A2 MATHEMATICS HOMEWORK C3

Transcription

1 Name Teacher A2 MATHEMATICS HOMEWORK C3 Mathematics Department September 2016 Version 1

2 Contents Contents... 2 Introduction... 3 Week 1 Trigonometric Equations Week 2 Trigonometric Equations Week 3 The Exponential Functions... 8 Week 4 Differentiation chain, product and Quotient Week 5 Numerical Methods Week 6 The Modulus Function Week 7 Applications of Trig C3 Revision Test June Page

3 Introduction Aim to complete all the questions. If you find the work difficult then get help [lunchtime workshops in room 216, online, friends, teacher etc]. To learn effectively you should check your work carefully and mark answers? If you have questions or comments, please write these on your homework. Your teacher will then review and mark your mathematics. If you spot an error in this pack please let your teacher know so we can make changes for the next edition! Homework Tasks These cover the main topics in C3. Your teacher may set homework from this or other tasks. Week HW Topic HW1 Trigonometric Equations 1 Date completed Mark HW2 Trigonometric Equations 2 HW3 The Exponential Function HW4 Differentiation Chain, Product and Quotient HW5 Numerical Methods HW6 The Modulus Function HW7 Applications of Trigonometry HW8 Revision Test for Mock Exam 3 Page

4 Week 1 Trigonometric Equations 1 Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words principal value, secondary value, repeat period, radians, interval Exercise A 1. Read pages and make sure you understand. Try to link this with what you are learning about mappings and functions. Learn the following 2. Starting with, prove the following identities a) b) 3. Sketch the graphs of a) b) c) 4. Solve each of the following equations giving your answers in the specified interval. a) b) c) 5. Solve each of the following equations giving your answers in the range specified. a) b) ( ) Exercise B - Exam questions 1. [C3 Jun 06 Q6] a) Using show that b) Hence, or otherwise, prove that c) Solve, for (6) 2. [C3 Jan 07 Q8] Prove that Exercise C - Challenge 1. Solve 4 Page

5 2. [C3 Jan 08 Q7] Given that and a) Express in terms of b) Hence evaluate. give your answer in terms of Answers Exercise A 3) Sec x cot x 4. a) (- ),,,, b) 0.201, 2.94 c) 1.11, 4.25, 5. a) 63.4, 135, 243.4, 315 b) Exercise B 1 Exercise C 1 2 a) ( ), b) 5 Page

6 Week 2 Trigonometric Equations 2 Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words compound angle formula, double angle formula Exercise A 1. Read pages and make sure you understand [key points on p79,80] Learn the identities. Sketch a graph of Sketch a graph of and show that and show that 2. Use the compound angle formula to expand simplifying your answers without a calculator: a) b) c) 3. Solve the following; a) in the range b) ( ) 4. Start with the compound angle formulae and prove the following: a) [eg start with ] b) c) d) 6 Page

7 Exercise B - Exam questions 1. [C3 Jun 10 Q1]. a) Show that b) Hence find, for all the solutions of Give your answers to 1 decimal place. 2. [C3 Jan 2011 Q3]. Find all the solutions of in the interval (6) 3. [C3 Jan 2012 Q8] a) Starting from the formulae for and prove that (4) b) Deduce that ( ) c) Hence, or otherwise, solve, for Give your answers as multiples of. (6) Exercise C - Challenge 1. Prove 2 Prove a similar identity for 3. What about? Or? Is there a pattern or rule you can define? Answers: Exercise A 2. a) b) c) 3. a) 165 b) 2.64 radians (This is a hard question expand both sides then use Exercise B Exam questions Page

8 Week 3 The Exponential Function Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words: Exponentials, Natural logarithms, exponential growth/decay, rates, modelling Exercise A 1. Sketch the following graphs on different axis a) and (on the same axis) b) c) d) 2. Find the exact solutions of a) b) c) d) e) =2 3. Find the exact solutions for for the following equations a) b) c) d) e) f) 4. [Created by an AS student] A maths class is learning Calculus for the first time. After minutes, their stress levels, micrograms of cortisol per deciliter, is given by: a) What are their stress levels when they enter the class? b) After 10 minutes their stress level is 12 micrograms per deciliter. Show that to three significant figures. 8 Page

