( ) 4 ( )! x f) h(x) = 2cos x + 1

Transcription

1 Chapter Prerequisite Skills BLM -.. Identifying Types of Functions. Identify the type of function (polynomial, rational, logarithmic, etc.) represented by each of the following. Justify your response. a) f(x) = x 4 x + 4 b) y = x + 4 x + c) y = x d) g(x) = x x + e) y = log 8 x f) h(x) = cos x + Analysing Polynomial Graphs 7. For the graph below, determine the intervals, or values of x, over which: a) the function is increasing and decreasing b) the function is positive and negative c) the curve has zero slope, positive slope, and negative slope Determining Slopes of Perpendicular Lines. For each function, state the slope of a line that is perpendicular to it. a) y = x b) x + y = 4 c) x y 8 = 0 d) x = 6 Using the Exponent Laws. Express each radical as a power. ( ) 4 a) x b) 5 x c) x 4. Express each term as a power with a negative exponent. a) x b) x c)! 5 x 5. Express each quotient as a product by using negative exponents. a) x ( x + ) b) x 4 x + c) ( 5! x) ( x! ) Simplify Expressions with Negative Exponents 6. Simplify. Express answers using positive exponents. a) (x 4 ) b) x4 ( )! c) (x + ) x + d) x x 7 " $ 4 x %!' & Solve Equations, Factor Polynomials 8. Solve. a) x 5x 4 = 0 b) x 9x = 0 c) 6x 7x = 0 d) 5x + x 6 = 0 e) x 4x + = 0 f) x 5 = 0 9. Solve using the factor theorem. a) x 7x 6 = 0 b) x 9x + 7x + 6 = 0 c) 5x + 6x 9x 6 = 0 d) x 4 7x x + x + 6 = 0 Simplify Expressions 0. Expand and simplify. a) (x + ) 5x(x + ) b) (x )(4x + ) (5x )(4x + ). Factor first and then simplify. a) 4(x ) (x 4) + (x ) (x 4) b) x(x + ) 4 (x + x) + 6x(x + ) (x + x) 4 Creating Composite Functions. Given f(x) = 4x +, g(x) = x and h(x) = x + 4, determine a) f o g(x) b) g o f (x) c) h(g(x)) d) f (h(x)) Calculus and Vectors : Teacher s Resource BLM - Prerequisite Skills Copyright 008 McGraw-Hill Ryerson Limited

2 . Derivative of a Polynomial Function BLM -... With the help of a diagram, explain why if y = a, where the tangent to a is horizontal, then dy dx = 0.. For each function, determine dy dx. a) y = x b) y = x c) y = x 4 d) y = x e) y = x f) y = g) y = x x 4 h) y = 4 x 4. Determine the slope of the tangent to the graph of each function at the indicated x-value. a) y = π, x = 5 b) f(x) = x 4, x = c) y = x, x = 5! d) g(x) = x, x = 4. Determine the derivative of each function. State the derivative rules used. a) f(x) = x 4 + x b) y = x + 5x c) h(t) = 4.9t + 0t d) g(x) = x + 4x e) y = 4 x 8 5. Determine the equation of the tangent line to the polynomial y = x x 4x + at each of the following points: a) (, 5) b) (, ) c) (0, ) 6. Determine the point(s) where the slope of the tangent to y = x x 8x + is zero. 7. Simplify, and then differentiate. a) f(x) = x (x ) b) g(x) = 8x4! x x c) h(x) = 4x(x + ) 8. A ball is thrown in the air so that its height, in metres, at t seconds is described by h(t) = 4.9t + t +. a) Determine the rate of change of the height of the ball after s. b) When does the ball hit the ground? c) What is its velocity, or rate of change, when it hits the ground? 9. Determine the equation of the tangent to the graph of y = x 4x + 5 when x =. 0. A platform diver s height above the water is given by the equation h(t) = 4.9t + t + 0. a) Determine the diver s height after 0.5 s. b) Determine the rate of change of the height of the diver after 0.5 s, s, and s. c) When does the diver reach the water (h = 0)?. a) Determine the coordinates of the point(s) on the graph of f(x) = x + where the slope of the tangent is. b) Determine the equation(s) of the tangent(s) to the graph of f(x) at the point(s) found in part a).. Determine the equation(s) of the lines that are tangent to y = x + x + and pass through the origin.. Determine the equation of the normal to the curve y = x + 4x x + at x =. 4. Determine the values of a and b for the function f(x) = x ax + bx + 4 such that f!(x) = 6x 4x + 5. Calculus and Vectors : Teacher s Resource BLM - Section. Derivatives of a Polynomial Function Copyright 008 McGraw-Hill Ryerson Limited

