Angle Measure 1. Use the relationship π rad = 180 to express the following angle measures in radian measure. a) 180 b) 135 c) 270 d) 258

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1 Chapter 4 Prerequisite Skills BLM Angle Measure 1. Use the relationship π rad = 180 to express the following angle measures in radian measure. a) 180 b) 135 c) 70 d) 58. Use the relationship 1 =! rad to 180 express the following radian measures in degree measure. a) π b) 3! c)! 7 4 d) 5 Trigonometric Functions and Radian Measure 3. Graph each function over the interval π x π. a) y = 3sin x b) y = cos x Determine the amplitude and period of the function shown. Differentiation Rules 6. Differentiate each function with respect to the variable indicated. a) y = 3x 5x + 3 b) y = x 3 + 5x c) y = (x + 5) 6 d) f(t) = t(4t 5) 3 e) d(x) = (4x 1) (3x + 4) 5 7. Copy and complete the statement. If h(x) = f(g(x)), then... Applications of Derivatives 8. Find the slope of the line tangent to the curve y = x 3 3x + 4 at x =. 9. Find the equation of the tangent line to the curve y = (x + 5) 4 at x = Determine the point where the tangent line to y = 3x + 4x is horizontal. 11. Determine the coordinates of all local maxima and minima for the function y = x 3 + x 8x + and state whether each is a local maximum or minimum. Trigonometric Identities 1. Use the sum or difference identity to find another expression for: a) sin x b) cos x 5. Use special triangles to express the following as exact values. a) cos! % 3 b) sin! % c) sin! % 3 + cos! % 6 d) sin! % 4 + cos! % % Prove each identity. a) sec θ 1 = tan θ b) cos(x + y) cos(x y) = cos x sin y 14. a) Use a sum or difference identity to prove that sin θ = cos! +! %. b) Sketch a graph of y = cos θ. Then use transformations of this graph to sketch a graph illustrating this identity. Calculus and Vectors 1: Teacher s Resource BLM 4-1 Prerequisite Skills Copyright 008 McGraw-Hill Ryerson Limited

2 4.1 Instantaneous Rates of Change BLM of Sinusoidal Functions 1. Examine the graph. a) Identify all points where the slope of y = sin x is: i) zero ii) a local maximum iii) a local minimum b) Over which intervals is the curve i) concave up? ii) concave down? c) What are the maximum and minimum values of the slope? d) Sketch a graph of the instantaneous rate of change of y = sin x as a function of x. b) Identify the intervals for which the instantaneous rate of change is: i) increasing ii) decreasing c) Identify the intervals for which the graph is: i) concave up ii) concave down 3. a) Sketch a graph of y = cos x. b) Sketch a graph of the instantaneous rate of change of y = cos x as a function of x. c) Sketch a graph of y = 3cos x. d) Sketch a graph of the instantaneous rate of change of y = 3cos x as a function of x. e) How does your graph in part d) compare to your graph in part b)? 4. Use Technology a) Sketch a graph of y = tan x. b) Determine the instantaneous rate of change at several points. c) Sketch a graph of the instantaneous rate of change of y = tan x as a function of x. Identify the key features of the graph.. The graph of y = cos x is shown. a) Identify the intervals for which the function is: i) increasing ii) decreasing Calculus and Vectors 1: Teacher s Resource Copyright 008 McGraw-Hill Ryerson Limited BLM 4-3 Section 4.1 Instantaneous Rates of Change of Sinusoidal Functions

