Math 611b Assignment #6 Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

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1 Math 611b Assignment #6 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find a formula for the function graphed. 1) 1) A) f(x) = 5 + x, x < - x > C) f(x) = 5 + x, x - x > B) f(x) = 5 - x, x - x > D) f(x) = 5 - x, x < - x ) Assume that a watermelon dropped from a tall building falls y = 16t ft in t sec. Find the watermelon's average speed during the first 5 sec of fall. A) 40 ft/sec B) 80 ft/sec C) 160 ft/sec D) 100 ft/sec ) ) lim x 7 x + 49 x + 7 A) 14 B) Does not exist C) 0 D) 7 ) Evaluate. 4) lim x 0 x + 1x - 5x 5x A) -1 B) 5 C) 0 D) Does not exist 4) 1

2 Evaluate. 5) lim f(x) x -1/ 5) A) - B) Does not exist C) 0 D) -1 6) lim f(x) 6) x 1- A) Does not exist B) 1 C) D) -1 Find the limit. 7) Let lim f(x) = -4 and x lim g(x) = -5. i) Find x ii) Find iii) Find lim [f(x) - g(x)]. x lim x lim x [f(x) g(x)]. iii f(x) g(x). A) -9, 0, 1.5 B) 1, 0, 0.8 C) -4, -0, 0.5 D), -0, 1.5 7)

3 Evaluate or determine that the limit does not exist for each of the limits (a) for the given function f. 8) -x -, for x < 1, f(x) = 1, for x = 1, x - 10, for x > 1 lim x 1- f(x), (b) lim x 1+ f(x), and (c) 8) lim x 1 f(x) A) (a) -4 (b) -7 (c) -11 C) (a) -7 (b) -4 (c) -11 B) (a) -7 (b) -4 (c) Does not exist D) (a) -4 (b) -7 (c) Does not exist Find the limit. 9) lim x 7x - x + x -x - x + 5 9) A) B) C) 7 D) -7 10) lim x -4- x- x +x-1 ii) Identify and name any points of discontinuity. 10) iii) Write an equation for the extended function that is continuous for all values of x 0 A) 1 B) 5 C) 0 D)

4 Match the function with the graph of its end behavior model. 11) y = -4x + 5x + 1 x + 9 A) B) 11) C) D) Find a simple basic function as a right-end behavior model and a simple basic function as a left-end behavior model. 1) y = ex + x 1) A) y = ex; y = x B) y = -e-x; y = x C) y = -e-x; y = x D) y = ex; y = x Use the definition f'(a) = lim h->0 f(x + h) - f(x) h to find the derivative of the given function for any value of x. 1) f(x) = 17-9x 1) A) -17 B) -9 C) 17 D) 8 14) Find d dx (x - ). 14) A) 4x - B) 4x C) 4x - D) x 4

5 The graph of a function is given. Choose the answer that represents the graph of its derivative. 15) 15) A) B) C) D) 16) Find the equation of the normal line to the curve y = 4x - x at the point (, 0). 16) A) x - 4y = 0 B) x + 1y - = 0 C) x - 4y - = 0 D) x + 1y = 0 5

6 Find the values where the function is not differentiable. 17) 17) A) x = 0 B) x = 1 C) x = -1 D) x = 18) The graph shows the yearly average interest rates for 0-year mortgages for years since 1988 (Year 0 corresponds to 1988). Sketch a graph of the rate of change of interest rates with respect to time. 18) A) 6

7 B) C) D) Determine the values of x for which the function is differentiable. 1 19) y = 19) x - 64 A) All reals except 64 B) All reals except -8 and 8 C) All reals except 8 D) All reals 7

8 Find dy/dx. 8x - 7 0) y = 0) x - 8x + 9 A) 8x - 80x + 150x - 56 (x - 8x + 9) B) -8x + 14x + 16 (x - 8x + 9) C) 4x - 14x + 18 (x - 8x + 9) D) 8x + 14x + 16 x - 8x + 9 1) y = ln x 8 1) A) 1 x B) 1 8x C) 8 x D) 1 x - ln 8 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. ) u(1) =, u (1) = -6, v(1) = 7, v (1) = -4. ) d u dx v at x = 1 A) B) C) D) Find the slope of the line tangent to the curve at the given value of x. ) y = x - 5x; x = ) A) -6 B) 6 C) 1 D) -9 Find the fourth derivative of the function. 4) y = x - x-1 4) A) 18x-4 B) 7x-5 C) -6x-5 D) -7x-5 5) Find an equation of the tangent to the curve y = x - x + 1 that has slope. 5) A) y = x B) y = x - 1 C) y = x + 1 D) y = x + 6) The population P, in thousands, of a small city is given by P(t) = 900t t + 1, where t is the time, in months. Find the growth rate, dp dt. A) dp dt = 900(t - 1) (t + 1) C) dp dt = 900(1 - t ) t + 1 B) dp dt = 900(1 + 6t ) (t + 1) D) dp dt = 900(1 - t ) (t + 1) 6) 8

