The graph of a function is given. Choose the answer that represents the graph of its derivative. 1) 1)

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1 Assignment 4 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a function is given. Choose the answer that represents the graph of its derivative. 1) 1) A) B) C) D) Calculate the derivative of the function. Then find the value of the derivative as specified. 2) g(x) = 3x2-4x; g (3) 2) A) g (x) = 6x; g (3) = 18 B) g (x) = 6x - 4; g (3) = 14 C) g (x) = 3x - 4; g (3) = 5 D) g (x) = 2x- 4; g (3) = 2 1

2 Given the graph of f, find any values of x at which f is not defined. 3) 3) A) x = 0, 3 B) x = 3 C) x = 0 D) Defined for all values of x 4) 4) A) x = -1 B) x = 2 C) x = 0 D) x = 1 Calculate the derivative of the function. Then find the value of the derivative as specified. 5) ds dt t =-1 if s =t 2 - t 5) A) ds dt C) ds dt = 2 - t; ds dt t =-1 = 3 = t - 1; ds dt t =-1 = -2 ds ds B) = 2t + 1; dt dt t =-1 = -1 ds ds D) = 2t - 1; dt dt t =-1 = -3 Find an equation of the tangent line at the indicated point on the graph of the function. 6) y = f(x) = x - x2, (x, y) = (1, 0) 6) A) y = 3x + 1 B) y = -x + 1 C) y = x + 1 D) y = 3x - 1 2

3 The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? 7) x = 0 7) A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable Find an equation of the tangent line at the indicated point on the graph of the function. 8) y = f(x) = x2 + 3, (x, y) = (-3, 12) 8) A) y = -6x - 12 B) y = -6x - 15 C) y = -6x - 6 D) y = -3x - 6 9) s = h(t) = t3-16t - 4, (t, s) = (4, -4) 9) A) s = 28t B) s = 32t C) s = -4 D) s = 32t - 4 Given the graph of f, find any values of x at which f is not defined. 10) 10) A) x = 5 B) x = 2, 5 C) x = 2 D) Defined for all values of x SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 11) Find the derivative of y = 5x(x2-3x) by using the Product Rule and by rewriting and then 11) using the Constant Multiple Rule. Show that your answers are equivalent. 3

4 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the derivative of the function. 12) z = 3x2ex 12) A) dz dx = 3xe x + 3x2ex C) dz dx = 6xe x + 3x2ex B) dz dx = 6xe x - 3x2ex D) dz dx = 3xe x + 6x2ex Find the derivative. 13) y = 1 x2.4 - x 13) A) -2.4x x -3/2 B) -3.4x x -3/2 C) -2.4x x -3/2 D) -2.4x x-3/2 Find the second derivative of the function. 14) y = x x2 14) A) d 2y 10 = 2x - dx2 x3 B) d 2y dx2 = x4 C) d 2y dx2 = x4 D) d 2y dx2 = 2-30 x4 Find the derivative. 15) y = 3x2e-x 15) A) 3xe-x(2 - x) B) 3xex(2 - x) C) 3xe-x(x + 2) D) 6xe-x(1 - x) Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 16) u(2) = 10, u (2) = 3, v(2) = -1, v (2) = ) d u dx v at x = 2 A) 47 B) - 53 C) D) ) u(2) = 8, u (2) = 3, v(2) = -2, v (2) = ) d (2u - 4v) at x = 2 dx A) 26 B) 8 C) 24 D) -14 4

5 Find the derivative. 18) y = 7 x6 +x6e 18) A) 6 7 x1/7 + 6ex6e-1 B) 6 7 x-1/7 + 6ex6e-1 C) 6 7 x1/7 + 6x6e-1 D) 6 7 x-1/7 + 6x6e-1 19) y = 1 13x x 19) A) 2 13x x2 B) x3-1 3x2 C) x - 1 3x2 D) x x2 20) s = 3t2-9t ) A) 6t - 9 B) 6t2-9 C) 3t2-9 D) 3t - 9 The figure shows the velocity v or position s of a body moving along a coordinate line as a function of time t. Use the figure to answer the question. 21) v (ft/sec) 21) t (sec) When does the body reverse direction? A) t = 7 sec B) t = 4 sec C) t = 2, t = 3, t = 5, t = 6, t = 7 sec D) t = 4, t = 7 sec Solve the problem. 22) A runner is competing in an 8-mile race. As the runner passes each miles marker (M), a steward 22) records the time elapsed in minutes (t) since the beginning of the race, as shown in the table. What is the runner's average speed over the first 4 miles? Round your answer to four decimal places. M t A) miles/min B) miles/min C) miles/min D) miles/min 5

6 The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 23) s = - t3 + 6t2-6t, 0 t 6 23) Find the body's displacement and average velocity for the given time interval. A) -36 m, -6 m/sec B) -36 m, -12 m/sec C) 396 m, 66 m/sec D) 36 m, 6 m/sec The graphs show the position s, velocity v = ds/dt, and acceleration a = d2s/dt2 of a body moving along a coordinate line as functions of time t. Which graph is which? 24) 24) A) A = position, B = velocity, C = acceleration B) C = position, A = velocity, B = acceleration C) A = position, C = velocity, B = acceleration D) B = position, A = velocity, C = acceleration 25) 25) A) C = position, B = velocity, A = acceleration B) C = position, A = velocity, B = acceleration C) B = position, C = velocity, A = acceleration D) B = position, A = velocity, C = acceleration 6

7 Solve the problem. 26) The size of a population of mice after t months is P = 100( t t2). Find the growth rate at t 26) = 13 months. A) 144 mice/month B) 36 mice/month C) 72 mice/month D) 172 mice/month The graphs show the position s, velocity v = ds/dt, and acceleration a = d2s/dt2 of a body moving along a coordinate line as functions of time t. Which graph is which? 27) 27) A) C = position, A = velocity, B = acceleration B) B = position, A = velocity, C = acceleration C) A = position, B = velocity, C = acceleration D) A = position, C = velocity, B = acceleration Solve the problem. 28) Suppose that the dollar cost of producing x radios is c(x) = x - 0.2x2. Find the marginal cost 28) when 50 radios are produced. A) $50 B) $1600 C) $10 D) -$1600 7

8 The figure shows the velocity v or position s of a body moving along a coordinate line as a function of time t. Use the figure to answer the question. 29) v (ft/sec) 29) t (sec) What is the body's speed when t = 2 sec? A) 0 ft/sec B) -2 ft/sec C) 2 ft/sec D) 3 ft/sec Solve the problem. 30) At time t 0, the velocity of a body moving along the s-axis is v = t2-6t + 5. When is the body's 30) velocity increasing? A) t < 5 B) t > 5 C) t < 3 D) t > 3 8

Average rate of change of y = f(x) with respect to x as x changes from a to a + h:

Average rate of change of y = f(x) with respect to x as x changes from a to a + h: L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,