3 4. Derivatives of the Sine and Cosine Functions BLM Find the derivative with respect to x for each function. a) y = cos x b) y = 6sin x c) y = 5 cos x d) y = 4sin x. Differentiate with respect to x. a) f(x) = 3cos x b) f(x) = sin x cos x c) f(x) = 5x 3 + cos x d) f(x) = 3x + 4sin x 1 cos x e) f(x) = (π + 1)sin x f) f(x) =! cos x 3sin x Differentiate with respect to θ. 7. Find the equation of the tangent line to the function y = cos x + that passes through the! point 3, 3 %. 8. Find the equation of the tangent to y = cos x at x =!. 9. Find all points, in the domain π θ π, on the curve y = 5sin θ such that the slope of the tangent line is Describe a method for determining the nth derivative of y = sin x. 11. Examine the function. a) f(θ) = 4sin θ 3 cos θ b) f(θ) = πcos θ + 4sin θ a) Sketch the graph of y = 3cos x for π x π. b) Find the slope of the tangent line to y = 3cos x at x =! 4. c) Find another point on the graph that will have the same slope as in part b). Check your answer by substituting it into the derivative. 5. Find the slope of the graph of y = sin θ + 1 at θ =! a) Sketch the graph y = sin x 3. b) Find the equation of the tangent line to the function y = sin x 3 when x =! 6. c) Sketch the tangent line on your graph. a) Write the sinusoidal function for the graph shown. b) Find the derivative of the function and sketch it. c) Where does the slope of the tangent have a maximum value? 1. Determine for what values of x, where π x π, is the tangent line to y = sin x + cos x horizontal. 13. Use Technology Use graphing technology to graph y = sin x and y = sin x + 3. Use the graphs to explain why the derivatives of y = sin x and y = sin x + 3 are the same. Calculus and Vectors 1: Teacher s Resource BLM 4-4 Section 4. Derivatives of the Sine and Cosine Functions Copyright 008 McGraw-Hill Ryerson Limited

4 4.3 Differentiation Rules for Sinusoidal Functions BLM Find dy for each function. a) y = cos 3x b) y = sin(x + π) c) y = sin(x + 3) d) y = cos( πx + 3). Differentiate with respect to θ. a) y = 4cos θ b) y = 3sin(θ π) c) y =! cos! % ( ) d) y = πsin 4θ 3. Find the derivative with respect to x. a) y = cos x b) f(x) = sin 4 x 1 c) y = sin x d) f(x) = sin 3 x cos x 4. Differentiate with respect to t. a) y = sin (t 3) b) y = cos (4t 1) c) f(t) = sin 3t + cos 4t d) f(t) = sin(cos t) 5. Find dy for each of the following: a) y = xsin 3x b) y = x cos x c) y = 4xsin(x ) d) y = πxsin (x + π) 6. For the function y = (sin x) 4, determine the a) first derivative b) second derivative c) third derivative 7. Find the slope of the tangent line to y = sin x cos x at x = π. 8. a) Verify that the point (π, 0) is on the function y = x sin x. b) Find the slope of the tangent line to y = x sin x at this point. c) Find the equation of the tangent line at this point. 9. Find the equation of the line tangent to y = sin 3x at x =! a) Determine the equation of the tangent line to y = xsin x at x =!. b) The normal to a graph is a line that passes through a point and is perpendicular to the tangent line at that point. Find the equation of the normal to y = xsin x at x =!. 11. Use differentiation rules and trigonometric identities to show that if y = sin x, then dy = sin x. 1. Using trigonometric identities, y = sin x and y = sin x cos x should have the same derivatives. Prove that this is true. 13. a) Write y = cot x as the product of two functions you can differentiate. b) Show that dy = sec x Calculus and Vectors 1: Teacher s Resource BLM 4-6 Section 4.3 Differentiation Rules for Sinusoidal Functions Copyright 008 McGraw-Hill Ryerson Limited