9 7) The function V = r describes the volume of a right circular cylinder of height feet and radius r feet. Find the (instantaneous )rate of change of the volume with respect to the radius when r = 6. Leave answer in terms of. A) 6 ft/ft B) 1 ft/ft C) 6 ft/ft D) 18 ft/ft 7) 8) The dollar profit from the expenditure of x thousand dollars on advertising is given by P(x) = x - x. Find the marginal profit when the expenditure is x = 9. A) 5 thousand dollars B) 800 thousand dollars C) 189 thousand dollars D) -11 thousand dollars 8) 9) At time t, the position of a body moving along the s-axis is s = t - 7t + 40t m. Find the body's acceleration each time the velocity is zero. A) a(0) = 10 m/sec, a(16) = 0 m/sec B) a(10) = 6 m/sec, a(8) = -6 m/sec C) a(10) = 0 m/sec, a(8) = 0 m/sec D) a(10) = -6 m/sec, a(8) = 6 m/sec 9) Find dy/dx. 0) y = 4 tan6x 0) A) 4 tan7x B) 4 tan5x secx C) 4 tan5x D) 4 tan6x sec x The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). 1) s = - + cos t Find the body's speed at time t = / sec. A) - m/sec B) m/sec C) m/sec D) - m/sec 1) Find the indicated derivative. ) Find y if y = 4x sin x. ) A) y = 8 cos x - 4x sin x B) y = - 8 cos x + 4x sin x C) y = 4 cos x - 8x sin x D) y = - 4x sin x Find dy/dx. ) y = sin6 x - cos 5x ) A) 0 sin5 x cos x sin 5x B) 6 sin6 x cos x - 5 sin 5x C) 6 sin5 x + sin 5x D) 6 sin5 x cos x + 5 sin 5x Find dr/d. 4) r = cot (5-8 ) 4) A) -csc(5-8 ) B) 8 csc (5-8 ) cot (5-8 ) C) 8 csc(5-8 ) D) -csc (5-8 ) cot (5-8 ) 5) The position of a particle moving along a coordinate line is s = 5 + 4t, with s in meters and t in seconds. Find the particle's velocity at t = 1 sec. 5) A) 4 m/sec B) - 1 m/sec C) 1 6 m/sec D) m/sec 9

10 Find y. 6) y = cot x 10 A) 6 csc x 10 cot x 10 C) -6 csc x 10 B) 50 csc x 10 cot x 10 D) - 10 csc x 10 6) Find dy/dx. 7) y = 7 x 7) A) dy dx = 7 x -4/7 B) dy dx = 7 x 4/7 C) dy dx = 7 x 4/7 D) dy dx = 7 x -4/7 Use implicit differentiation to find dy/dx and dy/dx. 8) xy - x + y = 4 8) A) dy dx = y x + 1 ; d y dx = y - (x + 1) C) dy dx = y x + 1 ; d y dx = y + 1 (x + 1) B) dy dx = y + 1 x + 1 ; d y dx = y + (x + 1) D) dy dx = 1 - y 1 + x ; d y dx = y - (x + 1) Find where the slope of the curve is defined. 9) xy - 6xy = 10 9) A) dy dx = xy - 6y 1y - x ; defined at every point except where y = x 1 B) dy dx = xy - 6y x(1y + x) ; defined at every point except where x = 0 or y = - x 1 C) dy dx = xy - 6y x(1y - x) ; defined at every point except where x = 0 or y = x 1 D) dy dx = 6y + 1xy ; defined at every point except where x = 0 x 40) Given (x - ) + (y + ) = 106, find dy/dx and the slope of the curve at the point (-6, ). 40) A) dy dx = - y + x - ; 5 B) dy 9 dx = x - y + ; C) dy dx = - x - y + ; 9 5 D) dy dx = y + x - ; Find an equation for the tangent to the graph of y at the indicated point. Round to the nearest thousandth when necessary. 41) y = tan-1 x, x = 6 41) A) y = 1 7 x B) y = 1 6 x C) y = 1 x D) y = 0.169x

11 Find the derivative of y with respect to the appropriate variable. 4) y = sin-1 (5x) 4) 0x 0x 0x A) B) C) D) 1-5x6 1-5x6 1-5x 1-5x6 Find dy/dx. 4) f(x) = 7e-x 4) A) 1e-x B) -1e-x C) -e-x D) 7e-x 44) y = ln x 44) x 1 A) B) C) 6 D) x + x + x x 45) y = x ) A) + 4)x + B) x + 4 C) x + 4 D) + )x + Use logarithmic differentiation to find dy/dx. 46) y = (cos x)x 46) A) (cos x)x (ln cos x - x tan x) B) ln x(cos x)x - 1 C) ln cos x - x tan x D) (cos x)x (ln cos x + x cot x) 47) Suppose that the amount in grams of a radioactive substance present at time t (in years) is given by A(t) = 50e-.6t. Find the rate of decay of the quantity present at the time when t =. A). grams per year B) -. grams per year C) 6. grams per year D) -6. grams per year 47) Find the extreme values of the function on the interval and where they occur. Identify any critical points that are not stationary points. 48) g(x) = -x + 1x -, 4 x 8 48) A) Local maximum at 7, 4 ; minimum value is 0 at x = 8 and at x = 4 B) Local maximum at 6, 4 ; minimum value is 0 at x = 8 and at x = 4 C) Local maximum at 6, 4 ; minimum value is - at x = 0 D) Local maximum at 6, 68 ; minimum value is - at x = 0 Use the First Derivative Test to determine the local extrema of the function, and identify any absolute extrema. 49) y = 8x + 49) A) Absolute maximum at (0, ) B) Local minimum at (0, 0) C) None D) Local maximum at (0, 8) Use the Concavity Test to find the intervals where the graph of the function is concave up. 50) y = -x + 18x ) A) (-, ) B) (-, ) C) (, ) D) None Find all points of inflection of the function. 51) y = x - 1x + x ) A) (4, -105) B) (-4, -48) C) (4, -46) D) (4, -5) 11