9 Exercise B Exam question 1. [C3 June 2009 Q3] Rabbits were introduced onto an island. The number of rabbits,, years after they were introduced is modelled by the equation Extension a) Write down the number of rabbits that were introduced to the island. b) Find the number of years it would take for the number of rabbits to first exceed c) Find d) Find when 1. Create a model for exponential growth/decay for the following examples: (a) A cup of tea cooling down from 80C to room temperature. (b) Population of lizards on an island growing from 10 to 500 over 3 years. (1) (Total 8 marks) Do your models seem accurate for all situations? If not how could you alter it? 2. Try C3 Jan 2010 Q9 3. What is the maximum value of? Explain why. Answers 2.a) b) c) d) e) 3.a) b) c) d) e) or f) ( ) or 4.a) Exam 1.a) b) c) d) Page

10 Week 4 Differentiation Chain, Product and Quotient Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words: derivative, function, gradient function, natural log, exponential, substitution, trigonometric function, chain rule Read pages ; and make sure you understand. Try to link this with what you learnt in C1. Exercise A - Chain Rule 1. Differentiate the following: a) b) c) d) e) f) g) h) i) 2. Use the chain rule to differentiate the following: a) b) ( ) c) d) e) f) g) h) i) 3. Find the equation of the tangent to the curve at the point Exam questions 4. [C3 June 05] a) Differentiate to find., The curve with equation has a turning point at. The -coordinate of is. b) Show that. 5. [C3 June 06] Differentiate, with respect to, a) b) 6. [C3 Jan 06] The point lies on the curve with equation. The -coordinate of is. Find an equation of the normal to the curve at the point in the form, where and are constants. (5) 7. [C3 June 06] A heated metal ball is dropped into a liquid. As the ball cools, its temperature, minutes after it enters the liquid, is given by,. a) Find the temperature of the ball as it enters the liquid. C, 10 Page

11 b) Find the value of for which, giving your answer to 3 significant figures. (1) (4) c) Find the rate at which the temperature of the ball is decreasing at the instant when. Give your answer in C per minute to 3 significant figures. d) From the equation for temperature in terms of, given above, explain why the temperature of the ball can never fall to 25 C. (1) Answers 1 a) b) c) d) e) f) g) h) i) 2 a) b) ( ) ( ) c) d) e) f) a) g) h) i) 5. a) b) a) 425 ºC b) c) ± 1.64 d) as Key words: derivative, function, gradient function, natural log, exponential, chain rule, product rule, quotient, substitution Read pages ; and make sure you understand. Try to help this consolidate what you ve already learnt about differentiation Exercise B Product & Quotient Rule 1. Differentiate the following using the product rule: a) b) c) d) e) f) 2. Differentiate the following using the quotient rule: a) b) c) d) e) f) Exam questions 3. [C3 Jun 05] a) Differentiate with respect to i) ii) ( ) b) Given that, show that (6) 11 Page

12 4. [C3 Jan 06] a) Differentiate with respect to i) (4) ii) (4) b) Given that find in terms of. (5) Answers 1a), b) c) d) e) f) 2a) b) c) d) e) f) 3ai) aii) 4ai) aii) b) 12 Page

13 Week 5 Numerical Methods Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words: roots, interval, algebraic, accuracy, continuous function, domain, iteration, converge, diverge Exercise A 1. Sketch the following curves and say for which ones the change on sign method will work. If it doesn t work can you give a reason? a) ( ), y Figure 1 b), c), x 2. Figure 1 shows a sketch of a) How many roots does the equation have? b) Find an interval of unit length (e.g ) containing each root. c) Show that one root is 2.54 to 2 d.p. and find the other root. 3. What do you need to be true for the iteration method to work? 4. Find all the roots of to 1 d.p. [hint; it might be useful to make a sketch using and note all the roots are in the range [-3,3]] Exercise B Exam questions 1. [C3 Jan 07] f(x) = x 4 4x 8. (a) Show that there is a root of f(x) = 0 in the interval [ 2, 1]. (b) Find the coordinates of the turning point on the graph of y = f(x). (c) Given that f(x) = (x 2)(x 3 + ax 2 + bx + c), find the values of the constants a, b and c. (d) Sketch the graph of y = f(x). (e) Hence sketch the graph of y = f(x). (1) 2. [C3 Jun 08 Q7] a) Show that has a root,, between and. b) Show that the equation 0 can be written as c) Starting with, use the iteration ( ) to calculate the values of, and, giving your answers to 4 decimal places. d) By choosing a suitable interval, show that is correct to 3 d.p. 13 Page