5 4.4 Applications of Sinusoidal Functions BLM and Their Derivatives 1. An AC-DC coupled circuit produces a voltage described by the function V(t) = 5sin t + 8, where V is the voltage, in volts and t is the time, in seconds. a) Determine the first derivative of the voltage function. b) Find the maximum and minimum voltages. At what time do these values occur?. A simple pendulum has a length of 0 cm and a maximum horizontal displacement of 8 cm. a) Determine a function that gives the horizontal position of the bob as a function of time. b) Determine a function that gives the velocity of the bob as a function of time. c) Find the maximum velocity of the bob. d) Repeat parts a) to c) for a pendulum with a length of 40 cm and a maximum horizontal displacement of 8 cm. e) How did the maximum velocity of the 40 cm pendulum compare with the maximum velocity of the 0 cm pendulum? 3. Suppose the number of hours of daylight in a particular area is given by the formula h(n) = 3.8sin! ( 365 n 80.5 % )( +11 where n is the nth day of the year. a) When is the number of hours of daylight increasing the fastest? b) When is the number of hours of daylight decreasing the fastest? 4. A geometric figure is bouncing on a spring that is hanging from the ceiling. The figure s distance from the ceiling, d, varies sinusoidally with time, t. The figure starts at its closest point to the ceiling, d = 0.5 m and makes a complete cycle every second. The figure s farthest point from the ceiling is 1.5 m. a) Sketch a graph that shows the figure s height from the ceiling over time for 1 cycle. b) Determine the function for the height, h, in terms of time, t. c) How far from the ceiling is the figure when t =? d) Determine the function for the velocity of the figure in terms of time. e) How fast is the figure moving when t =? t = 3? t = 1.3? f) What is the fastest speed of the figure? g) When is its velocity equal to zero? 5. The graph shows a rider s height, in metres, on a Ferris wheel in terms of time, in seconds. a) What are the maximum and minimum heights for the rider? b) Write an equation for the rider s height, h(t), in terms of time. c) Write an equation for h!(t). d) At what rate is the rider s height changing when t = 10? t = 0? e) Is the rider s height increasing or decreasing when t = 45? How quickly is it changing? f) What is the maximum rate at which the rider s height is changing? Calculus and Vectors 1: Teacher s Resource Copyright 008 McGraw-Hill Ryerson Limited BLM 4-7 Section 4.4 Applications of Sinusoidal Functions and Their Derivatives

6 Chapter 4 Review BLM Instantaneous Rates of Change of Sinusoidal Functions 1. The graph of y = sin x! % 4 is shown. 4.3 Differentiation Rules for Sinusoidal Functions 7. Differentiate. a) y = sin x 3 b) y = sin 3 x c) f(θ) = 3cos θ + 4sin(θ + π) d) f(t) = 3cos(sin t) 8. Determine the derivative of each function with respect to the variable indicated. a) y = (sin 3x)(cos 4x) b) f(θ) = sin θ cos θ c) f(t) = t 3 sin(4t π) d) y = sin (cos θ) 9. a) Determine the y-coordinate that a) Identify the points over the interval π x π where the slope is: i) zero ii) a maximum iii) a minimum b) Sketch a graph of the instantaneous rate of change of this function with respect to x.. a) Sketch a graph of the function y = cos x. b) Sketch a graph of the instantaneous rate of change of this function with respect to x. 4. Derivatives of the Sine and Cosine Functions 3. Find the derivative of each function with respect to x. a) y = 3sin x b) f(x) = 4cos x + π c) y = 3sin x + cos x d) f(x) = π(7sin x cos x) 4. Determine the slope of the tangent line to the function y = 3cos x at x = 3! Find the equation of the line tangent to the curve y = cos θ sin θ at θ =!. 6. Determine the point(s) where the tangent line to y = 4cos θ has a slope of for the domain 0 θ π. corresponds to x =! for the function 3 y = x sin 4x. b) Find the equation of the tangent line to y = x sin 4x at x =! Applications of Sinusoidal Functions and Their Derivatives 10. The graph shows the height of a rider, over time, as he sits on a ride. a) Determine the function h(t) that models the rider s height, in metres, in terms of time, t, in seconds. b) What is the rider s height when t = 5? c) What is the rider s velocity when t = 5? t = 60? 11. The movement of an engine piston can be modelled by h = 4sin t where h is the height of the piston, in centimetres, above the neutral position and t is time, in seconds. a) Determine the velocity of the piston when t = 5. b) Determine the acceleration of the piston when t = 5. Calculus and Vectors 1: Teacher s Resource BLM 4-8 Chapter 4 Review Copyright 008 McGraw-Hill Ryerson Limited