3 . The Product Rule BLM Differentiate each function using two methods. i) expand and simplify, and then differentiate ii) apply the product rule and then simplify a) f(x) = (x )(x ) b) f(x) = (x + )(x ) c) f(x) = ( x + )(x 4) d) f(x) = (x 4)( x). Use the product rule to differentiate each function. a) f(x) = (x + )(5x ) b) g(x) = (5 x)( x) c) h(t) = (4t )(t 7) d) d(t) = (t 4)(t + ). Differentiate. a) f(x) = ( + x)(x ) b) h(t) = (t + 4)(t + ) c) g(x) = (.4x + 4)(x ) d) p(u) = (u + )(u + ) 4. Determine f!() for each function. a) f(x) = (x + 4)(x ) b) f(x) = (x )(x ) c) f(x) = ( x + 4)(x + ) d) f(x) = (x + )(4 x ) 5. Determine the equation of the tangent to each curve at the indicated value. a) f(x) = (x + )(x ), x = b) g(x) = (x 4)(x + ), x = c) h(x) = (x )(x + 4), x = d) p(x) = ( x + )(x 5), x = 6. Determine the point(s) on each curve that correspond to the given slope of the tangent. a) y = (x )(x + ), m = 7 b) y = (4x )(x + 5), m = 6 c) y = (4 x)(x + ), m = d) y = (x )(x + ), m = 0 7. Differentiate. a) y = (4x x + )(x ) b) y = ( x )(x + x ) c) y = (x ) d) y = x (x + )(5x ) 8. The student council is selling tickets for a dance. In the past, they have found that if they charge $8 for tickets they will sell about 00 tickets. For every $ increase in price they sell 50 fewer tickets. a) Write an equation that models the council s revenue, R, as a function of x, where x represents the number of $ increases in the price. b) Use the product rule to determine R!(x). c) Evaluate R!() and interpret its meaning for this situation. d) What price will result in the maximum revenue? 9. a) Determine the point on the curve f(x) = ( x) where the tangent line is horizontal. b) Sketch the curve and the tangent. 0. Use the product rule to differentiate. a) f(x) = (x + )(x )(x ) b) f(x) = (x )(x + 5)(x + ). Given f(x) = (ax + )(x + b), find values of a and b such that f!(x) = 8x 4. Calculus and Vectors : Teacher s Resource BLM -4 Section. The Product Rule Copyright 008 McGraw-Hill Ryerson Limited

4 . Velocity, Acceleration, and Second Derivatives BLM -7.. (page ). Determine the first and second derivatives of each function. a) y = x x + b) h(t) = t + t 4t + c) f(x) = 4.9x + 8x For each distance time graph, sketch the velocity function and the acceleration function. a) d) g(x) = x + x4! e) d(t) = t 5 + t 4 6t 4 f) h(x) = (x )(x + ). Determine f!!( ) for each function. a) f(x) = x x b) f(x) = 4 x4! x + 5x c) f(x) = (x ) d) f(x) = (x 5)(x ). Determine the velocity and acceleration functions for each position function s(t). Where possible, simplify the functions before differentiating. a) s(t) = 4t t + t b) s(t) = (t 5)(4 t) c) s(t) = 4.9t + 5t d) s(t) = 5t6! t 4 + t t 4. Determine the velocity and acceleration at t = 4 for each position function s(t), where s is in metres and t is in seconds. a) s(t) = t t + 4t b) s(t) = 4.9t + 0t + c) s(t) = (t + 5)( t) d) s(t) = t (4t )( + t) b) c) Calculus and Vectors : Teacher s Resource BLM -7 Section. Velocity, Acceleration, and Second Derivatives Copyright 008 McGraw-Hill Ryerson Limited

