MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Transcription

1 Eam 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. ) ) A) B) C) D)

2 ) ) A) B) C) D)

3 3) 3) A) B) C) D)

4 4) 4) A) B) C) D)

5 ) ) A) B) C) D)

6 6) 6) A) B) C) D) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 7) Find ds/dt when q = p/4 if s = sin q and dq/dt = 0. 7) Solve the problem. 8) Let Q() = bo + b( - a) + b( - a) be a quadratic approimation to f() at = a with the properties: i. Q(a) = f(a) ii. Q (a) = f (a) iii. Q (a) = f (a) (a) Find the quadratic approimation to f() = at = (b) Do ou epect the quadratic approimation to be more or less accurate than the linearization? Give reasons for our answer. 8) 6

7 Provide an appropriate response. 9) What is wrong with the following? = (3)() d d = (3 )(4) = 6 What is the correct derivative? 9) 0) Find d d ( 9/) b rewriting 9/ as 4 œ / and using the Product Rule. 0) ) Can a tangent line to a graph intersect the graph at more than one point? If not, wh not. If so, give an eample. ) ) Is there anthing special about the tangents to the curves = and - = at their point of intersection in the first quadrant? Eplain. ) Find a parametrization for the curve. 3) The line segment with endpoints (-4, 3) and (-, -6) 3) Provide an appropriate response. 4) Find the tangent to the curve = 4 cot p the curve can ever have on the interval - < < 0? at =. What is the largest value the slope of 4) ) If g() = f() + 3, find g (4) given that f (4) =. ) Find the derivatives of all orders of the function. 6) = ) Provide an appropriate response. 7) Find the derivative of = b using the Quotient Rule and b simplifing and 4 then using the Power Rule for Negative Integers. Show that our answers are equivalent. 7) 8) Find d998/d998 (sin ). 8) 9) Given that ( - 3) + = 9, find d two was: () b solving for and differentiating the d resulting functions with respect to and () b implicit differentiation. Show that the results are the same. 0) Suppose that u = g() is differentiable at =, = f(u) is differentiable at u = g(), and (f g) () is positive. What can be said about the values of g () and f (g())? Eplain. 9) 0) ) Rewrite tan and use the product rule to verif the derivative formula for tan. ) ) Graph = - tan and its derivative together on - p, p = - tan ever positive? Eplain. 7. Is the slope of the graph of )

8 3) Over what intervals of -values, if an, does the function = decrease as increases? For what values of, if an, is negative? How are our answers related? 3) Find a parametrization for the curve. 4) The upper half of the parabola + = 4) Provide an appropriate response. ) Find d/dt when = if = and d/dt = /. ) 6) Suppose that r is a differentiable function of s, s is a differentiable function of t, and t is a differentiable function of u. Write a formula for dr/du. 6) 7) Eplain wh the curve + = has no horizontal tangents. 7) 8) Find the derivative of = 3(3-4) b using the Product Rule and b using the Constant Multiple Rule. Show that our answers are equivalent. 8) 9) Find a value of c that will make f() = sin 4, 0 c, = 0 continuous at = 0. 30) Does the curve = ever have a negative slope? If so, where? Give reasons for our answer. 9) 30) 3) Which of the following could be true if f () = -/4? i) f () = 4-3/4 3) ii) f () = 4 3 3/4 + 3 iii) f() = 6 7/4 + iv) f() = 4 3 3/4 - Solve the problem. 3) For functions of the form = an, show that the relative uncertaint d variable is alwas n times the relative uncertaint d in the dependent in the independent variable. 3) Provide an appropriate response. 33) What is wrong with the following application of the chain rule? What is the correct derivative? d d ( - 3) 4 = 43( - 3) 33) 8