7 Chapter 4 Test BLM For questions 1 5, select the best answer. 1. The graph shows the instantaneous rate of change for which of the functions given? 6. Differentiate with respect to the variable indicated. a) y = sin (4x) + 1 b) y = cos θ + 3sin θ c) f(x) = (cos x + ) d) f(t) = πsin t + cos t e) y = x cos x f) f(θ) = sin θ cos θ 7. Find the slope of the line tangent to the curve y = sin(cos x) at x =!. A y = sin x C y = sin x B y = cos x D y = cos x 8. Find the equation of the line tangent to the curve y = x + sin 4x at x =! 3.. Which of the following is the derivative of y = cos 4x with respect to x? A y = 4cos 4x B y = 4cos 4x sin 4x C y = 4cos 4x sin 4x D y = 4sin 4x 3. What is the slope of the tangent to y = sin x when x =!? A 0 B C D undefined 4. Which of the following is the derivative of y = sin x? A dy C dy = cos x B = sin x D dy = sin x dy = cos x sin x 5. The tangent line to y = sin x is horizontal when: A x = k! (k +1)! B x = C x = k! D x = (k +1)! 9. a) Determine the point(s) where the tangent line to y = sin x 4x has a slope of 3 in the domain 0 x π. b) Find the equation of this tangent line. 10. The function for a current is I(t) = 9cos 10πt where I is the current, in amperes, and t is the time in seconds. a) Find the current when t = 6 s. b) The electromotive force, E(t), in a circuit, is given in volts by finding the first derivative of the current function I(t). Find the function for the electromotive force E(t). c) Determine the electromotive force when t = 3. d) Find the maximum and minimum electromotive forces and the times that they occur. 11. Sketch the function y = sin x + cos x and its derivative. 1. Give an example of a function such that f!!(x) = f (x). Provide support for your answer. Calculus and Vectors 1: Teacher s Resource BLM 4-10 Chapter 4 Test Copyright 008 McGraw-Hill Ryerson Limited

8 Chapter 4 Practice Masters Answers BLM (page 1) Prerequisite Skills 1. a) π b) 3! c)! 3 43! d) a) 360 b) 70 c) 315 d) a) 4.1 Instantaneous Rates of Change of Sinusoidal Functions ( k +1)! 1. a) i), k Z ii) kπ, k Z iii) (k + 1)π, k Z b) i) (k + 1)π < x < kπ, k Z ii) kπ < x < (k + 1)π, k Z c) maximum value =, minimum value = d) b) 4. amplitude = 4, period = π 5. a) 1 b) 1 c) 3 d) 6. a) y! = 6x 5 b) y! = 6x + 10x c) y! = 1(x + 5) 5 d) f!(t) = (4t 5) 3 + 4t (4t 5) e) d!(x) = 16x(4x 1)(3x + 4) (3x + 4) 4 (4x 1) 7. h!(x) = f!(g(x)) g!(x) y = 16x + 97! 10. 3,!31 % (, 14), maximum 4 3,! 4 14 % 7, minimum 1. a) sin x cos x b) cos x sin x 13. Proofs may vary. 14. Proofs may vary.. a) i) (k 1)π < x < kπ, k Z ii) kπ < x < (k + 1)π, k Z ( 4k + 1)! (4k + 3)! b) i) < x <, k Z ( 4k! 1) (4k + 1) ii) < x <, k Z ( 4k + 1)! (4k + 3)! c) i) < x <, k Z ( 4k! 1) (4k + 1) ii) < x <, k Z 3. a) b) Calculus and Vectors 1: Teacher s Resource BLM 4-1 Chapter 4 Practice Masters Answers copyright 008 McGraw-Hill Ryerson Limited