5 . Velocity, Acceleration, and Second Derivatives BLM -7.. (page ) 6. The graph shows the position function of an object. a) At what point(s) on the graph is the velocity 0? b) During which intervals is the velocity positive? c) During which intervals is the velocity negative? d) During which intervals is the acceleration positive? e) During which intervals is the acceleration negative? 7. At a fall fair, a pumpkin is catapulted. The height of the pumpkin, in metres, t seconds after it is launch is given by the function h(t) = 4.9t + 0t +, t 0. a) Determine the height of the pumpkin after s. b) Determine the velocity of the pumpkin at s and at s. c) What is the velocity of the pumpkin at t = 0? What is the significance of this point? d) What is the acceleration of the pumpkin at s and at s? e) When will the pumpkin hit the ground? f) What is the velocity of the pumpkin when it hits the ground? 8. The position function of an object moving along a straight line is represented by the function s(t) = t t 6t + 6, where t is in seconds and s is in metres. a) What is the position of the object after s and after 5 s? b) What is the velocity of the object after s and after 5 s? c) When is the object stopped? What is its position at this time? d) When is the object moving in a positive direction? e) Determine the total distance travelled by the object during the first 0 s. Calculus and Vectors : Teacher s Resource BLM -7 Section. Velocity, Acceleration, and Second Derivatives Copyright 008 McGraw-Hill Ryerson Limited

6 .4 The Chain Rule BLM Determine the derivative of each function by using the following methods. i) simplify first and then differentiate ii) use the chain rule and then simplify a) y = (x 4 )! b) y = " c) y = ( 5x ) 4 5 x4 $ % & d) y = ( x) 5. Differentiate. Express each answer using positive exponents. a) f(x) = (x ) b) f(x) = (4x ) c) g(x) = (x + 5) d) g(x) = (4x + 7) e) y = (x x) f) y = ( 4x 4 ). Express each function as a power with a rational exponent, and then differentiate. Express each answer using positive exponents. a) f(x) = x! 5 b) g(x) =!x + 4 c) h(x) = x + 5x 4 5 d) p(x) = 7x + x! 4 4. Express each as a power with a negative exponent, and then differentiate. Express each answer using positive exponents. a) f(x) = x + b) g(x) = c) h(x) = d) f(x) = ( ) ( 4x! x) x + x x + 5x 5. Determine f!( ). a) f(x) = (x + x ) b) f(x) = (4x + ) 4 c) f(x) = x 4! 5x d) f(x) = x + 4 ( ) 6. Using Leibniz notation, apply the chain rule to determine dy at the indicated dx value of x. a) y = u, u = x 5, x = b) y = u, u = x x +, x = c) y = u, u = x + 4x, x = d) y = u u, u = x x, x = 7. Determine the slope of the tangent to the curve y = (5x x) 4 at x =. 8. Determine the slope of the tangent to the curve y = at x =. x! x ( ) 9. Determine the equation of the tangent to the curve y = (x + x ) at x = Determine the point(s) on the curve y = (4x ) where the tangent line is horizontal.. Find the second derivative of y = (x + x).. If f(x) = x, g(x) = x + and h(x) = x, determine the derivative of each composite function. a) y = f o g(x) b) y = ho g(x) c) y = f o g o h(x). Use the product rule and chain rule together to differentiate the following. a) f(x) = (x ) (x ) b) g(x) = (x + x) (4x + ) Calculus and Vectors : Teacher s Resource BLM -9 Section.4 The Chain Rule Copyright 008 McGraw-Hill Ryerson Limited