9 34) Over what intervals of -values, if an, does the function = increase as increases? For what values of, if an, is positive? How are our answers related? 3) Find d d 3 - our answers are equivalent. b using the Quotient Rule and b using the Product Rule. Show that 34) 3) 36) Find the derivative of = 3 b using the Quotient Rule and b using the Power Rule for 36) Negative Integers. Show that our answers are equivalent. 37) Which of the following could be true if f () = --/? i) f () = / 37) ii) f () = -3/ iii) f() = -/ iv) f() = - / 38) What is the range of values of the slope of the curve = 3 + -? 38) Find a parametrization for the curve. 39) The ra (half line) with initial point (-6, ) that passes through the point (-, -) 39) Provide an appropriate response. 40) Find d997/d997 (sin ). 40) 4) Is there an difference between finding the derivative of f() at = a and finding the slope of the line tangent to f() at = a? Eplain. 4) 4) Suppose that u = g() is differentiable at = and that = f(u) is differentiable at u = g(). If the tangent to the graph of = f(g()) at = is not horizontal, what can we conclude about the tangent to the graph of g at = and the tangent to the graph of f at u = g()? Eplain. 4) 43) Graph = - tan and its derivative together on - p, p. Does the graph of = -tan 43) appear to have a smallest slope? If so, what is it? If not, eplain. Solve the problem. 44) Consider the functions f() = and g() = 3 and their linearizations at the origin. Over some interval - e e, the approimation error for g() is less than the approimation error for f() for all within the interval. Derive a reasonable approimation for the value of e. Show our work. (Hint, the absolute value of the second derivative of each function gives a measure of how quickl the slopes of the function and its linear approimation are deviating from one another.) 44) 9

10 Provide an appropriate response. 4) Does the curve = have a tangent whose slope is -? If so, find an equation for the line and the point of tangenc. If not, wh not? 46) Find equations for the tangents to the curves = tan and = - tan (/) at the origin. How are the tangents related? 4) 46) 47) Suppose that the function v in the Quotient Rule has a constant value c. What does the Quotient Rule then sa? 47) 48) Assume = f(u) is a differentiable function of u and u = g() is a differentiable function of. If changes m times as fast as u and u changes n times as fast as, then changes how man times as fast as? Find the derivatives of all orders of the function. 7 49) = 0,080 48) 49) Provide an appropriate response. 0) Show that the derivative of =, 0, is = b writing = = and then using 0) the chain rule. ) Does the curve = ( + 3)3 have an horizontal tangents? If so, where? Give reasons for our answer. ) Find d d b using the Quotient Rule and b using the Power Rule for Negative Integers. Show that our answers are equivalent. ) ) 3) Graph = - tan and its derivative together on - p, p. Does the graph of = -tan 3) appear to have a largest slope? If so, what is it? If not, eplain. 4) Find the derivative of = ( - 3) b using the Product Rule and b rewriting and then using the Constant Multiple Rule. Show that our answers are equivalent. 4) ) If g() = - f() - 3, find g (4) given that f (4) =. ) 6) What is wrong with the following? = 3 d d = 3 4 = 3 What is the correct derivative? 7) Given 3 + = 0, find both d/d (treating as a differentiable function of ) and d/d (treating as a differentiable function of ). How are d/d and d/d related? 6) 7) 0

11 8) Find d998/d998 (cos ). 8) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use implicit differentiation to find d/d. 9) = sec(7) A) 7 sec(7) tan(7) B) 7 sec(7) tan(7) 9) C) cos(7) cot(7) D) 7 cos(7) cot(7) Suppose that the functions f and g and their derivatives with respect to have the following values at the given values of. Find the derivative with respect to of the given combination at the given value of. f() g() f () g () 60) ) /g(), = 4 A) B) 0 7 C) D) - 7 Solve the problem. 6) Suppose that the dollar cost of producing radios is c() = Find the marginal cost when 0 radios are produced. A) $60 B) $0 C) -$300 D) $300 6) The function f() changes value when changes from 0 to 0 + d. Find the approimation error f - df. Round our answer, if appropriate. 6) f() =, 0 = 3, d = 0.6 6) A) B) C) D) Find the value of d/d at the point defined b the given value of t. 63) = t + 3, = - t, t = 3 A) B) - C) - 4 D) 8 63)