9 Chapter 4 Practice Masters Answers BLM (page ) c) 4. Derivatives of the Sine and Cosine Functions 1. a) y! = sin x b) y! = 6cos x c) y! = 5 sin x d)! y = 4cos x. a) f (x) = 3sin x b) f (x) = cos x + sin x c) f (x) =15x sin x d) d) f (x) = 3 + 4cos x + 1 sin x e) f (x) = (π + 1)cos x f) f (x) =! sin x 3cos x 3. a) f (θ) = 4cos θ + 3 sin θ e) The graph in part d) is the graph from part b) stretched vertically by a) b) f (θ) = πsin θ + 4cos θ 4. a) b) When x = 0, π, π,, the instantaneous rate of change = 1. This is the minimum instantaneous rate of change. When b)!3 c)!7 4, % 3 ( a) x =! 4, 3! 4, 5! 4,, the instantaneous rate of change =. c) b) y = 3x! 1 +! 3 ( ) 6 Features: minimum = 1, no maximum Calculus and Vectors 1: Teacher s Resource BLM 4-1 Chapter 4 Practice Masters Answers copyright 008 McGraw-Hill Ryerson Limited

10 Chapter 4 Practice Masters Answers BLM (page 3) c) 7. y =! 3 x y = x π 9. ( π, 0), (0, 0), (π, 0) 10. If n = 4k, where k Z then y = sin x. If n = 4k 3, where k Z then y = cos x. If n = 4k, where k Z then y = sin x. If n = 4k 1, where k Z then y = cos x. 11. a) y = 3sin x b) y = 3cos x c) (k)π, k Z 1.! 7 4,! 3 4, 4, The tangent lines to the graph of y = sin x + 3 will be parallel to the tangent lines to the graph of y = sin x. So, the slopes of the tangents will be the same. 4.3 Differentiation Rules for Sinusoidal Functions 1. a) dy = 3sin 3x c) dy dy b) dy = cos(x + 3) d) = cos(x + π) = πsin( πx + 3). a) y = 8sin θ b) y = 6cos(θ π) c) y = sin!! % ( d) y = 4πcos 4 θ 3. a) y = (cos x)(sin x) b) f (x) = 8sin 3 x cos x c) y =! cos x or y = (cot x)(csc x) sin x d) f (x) = 3sin x cos x + sin x 4. a) y = sin(t 3) cos(t 3) b) y = 16cos(4t 1)sin(4t 1) c) f (t) = 6tcos(3t ) 4sin(4t) d) f (t) = (cos(cos t))(cos t)(sin t) or f (t) = (sin(t))(cos(cos t)) 5. a) dy b) dy = sin 3x + 3xcos 3x = x(cos x xsin x) c) dy = 4[sin(x ) + x cos(x )] d) dy = π[sin (x + π) + xsin(x + π)cos(x + π)] dy = π[sin (x + π) + xsin(x + π)] 6. a) y = 4(sin 3 x)(cos x) b) y = 4sin x(3cos x sin x) c) y = 8sin x cos x (3cos x 5sin ) 7. slope of tangent = 8. a) Substitute x = π. b) π c) y = π x π 3 9. y = 3x + + 3! a) y = x b) y = x + π 11. y = sin x cos x = sin x 1. If y = sin x then y = cos x = (cos x sin x). If y = sin x cos x then y = (cos x sin x). 13. a) y = cos x(sin x) 1 Calculus and Vectors 1: Teacher s Resource BLM 4-1 Chapter 4 Practice Masters Answers copyright 008 McGraw-Hill Ryerson Limited