7 .5 Derivatives of Quotients BLM -... Express each quotient as a product. State the domain of the function.!4 a) f(x) = b) g(x) = x + x! 5 c) p(x) = d) q(x) = 5x! 8! x. Differentiate each function in question. Do not simplify your answers.. Express each quotient as a product, and state the domain of the function. x a) f(x) = b) g(x) =!4x x! 5 x + 8 x 5x c) h(x) = d) p(x) = x + x! 8 4. Differentiate each function in question. Do not simplify. 5. Differentiate. x! a) y = x + 5 c) y = x! 4 x + 5 b) y = 5! x x + x d) y = x + x! 4 6. Determine the slope of the tangent to each curve at the indicated value of x. x a) y = 4x +, x = x b) y = x!, x = x + c) y = x!, x = d) y = e) y = x! 5 x + x!, x = x x! 4x +, x = 4 7. a) Explain the two different methods that could be used to differentiate y = x + 4x + x! 4 x. b) Differentiate the function using both methods. Which method do you prefer? Explain why. 8. Determine the points on the curve y = where the slope of the tangent is. 4 x + 9. Determine the equation of the tangent to the x! curve y = at the point where x =. x + 0. Determine the equation of the tangent to the x curve y = at the point where x =. x! ( ). The value of a company machine, t years after it is purchased, is given by the formula V(t) = t ( ). a) What is the value of the machine after years? b) At what rate is the value of the machine decreasing after year? c) At what rate is the value of the machine decreasing after 5 years?. Determine the point(s) on the function x f(x) = where the tangent line is x! 4 horizontal.. Given the functions f(x) = x! and g(x) =, determine the derivative of x each composite function and state its domain. a) y = f o g(x) b) y = g o f (x) Calculus and Vectors : Teacher s Resource BLM - Section.5 Derivatives of Quotients Copyright 008 McGraw-Hill Ryerson Limited

8 .6 Rate of Change Problems BLM -... A store sells 50 pairs of Brand X running shoes each month when priced at $00 per pair. It has been determined that for every $5 decrease in price, an additional 0 pairs will be sold. a) Determine the demand or price function. b) Determine the revenue function. c) Find the revenue when sales are 50 pairs and 90 pairs of shoes per month. d) Determine the marginal revenue when sales are 90 pairs of shoes per month.. A theatre has found that if the price of a ticket is $4 then 400 tickets will be sold. For every $ decrease in price, an additional 50 tickets will be sold. a) Determine the demand or price function based on the number of price decreases. b) Determine a revenue function based on the number of price decreases. c) Determine the marginal revenue for the revenue function developed in part a). d) When is this marginal revenue function equal to zero? What is the total revenue at this time? How can the owners of the theatre use this information? e) If the theatre has a maximum capacity of 450, what price will maximize revenue?. The mass, in grams, of the first x metres of a wire is represented by the function f(x) = 4x!. a) Determine the average linear density of a segment of the wire from x = to x = 7. b) Determine the linear density at x = 4 and x = 0. What do these values confirm about the wire? 4. Given the revenue function R(x) = 0x 0.05x and the cost function C(x) = x + 5, where x is the number of items being sold, determine a) the revenue and marginal revenue when 500 items are sold b) the cost and marginal cost when 500 items are produced c) the profit and marginal profit from sales of 500 items 5. The cost function of producing yo-yos is C(x) = x x. a) Find the marginal cost of producing 000 yo-yos. b) The revenue from selling x number of yo-yos is R(x) = 6x. Determine the marginal profit from selling 000 yo-yos. 6. The volume of water in a tank that is draining from the bottom is given by the " function V(t) = 50! t % $ 5& ', 0 t 5, where V is in litres and t is in minutes. a) At what rate is the water draining from the tank after min? after min? b) When is the water completely drained from the tank? Explain how you determined this. 7. The mass, in grams, of a wire is given by the function f (x) = x( + x ) where x is the length measured in metres from one end. a) Determine the mass of the wire when x = 5. b) Determine the linear density of the wire when x = m and when x = 5 m. 8. In a certain electrical circuit, the resistance, R, in ohms, is represented by the function R = 40, where I is the current, in amperes. I Determine the rate of change of the resistance with respect to the current when the current is 8 A. Calculus and Vectors : Teacher s Resource BLM - Section.6 Rate of Change Problems Copyright 008 McGraw-Hill Ryerson Limited