14 3. [C3Jan 06 Q5] a) Show that the equation can be written as The equation has a root between and. b) Use the iteration formula with, to find, to 2 decimal places, the value of and. The only real root of is. c) By choosing a suitable interval, prove that, to 3 decimal places. 4. [C3 Jun 05 Q4] Consider The iterative formula is used to find an approximate value for. a) Calculate the values of and, giving your answers to 4 decimal places. b) By considering the change of sign of in a suitable interval, prove that correct to 4 decimal places. Extension Find the solutions to the simultaneous equations and. ANSWERS Exercise A 2.a) 2b),, c), and continuous 3. successive iteration must converge, and must converge to the root you are looking for. 4. Exercise B 1. a). Change of sign (and continuity) root in interval, b) 1, 11, c) a 2, b 4, c 4 d) e) see class / workshop/ exam sol. 2. a) < 0 > 0 Change of sign (and continuity) b), c) Choosing the interval d) Due to change of sign (and continuity) 3. b). 4. c) c) Choosing (1.3915, ),, Change of sign (and continuity) d) Using 14 Page

15 Week 6 The Modulus Function Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words Modulus Function, Absolute value Transformations Exercise A 1. Calculate the following: a) b) c) d) e) f) 2. True or false; which of the following statements are correct. a) b) c) d) e) 3. Sketch the following graphs: a) b) c) d) e) f) g) 4. Solving the following equations (sketching the graphs is very important) a) b) c) d) e) Exercise B Exam Questions 1. [C3 Jun 05] Figure 1 y 1 O b 3 x (1, a) Figure 1 shows part of the graph of y = f(x), x R. The graph consists of two line segments that meet at the point (1, a), a < 0. One line meets the x-axis at (3, 0). The other line meets the x-axis at ( 1, 0) and the y-axis at (0, b), b < 0. In separate diagrams, sketch the graph with equation a) y = f(x + 1), b) y = f( x ). Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that f(x) = x 1 2, find c) the value of a and the value of b, 15 Page

16 d) the value of for which. (4) 2. [C3 Jan 06] Figure 1 y M (2, 4) 5 O 5 x Figure 1 shows the graph of x. The point M (2, 4) is the maximum turning point of the graph. Sketch, on separate diagrams, the graphs of a), b) y = f(x), c) y = f( x ). Show on each graph the coordinates of any maximum turning points. 3. [C3 Jun 07] Find the exact values of for which. Extension 1) Go to (username: cityisli, password: ask teacher) for further questions. Using the modulus function and by limiting the domain of different functions can you create a set of functions which combine to look like: a) A house b) The Batman symbol Answers Exercise A 1a) 3 b) 5 c) 3 d) 1 e) -3 f) 3 2a) T b) T c) T d) F e) T 3 Check using graph plotter or teacher will check. 4a) b) c) only sketch the graph d) e) For d) & e) you can use the squaring method. See Modulus equations on examsolutions Exercise B Exam questions - see 1. c) d) Page

17 Week 7 Applications of Trig Complete on a separate sheet of paper. Show clear working. Mark your answers. Key words compound angle formula, double angle formula Read pages and have a look at the examples. Exercise A 1. Express each of the following in the form shown, where and a) b) c) d) 2. Sketch the graph of and mark on this the maximum and minimum points on the graph. Use this graph to help solve the equation 3. For the following functions find the max value of the function and the first positive value of for which it occurs a) b) c) Exercise B Exam questions 1. [C3 Jun 07 Q6] a) Express in the form where and. (4) b) Hence find the greatest value. c) Solve, for, the equation, giving your answers to 3 decimal places. (5) 17 Page

18 2. [C3 Jun 2008 Q2] Given that where and a) find the value of R and the value of α to 3 decimal places. (4) b) Hence solve the equation for (5) c) i) Write down the maximum value of. (1) ii) Find the smallest positive value of for which this maximum value occurs. 3. [C3 Jan 09 Q8] a) Express in the form, where and are constants, and (4) b) Hence find the maximum value of and the smallest positive value of θ for which this maximum occurs. The temperature,, of a warehouse is modelled using the equation where t is the time in hours from midday and. c) Calculate the minimum temperature of the warehouse as given by this model. d) Find the value of t when this minimum temperature occurs. Extension For find the max and minimum value of this function Also see page 77 in text book for more questions. Answers: 1a sin b 5sin c cos d cos( 2 3a f(x) max = 13, b f(x) max = 3, c y max = 3, Jun 07, 6a, = (Allow 33.7 ), b 169, c x = or x = (awrt)both (radians only) Jun 08, 2a R =13, b 2.3, awrt or c R max =13 at max Jun 09, 8a R=5,. b max value = 5 and this occurs at. c min temp is 5 D t = 15.5 Extension 5 max = 2, min = 1 / 3 18 Page

