x 2 7x) 4 Calculate dy dx using the chain rule WB4 Given that y = 1 6x 3 Calculate dy dx using the chain rule

70 C3WB: Differentiation WB5 Calculate dy for the following equation dx y3 + y = x and find the gradient at the point (2,1)

71 C3WB: Differentiation Product rule - Notes BAT differentiate a product of two functions

72 C3WB: Differentiation WB6 differentiate y = x 2 (3x 9) WB7 Differentiate y = (x 2 + 3)(4x + 1)

73 C3WB: Differentiation WB8 Given that: f(x) = x(2x + 1) 3, find f (x) WB9 Given that: f(x) = x 2 3x 1, find f (x)

74 C3WB: Differentiation Chain, Product and Quotient rules - Notes BAT differentiate a quotient of two functions BAT differentiate complex functions using the chain rule, product rule and quotient rules efficiently

75 C3WB: Differentiation WB10 Given that y = x 2x+5 Calculate dy dx WB11 Given that y = 3x (x+4) 2 Calculate dy dx

76 C3WB: Differentiation Exponential and natural logarithm functions- Notes BAT differentiate the exponential and natural logarithm functions BAT apply the chain. Product and Quotient rules BAT solve geometry problems using what we have learned so far

77 C3WB: Differentiation WB12 Chain rule a) Find the gradient of y = 5 ln x + e x 4 ln 2x when x = 4 b) Find the gradient of y = 3 ln(3 x 2 + 6x) when x = 1 2

78 C3WB: Differentiation WB13 Product rule Differentiate the following: a) y = x e x b) y = x 3 ln x

79 C3WB: Differentiation WB14 Quotient rule Differentiate the following: a) y = e3x 1 x b) y = ln5x x

80 C3WB: Differentiation WB15 Geometry Find the equation of the tangents to the curve y = 3e 5x at the point where x = 0 WB16 Geometry Find the equation of the normal to the curve y = 1 4 ln x at the point where x = 1

81 C3WB: Differentiation Trig functions - Notes BAT differentiate trig functions using what we have learned so far BAT differentiate mixed trig, exponential, logarithm and other functions

82 C3WB: Differentiation WB17 Differentiate: a) y = sin 3x b) y = sin 2 3 x c) y = sin2 x WB18 Differentiate: a) y = cos(4x 3) b) y = cos 3 x c) y = 3 cos x

83 C3WB: Differentiation WB19 Find the derivative of y = tan x WB20 Find the derivative of y = tan 4x

84 C3WB: Differentiation WB21 Find the derivative of y = x tan 2x WB22 Find the derivative of y = ln x sin x

85 C3WB: Differentiation WB23 Find the derivative of y = e x sin x

86 C3WB: Differentiation Reciprocal Trig functions - Notes BAT differentiate reciprocal trig functions using what we have learned so far BAT differentiate mixed trig, exponential, logarithm and other functions

87 C3WB: Differentiation WB24 Find the derivative of y = cot x Find the derivative of y = sec x Find the derivative of y = cosec x

88 C3WB: Differentiation WB25 Find the derivative of y = sec 3x WB26 Find the derivative of y = cosec 2x x 2

89 C3WB: Numerical methods Notes BAT Show that a root of an equation lies between two values BAT Approximate solutions using an iterative formula BAT check solutions with an appropriate method

90 C3WB: Numerical methods WB1 Show that the equation f(x) = ln x 1 Has a root between 2 and 3 WB2 Find a root of the equation x 3 12x + 12 = 0

91 C3WB: Numerical methods WB3 Find a root of the equation e x x = 0 WB4 α is the positive root of the equation x 3 1 x 1 = 0 Show that to 3 decimal places α = 1.221

92 C3WB: Numerical methods WB5 Exam Q The curve y = 5 x intersects the line y = x + 2 at the point where x = α a) Show that α lies between 0 and 1 b) Show that the equation 5 x = x + 2 can be rearranged into x = ln(x+2) c) Use the iteration x n+1 = ln(x n+2) ln 5 with x 1 = 0 to find x 3 to two sf ln 5 WB6 Exam Q f(x) = x 3 + 7x 2 5 a) Show that the equation f(x) = 0 can be rewritten as x = 5 b) Starting with x 1 = 0.9, use the iteration x n+1 = 5 7 x n 7 x to calculate the values of x 2, x 3, and x 4 giving your answers to 4 decimal places c) Show that x = is a root of f(x) = 0 correct to 3 decimal places