9 Chapter Review BLM Derivative of a Polynomial Function. Differentiate each function. State the derivative rules used. a) f(x) = 5x + x b) g(x) = x 5 + 4x 8x c) h(t) = 4.9t 9t + d) p(x) = x 4 x + x e) d(t) = 4t! t 4 f) q(x) = (x 4)(x + ). The Product Rule. Determine the equation of the tangent to each curve at x =. a) f(x) = 4x x + b) g(x) =!4x5 + x! x x. Determine the equation of the tangent to the curve y = x 8 where the slope of the x tangent is Use the product rule to differentiate. a) f(x) = (x + )(x ) b) g(x) = (x 9)(4 x) c) h(t) = (4.9t 5)(t + ) d) d(t) = (6t )(4 t) 5. Determine the equation of the tangent to the graph of each curve at the point that corresponds to each value of x. a) y = (x + )(x ), x = 4 b) y = (5 x)(x + ), x = c) y = (x + 4)( 5x), x = d) y = (x + )( + x), x =. Velocity, Acceleration, and Second Derivatives 6. Determine the velocity and acceleration function for the position function d(t) = t t + 5t A ball is thrown upward from the roof of a school, 6 m high, with an initial velocity of 0 m/s. The height of the ball above the ground at time t is given by the function h(t) = 4.9t + 0t + 6, t 0. a) Find the height of the ball at t = and t =. b) Find the rate of change of the height of the ball at s and s. c) What is the maximum height of the ball? d) How fast was the ball travelling when it hit the ground?.4 The Chain Rule 8. Apply the chain rule, in Leibniz notation, to determine dy at the indicated value of x. dx a) y = u 4, u = x, x = b) y = u +, u = x 5, x = c) y = u, u = x, x = 6.5 Derivatives of Quotients 9. Find the equation of the tangent line to y = x 4! x at x =. 0. Differentiate each function. a) h(x) = x! 5x b) f(x) = x + 4x! ( x +) x! 5 c) y = d) q(x) = x! x + x! 8.6 Rates of Change Problems. When the price is $.75 each, 000 fruit bars will be sold. If the price of a fruit bar is raised to $.00, sales will drop to 500. a) Determine the demand, or price, function. b) Determine the marginal revenue from the sale of 700 bars. c) The cost for the bars is given by the function C(x) = x. Determine the marginal cost of purchasing 000 bars. d) Determine the marginal profit function for the sale of the fruit bars. e) Determine the marginal profit from the sale of 000 bars. Calculus and Vectors : Teacher s Resource BLM -4 Chapter Review Copyright 008 McGraw-Hill Ryerson Limited

10 Chapter Test BLM -6.. (page ) For questions and, choose the best answer.. For which of the following does y! " 0? A y = 4 B y = π C y = x D y = 0 D. Which of the following graphs represents the derivative of y = f(x) shown below.. Differentiate. a) y = 6x 4 5 x + x b) g(x) = ( x +! x ) 4 A B C c) s(t) = ( + t ) 5 (4t 7 t 6 ) d) y = ( 4x! x ) 5 x + 4. Determine f!!(") for the function f(x) = ( x! ). 5. A toy rocket is shot into the air. Its height, in metres, after t seconds is given by h(t) = 4.9t + 8t +.. a) Determine the height of the rocket after s. b) Determine the rate of change of the height of the rocket after s and 4 s. c) How long does it take the rocket to hit the ground? d) How fast was the rocket travelling when it hit the ground? Explain your reasoning. 6. Determine the equation of the tangent to the curve y = (x )(x + ) at the point where x =. 7. Determine the coordinates of the point on the graph of f(x) = x + where the tangent line is perpendicular to the line x + y + 4 = 0. Calculus and Vectors : Teacher s Resource BLM -6 Chapter Test Copyright 008 McGraw-Hill Ryerson Limited