19 Revision for C3 test - June 2010 Check numerical answers and try to work out where you ve made errors. Working under timed conditions is the most effective way to prepare for examinations 1. a) Show that sin 2θ 1 cos 2θ = tan θ. b) Hence find, for 180 θ < 180, all the solutions of 2sin 2θ 1 cos 2θ Give your answers to 1 decimal place. 2. A curve C has equation y = 3 ( 5 3x) 2 = 1., x 3 5. The point P on C has x-coordinate 2. Find an equation of the normal to C at P in the form ax + by + c = 0, where a, b and c are integers. (7) 3. f(x) = 4 cosec x 4x +1, where x is in radians. a) Show that there is a root α of f(x) = 0 in the interval [1.2, 1.3]. b) Show that the equation f(x) = 0 can be written in the form 1 1 x = + sin x 4 c) Use the iterative formula 1 1 x n+ 1 = +, x0 = 1.25, sin x n 4 to calculate the values of x 1, x 2 and x 3, giving your answers to 4 decimal places. d) By considering the change of sign of f(x) in a suitable interval, verify that α = correct to 3 decimal places. 4. The function f is defined by f : x 2x 5, x R. a) Sketch the graph with equation y = f(x), showing the coordinates of the points where the graph cuts or meets the axes. b) Solve f(x) =15 + x. The function g is defined by c) Find fg. d) Find the range of g. g : x x 2 4x + 1, x R, 0 x Page

20 5. Figure 1 Figure 1 shows a sketch of the curve C with the equation y = (2x 2 5x + 2)e x. a) Find the coordinates of the point where C crosses the y-axis. b) Show that C crosses the x-axis at x = 2 and find the x-coordinate of the other point where C crosses the x-axis. d y c) Find. dx d) Hence find the exact coordinates of the turning points of C. (5) (1) 7. a) Express 2 sin θ 1.5 cos θ in the form R sin (θ α), where R > 0 and 0 < α < 2. Give the value of α to 4 decimal places. b) (i) Find the maximum value of 2 sin θ 1.5 cos θ. (ii) Find the value of θ, for 0 θ < π, at which this maximum occurs. Tom models the height of sea water, H metres, on a particular day by the equation 4 t H = sin 25 where t hours is the number of hours after midday. 4 t 1.5 cos, 0 t <12, 25 c) Calculate the maximum value of H predicted by this model and the value of t, to 2 decimal places, when this maximum occurs. d) Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres. (6) 20 Page

21 6. Figure 2 Figure 2 shows a sketch of the curve with the equation y = f(x), x R. The curve has a turning point at A(3, 4) and also passes through the point (0, 5). a) Write down the coordinates of the point to which A is transformed on the curve with equation i) y = f(x), ii) y = 2f( 2 1 x). b) Sketch the curve with equation y = f( x ). (4) On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the y-axis. The curve with equation y = f(x) is a translation of the curve with equation y = x 2. c) Find f(x). d) Explain why the function f does not have an inverse. (1) 8. a) Simplify fully Given that b) find x in terms of e. 2 2x 9x 5. 2 x 2x 15 ln (2x 2 + 9x 5) = 1 + ln (x 2 + 2x 15), x 5, (4) TOTAL FOR PAPER: 75 MARKS END 21 Page

22 Answers 1. (b) 26.6, x 18y + 52 = 0 3. (c) x 1 = , x 2 = , x 3 = (b) x = 3 (c) fg = 11 (d) 3 g(x) 6 5. (a) (0, 2) (b) x = 2 1 (c) (4x 5)e x (2x 2 5x + 2)e x (d) (1, e 1 ), 7, e 6. (a) (i) (3, 4) (ii) (6, 8) (c) f(x) = (x 3) (a) (b) (i) 2.5 (ii) 2.21 (c) 4.41 (d) 14:06, 18:43 8. (a) (2x 1) ( x 3) (b) x = 3e 1 e 2 22 Page