12 Use the graph of f to estimate where f' is 0, positive, and negative. 5) 5) A) Zero: x = ±1; positive: x = (-, -1) and (1, ); negative: x = (-1, 1) B) Zero: x = ±1; positive: x = (-, -1); negative: x = (-1, 1) C) Zero: x = ±1; positive: x = (-, -1) and (1, ); negative: x = (0, 1) D) Zero: x = ±1; positive: x = (1, ); negative: x = (-1, 1) Use the graph of f to estimate where f'' is 0, positive, and negative. 5) 5) A) Zero: x = ±1; positive: (-, 0) ); negative: (0, ) B) Zero: x = 0; positive: (0, ); negative: (-, 0) C) Zero: x = 0; positive: (-, 0) ); negative: (0, ) D) Zero: x = ±1; positive: (0, ); negative: (-, 0) Use the Second Derivative Test to find the local extrema for the function. 54) y = x + x ) A) Local minimum: -, - 1 B) Local maximum: - 4, C) Local minimum:, D) Local maximum:, 1 4 1

13 55) Select an appropriate graph of a twice-differentiable function y = f(x) that passes through the points (-,1), - following sign patterns. 6,5 9, (0,0), 6,5 9 and (,1), and whose first two derivatives have the 55) y : y : A) B) C) D) 56) From a thin piece of cardboard 0 in. by 0 in., square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume? Round to the nearest tenth, if necessary. A) 10 in. 10 in. 10 in.; 1000 in. B) 15 in. 15 in. 7.5 in.; in. C) 0 in. 0 in. 10 in.; 4000 in. D) 0 in. 0 in. 5 in.; 000 in. 56) 57) Find the number of units that must be produced and sold in order to yield the maximum profit, given the following equations for revenue and cost: R(x) = 40x - 0.5x C(x) = 4x +. A) 8 units B) 7 units C) 44 units D) 6 units 57) 1

14 58) The stiffness of a rectangular beam is proportional to its width times the cube of its depth. Find the dimensions of the stiffest beam than can be cut from a 10-in.-diameter cylindrical log. (Round answers to the nearest tenth.) 58) 10" A) w = 6.0; d = 9.7 B) w = 6.0; d = 7.7 C) w = 4.0; d = 9.7 D) w = 5.0; d = ) Water is falling on a surface, wetting a circular area that is expanding at a rate of 5 mm/s. How fast is the radius of the wetted area expanding when the radius is 145 mm? (Round approximations to four decimal places.) A) 18.1 mm/s B) mm/s C) mm/s D) mm/s 59) 60) One airplane is approaching an airport from the north at 0 km/hr. A second airplane approaches from the east at 00 km/hr. Find the rate at which the distance between the planes changes when the southbound plane is km away from the airport and the westbound plane is 16 km from the airport. A) 714 km/hr B) 979 km/hr C) 96 km/hr D) 499 km/hr 60) 5 61) Suppose that f(x) dx = -4. Find f(x) dx and 5 f(x) dx. 61) 5 5 A) -4; 4 B) 0; -4 C) 0; 4 D) 5; -4 6) Suppose that h is continuous and that h(x) dx = 6 and h(x) dx = -9. Find - h(t) dt. - 7 A) -15; 15 B) -; C) ; - D) 15; h(t) dt and - 6) Interpret the integrand as the rate of change of a quantity and evaluate the integral using the antiderivative of the quantity. 6) 9 sin x dx 6) 0 A) 18 B) 9 C) 16 D) Evaluate the integral. 8 64) - 1 dx 64) x 1/ A) 16 - ln 16 B) 15 - ln 4 C) 15 - ln 0.5 D) 15 - ln 16 14

15 Find the total area of the region between the curve and the x-axis. 65) y = x - 6x + 9; x 4 65) A) 4 B) C) 7 D) 1 Find the area of the shaded region. 66) f(x) = x + x - 6x 66) g(x) = 6x A) 81 1 B) 4 1 C) 160 D) Find the area of the regions enclosed by the lines and curves. 67) y = x + 4, y = x ) A) 9 B) 9 C) 9 D) 18 15