11 Chapter Test BLM -6.. (page ) 8. Apply the chain rule, in Leibniz notation, to determine dy at the indicated value of x. dx y = 4u( u), u = x, x = 9. A sunglasses shop sells 0 sports sunglasses per week at a price of $50 each. A customer survey indicates that for each $ decrease in price, sales will increase by sunglasses per week. a) Determine the revenue function. b) Determine the marginal revenue when the price is $45. What does this value represent? c) Solve R!(x) = 0. Interpret the meaning of this value for this situation. d) What is the price that corresponds to the value found in part c)? How can this information by used by the sales manager? 0. Determine the velocity and acceleration functions at t = s for each position function s(t), where distance, s, is in metres, and time, t, is in seconds. a) s(t) = t(t 4)(t + ) b) s(t) = (t + ). Water is being drained from the bottom of a tank. The equation that gives the volume V of water, in litres, remaining in the tank after " t minutes is V(t) = 000! t % $ 0& ', 0 t 0. a) Find the volume of water in the tank after 5 min. b) Find the rate at which the water is draining from the tank after i) min ii) 8 min Calculus and Vectors : Teacher s Resource BLM -6 Chapter Test Copyright 008 McGraw-Hill Ryerson Limited

12 Chapter Practice Masters Answers BLM -8.. (page ) Prerequisite Skills. a) polynomial b) exponential c) rational d) polynomial e) logarithmic f) trigonometric. a) b) c)! d) 0 5. a) x b) x c) x 4. a) x b) x 5 c)!x! 5. a) x(x + ) b) x 4 x + c) ( 5! x) ( x! )! 4 ( )! 6. a) b) c) x 8 x ( x + ) d) x 6! x 7. a) increasing: (,.), (.7, ); decreasing: (.,.7) b) positive: ( 4, ), (4, ); negative: (, 4), (, 4) c) zero slope: x =., x =.7; positive slope: (,.), (.7, ); negative slope: (.,.7) 8. a) x = 7, x = b) x = 4, x = c) x =!, x = d) x = 5, x = e) x = ± f) x = ± 5 9. a) x =, x =, x = b) x =, x =, x =! c) x =, x =! 5, x = d) x =, x =!, x =, x = 0. a) 4x 6x + 4 b) 8x x + 9x 6. a) (x ) (x 4) (x 6) b) x(x + ) (x + x) (x + 7x + ). a) 4x b) 6x + 8x c) x + d) x + 8 x Derivative of a Polynomial Function. The graph y = a is a horizontal line through (0, a). The slope of this line is zero. dy dy 4 dy. a) = b) = x c) = 8x dx dx dx dy d) = 6x dy dy e) = dx dx x! f) = x 5 dx dy dy 6 g) = x h) = x dx dx. a) 0 b) 96 c) 6 5 d) 4. a) f!(x) = x + x; sum rule, power rule, constant multiple rule b) y! = x + 5; sum rule, power rule, constant multiple rule c) h!(t) = 9.8t + 0; sum rule, power rule, constant multiple rule d) g!(x) =! 6 + 8x ; sum rule, power rule, x 4 constant multiple rule e) y! = 4 x! ; difference rule, power rule, constant rule, constant multiple rule 5. a) y = 7x + b) y = 0x + 9 c) y = 4x + 6. (4,! 77 ), (, ) 7. a) f!(x) = 5x 4 x b) g!(x) = 6x c) h!(x) = 48x + x a) h!() =. m/s b) t =.5 s c).8 m/s 9. y = 8x a) 0.75 m b).9 m/s; 6.8 m/s; 6.6 m/s c) after.5 s ( ), (!,!4 + ). a),4 + b) y = x 8 +, y = x y = 5x and y = x. y = 7 x a =, b = 5 Calculus and Vectors : Teacher s Resource BLM -8 Chapter Practice Masters Answers Copyright 008 McGraw-Hill Ryerson Limited