11 Chapter 4 Practice Masters Answers BLM (page 4) 4.4 Applications of Sinusoidal Functions and Their Derivatives 1. a) V!(t) = 5cos t b) Maximum voltage = 13, when ( 4k! 3) t =, k Z. Minimum voltage = 3, ( 4k!1) when t =, k Z.. a) h(t) = 8cos.πt b) v(t) = 17.6πsin.πt c) maximum velocity = 55.3 cm/s d) h(t) = 8cos 1.6πt, v(t) = 1.8πsin 1.6πt, maximum velocity = 40. cm/s e) The maximum velocity of the 40-cm pendulum was less than the maximum velocity of the 0-cm pendulum. 3. a) At day 80.5 (March 1st) it is increasing the fastest. b) At day 63 (September 0th) it is decreasing the fastest. 4. a) e) When t = 45 s the rider s height is decreasing at a rate of approximately 1.05 m/s. f) The maximum rate of change is approximately 1.05 m/s. Chapter 4 Review (4k!1)! 1. a) i), k Z ii) 4 (8k! 3)! iii), k Z 4 b). a) (8k +1)! 4, k Z b) b) h(t) = 0.5sin(π(t 0.5)) + 1 c) 0.5 m d) v(t) = πcos(π(t 0.5)) e) When t =, v = 0; when t = 3, v = 0; when t = 1.3, v = 3. f) π m/s or approximately 3.14 m/s g) When t = k, where k Z. 5. a) maximum height = m, minimum height = m b) h(t) = 10sin! % (x 15) c) h!(t) =! 3 cos! % (x 15) 30 d) When t = 10 s, v(t) = 0.91 m/s; when t = 0 s, v(t) = 0.91m/s. 3. a) y! = 3cos x b) f!(x) = 4sin x c) y = 3cos x sin x d) f!(x) = π(7cos x + sin x) 4. slope =! 3 5. y = x +!4 +! % 7! 6. 6,! 3 %, 11! 6, 3 % 7. a) y = 3x cos x 3 b) y = 3sin x cos x Calculus and Vectors 1: Teacher s Resource BLM 4-1 Chapter 4 Practice Masters Answers copyright 008 McGraw-Hill Ryerson Limited

12 Chapter 4 Practice Masters Answers BLM (page 5) c) f!() = 3sin θ + 4cos(θ + π) d) f!(t) = 3sin(sin t) cos t 8. a) y = 3cos 3x cos 4x 4sin 3x sin 4x b) f!() = sin θ cos θ cos θ c) f!(t) = 6t sin(4t π) + 8t 3 cos(4t π) d) y = 4sin(cos θ) cos(cos θ) sin θ 9. a)! 3 9 b) y =!!(3 3 4!) 9 % x + 6! 3 4 (3 3 ( 4!) 7!! 10. a) h(t) = 9sin (t 0) 40 % + 10 b) h(5) = 13.4 c) v(5) = 0.65 m/s d) v(60) = 0.71 m/s 11. a) 1.13 cm/s b) 3.8 cm/s f(x) = sin x Practice Test 1. D. D 3. A 4. C 5. B 6. a) y = 8cos 4x b) y = sin θ + 3cos θ c) f!(x) = (cos x + )(sin x) d) f!(t) = sin t(πcos t 1) e) y = x(cos x xsin x) f) f!() = sin θ(cos θ sin θ) y = 4!! 6 3 % x + 1!! 4!! ! 9. a) 3, 3 3! 4! % and 3 b) y = 3x + 3 3! and 3!3 3!17!% y = 3x + 3 5! 3,!3 3! 0! a) 9 amperes b) E(t) = 1080πsin 10πt c) 0 V 4k!1 d) Maximum = 3393 when t = 40, k Z. Minimum = 3393 when t = % 4k +1 40, k Z. Calculus and Vectors 1: Teacher s Resource BLM 4-1 Chapter 4 Practice Masters Answers copyright 008 McGraw-Hill Ryerson Limited