13 Chapter Practice Masters Answers BLM -8.. (page ). The Product Rule. a) f!(x) = x 5 b) f!(x) = 4x 5 c) f!(x) = 4x + d) f!(x) = 6x + 0. a) f!(x) = 0x + b) g!(x) = 4x c) h!(t) = 8t d) d!(t) = 8t + 6t 8. a) f!(x) = x + 4x b) h!(t) = t + 6t + 4 c) g!(x) = 4.x + 8x 4. d) p!(u) = 6u + 6u + 4. a) 4 b) 7 c) 5 d) a) y = x 8 b) y = 4x + 8 c) y = 78x 0 d) y = 0x 6. a) (, ) b) (!, 0) c) (, 9) d) (, ) and (, 0) 7. a) y! = x 6x + 5 b) y! = 4x 6x + 8x + c) y! = 6x x d) y! = 0x + 9x x 8. a) R(x) = (8 + x)(00 50x) b) R!(x) = 00x 00 c) R!() = 00. This means when there is a $ decrease, the revenue is falling at $00 per $ decrease in price. d) $7.00! 9. a) ", 0 $ % & b). Velocity, Acceleration, and Second Derivatives. a) y! = 6x ; y! = 6 b) h!(t) = 6t + t 4; h!!(t) = t + c) f!(x) = 9.8x + 8; f!!(x ) = 9.8 d) g!(x) = x + x ; g!!(x) = 4x + 6x e) d!(t) = 5t 4 + 8t 6; d!!(t ) = 60t + 4t f) h!(x) = x 7; h!!(x) =. a) 6 b) 44 c) 8 d) 4. a) s!(t) = t 6t + ; s!!(t) a(t) = 4t 6 b) s!(t) = t + ; s!!(t) = c) s!(t) = 9.8t + 5; s!!(t) = 9.8 d) s!(t) = 0t 4t + ; s!!(t) = 60t 4 4. a) 9; 46 b) 9.; 9.8 c) 6; d) 98; a) b) 0. a) f!(x) = 4x x 40x + b) f!(x) = 4x + 48x + x 4. a =, b = 5 Calculus and Vectors : Teacher s Resource BLM -8 Chapter Practice Masters Answers Copyright 008 McGraw-Hill Ryerson Limited

14 Chapter Practice Masters Answers BLM -8.. (page ) c) 6. a) B, D b) A to B and D to E c) B to D d) C to E e) A to C 7. a).4 m b) v() = 0. m/s; v() = 9.4 m/s c) 0 m/s; this is when the pumpkin is catapulted d) a() = 9.8 m/s ; a() = 9.8 m/s e) after 4. s f) v(4.) =.4 m/s 8. a) s() = 6, s(5) = b) v() = 4, v(5) = 84 c) stopped when t =, s() = 75 d) (, ) e) 50 m.4 The Chain Rule. a) 4x b) 9 x c) 50x 9 d) 8 5 x! 5. a) 8x b) 9x 96x +!8 c) 6x + 40x d) 4x + 7 ( ) e) 54x 5 45x 4 + x x f). a) f(x) = ( x! 5), f!(x) = b) g(x) = (!x + 4), g!(x) = c) h(x) = ( x + 5x ) 4, h!(x) = 6x (! 4x ) 4 x " 5 "x "x + 4 0x + ( ) x + 5x 4 d) p(x) = ( 7x + x! 4) 5, p!(x) = 4x + ( ) 5 5 7x + x " 4 4. a) f(x) = (x + ),! f (x) = b) g(x) = (4x x), g!(x) = ( )! c) h(x) = x + x ( )! d) f (x) = x + 5x 4 "4 ( x + ) (8x " ) ( 4x " x) 4, h!(x) = "(9x + ) x + x, f!(x) = ( ) "( + 0x) x + 5x ( ) 5. a) 80 b)!96 c) 0.00 d) !5!58 6. a) 6 b) c) d) does not exist, function not defined at x = y = 9x! 0. " 4, 0 $ % &. y!! = 6(x + x)(5x + 5x + 9). a) y! = 6(x + ) b) y! = ( ) x + c) y! = x x +. a) f!(x) = 6(x ) (x )(5x 4) b) g!(x) = (x + x) (4x + ) (6x + 4x + 6) 4 Calculus and Vectors : Teacher s Resource BLM -8 Chapter Practice Masters Answers Copyright 008 McGraw-Hill Ryerson Limited

15 Chapter Practice Masters Answers BLM -8.. (page 4).5 Derivatives of Quotients. a) f(x) = (x + ) ; x! " b) g(x) = 4(x ) ; x! c) p(x) = 5(5x ) ; x! ± 5 d) q(x) = (8 x ) ; x!. a) f!(x) = (x + ) b) g!(x) = (x ) c) p!(x) = 50x(5x ) d) q!(x) = 9x (8 x ). a) f(x) = x(x 5) ; x! 5 b) g(x) = 4x(x + 8) ; x! "8 c) h(x) = x (x + ) ; x! " d) p(x) = 5x (x 8) ; x! ± 4. a) f!(x) =(x 5) 4x(x 5) b) g!(x) = 4(x + 8) + 4x(x + 8) c) h!(x) = x(x + ) x (x + ) - d) p!(x) =0x(x 8) 0x (x 8) 9 5. a) y! = b) y! = x " 0x " (x + 5) (x + ) c) y! = x + 0x + 8 x a) 98 ( ) d)! b) 0.56 c) 8 "x " y = ( x + x " 4) d) not differentiable when x = e)! 7. a) Express the quotient as a product and use the product rule or quotient rule. b) y! = x + 8x 8. ( 5, ); (, ) 9. y = 7x not differentiable when x =. a) $ b) $455.5/year c) $048/year. (0,0). a) y! = b) y! = % $ x x " " ; x > x " ; x 0, x & ( '.6 Rate of Change Problems. a) p(x) = x b) R(x) = x( x) c) 50 shoes $5 000, 90 shoes $6 00 d) $7.50/pair of shoes. a) p(x) = 4 x b) R(x) = (4 x)( x) c) R!(x) = x d) when the price is $0, revenue is $0 000, the owners can use this information to choose the best price to charge for tickets e) $. a) g/m b) f!(4) = 0.56; f!(0) = 0.0; they confirm that the material of which the wire is composed of is not homogenous 4. a) $8750; $5/item b) $005; $/item c) $7745; $/item 5. a) $6.0/yoyo b) $0.0/yoyo 6. a) 6L/min; 4 L/min b) when t = 5 7. a) 6.8 g b) f!() =.; f!(5) = ohms/amp Chapter Review. Rules used may vary. a) 0x + b) 5x 4 + 8x 8 c) 9.8t 9 d) 6x e)!t!! 4 x t f) 8x + 4x. a) y = 67 x! 6 b) y =!6x y = x 4, y = x a) f!(x) = x b) g!(x) = 8x + 6x + 7 c) h!(t) = 9.8t d) d!(t) = 8t + 48t + 5. a) y = 47x 98 b) y = x + 7 c) y = 9x 4 d) y = 58x 6. v(t) = 9t 4t + 5, a(t) = 8t 4 Calculus and Vectors : Teacher s Resource BLM -8 Chapter Practice Masters Answers Copyright 008 McGraw-Hill Ryerson Limited

16 Chapter Practice Masters Answers BLM -8.. (page 5) 7. a). m; 6.4 m b) 0.; 0.4 c) 6.4 m d) 6.7 m/s 8. a) 8 b) undefined at x = c) 9. y =!9 7 x! a) h!(x) = "x + 6x + ( x + ) b) f!(x) = 0x " 0x 4x " ( )!(x + )(!x! 9) c) y = x! d) q!(x) = ( ) "x + 7x + ( ) x " 5 x + x " a) p(x) = x b) $0.55/bar c) $0.5/bar d) p!(x) = " 0.00x e) $0/bar 9. a) R(x) = (0 + x)(50 x) b) $40/decrease in price. This tells you that the revenue is decreasing by $40 per price decrease when the price is $45. c) x = 5. This means that the highest revenue occurs when there has been 5 price decreases or 5 price increases d) $55 this is the best price for the sunglasses to create the largest revenue 0. a) v() = 90, a() = 408 b) v() = 94, a() = 68. a) 750 L b) V!() = 40, V!(8) = Practice Test. C. A. a) y! = 4x + 0 x + x b) g!(x) = 4 "x + ( " x ) " " x( " x ) " & % ( $ ' c) s!(t) = (+ t ) 4 4t " 7 & $ % t 6 ' ( d) y! = 4x " x 64t & + 7t + 5 $ % t t 7 ' ( ( ) 4 ("57x + 88x + 40) ( x + ) 4. f!!(") = a) 4. m b) v() = 8.4 m/s; v(4) =. m/s c) 5.76 s d) 8.4 m/s 6. y = x 6 7. (4, ) 8. Calculus and Vectors : Teacher s Resource BLM -8 Chapter Practice Masters Answers Copyright 008 McGraw-Hill Ryerson Limited