12 The graphs show the position s, velocit v = ds/dt, and acceleration a = ds/dt of a bod moving along a coordinate line as functions of time t. Which graph is which? 64) C A B 64) t A) B = position, C = velocit, A = acceleration B) A = position, B = velocit, C = acceleration C) A = position, C = velocit, B = acceleration D) C = position, A = velocit, B = acceleration Find the value of d/d at the point defined b the given value of t. 6) = csc t, = 3 cot t, t = p 3 6) A) -9 3 B) -6 3 C) 3 3 D) 6 3 Given = f(u) and u = g(), find d/d = f (g())g (). 66) = u, u = - 3 A) 30 - B) - C) 0 D) ) Find the limit. 67) lim 7 cos ) A) B) 0 C) - D) Find. 68) = ) A) B) C) D) The equation gives the position s = f(t) of a bod moving on a coordinate line (s in meters, t in seconds). 69) s = cos t Find the bod's velocit at time t = p/3 sec. A) 9 3 m/sec B) m/sec C) 9 m/sec D) - 9 m/sec 69)

13 The figure shows the velocit v or position s of a bod moving along a coordinate line as a function of time t. Use the figure to answer the question. 70) v (ft/sec) t (sec) 70) When does the bod reverse direction? A) t = 4, t = 7 sec B) t = 4 sec C) t =, t = 3, t =, t = 6, t = 7 sec D) t = 7 sec Suppose u and v are differentiable functions of. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 7) u() =, u () = -, v() = 6, v () = -4. d (3v - u) at = d A) -7 B) -7 C) 3 D) 3 7) Use implicit differentiation to find d/d. 7) = 8 A) + + B) C) D) ) Find an equation for the line tangent to the curve at the point defined b the given value of t. 73) = t + cos t, = - sin t, t = p 6 73) A) = - - C) = 3-4 p + B) = p D) = p + 3 Solve the problem. Round our answer, if appropriate. 74) The radius of a right circular clinder is increasing at the rate of 6 in./sec, while the height is decreasing at the rate of 8 in./sec. At what rate is the volume of the clinder changing when the radius is 3 in. and the height is in.? A) -46p in.3/sec B) 0p in.3/sec C) -46 in.3/sec D) -40 in.3/sec 74) 3

14 The figure shows the velocit v or position s of a bod moving along a coordinate line as a function of time t. Use the figure to answer the question. 7) s (m) 7) 4 3 t (sec) When is the bod standing still? A) < t < 3, 4 < t <, 7 < t < 9 B) < t < C) t = 3, t =, t = 8 D) 8 < t 0 Write the function in the form = f(u) and u = g(). Then find d/d as a function of. 76) = ) A) = u0; u = ; d d = B) = 6u - 3 u - u; u = 0; d d = C) = u0; u = 6-3 D) = u0; u = ; d d = ; d d = Find the derivative of the function. 3 77) = - 77) A) = ( - ) B) = ( - ) C) = ( - ) D) = 3-3 ( - ) Solve the problem. 7 78) Find an equation for the tangent to the curve = at the point (, 9). + A) = -3 + B) = -6 + C) = D) = -6 78) 4

15 Suppose that the functions f and g and their derivatives with respect to have the following values at the given values of. Find the derivative with respect to of the given combination at the given value of. f() g() f () g () 79) ) f() œ g(), = 3 A) B) 79 C) D) Find the value of d/d at the point defined b the given value of t. 80) = 6t - 3, = t, t = 80) A) 4 B) - 48 C) 48 D) - 4 Given the graph of f, find an values of at which f is not defined. 8) 8) A) = -, B) = -, 0, C) = 0 D) Defined for all values of Solve the problem. 8) The graph of = f() in the accompaning figure is made of line segments joined end to end. Graph the derivative of f. 8) (3, ) (6, ) (-3, ) (-, 0) (0, -)

16 A) B) C) D) ) A charged particle of mass m and charge q moving in an electric field E has an acceleration a given b a = qe m, 83) where q and E are constants. Find d a dm. A) d a dm = qe m B) d a dm = qe m3 C) d a dm = qe m3 D) d a dm = - qe m 84) Estimate the volume of material in a clindrical shell with height 30 in., radius 7 in., and shell thickness 0.6 in. (Use 3.4 for p.) A) 79.3 in.3 B) 80.3 in.3 C) 38.8 in.3 D) 39.6 in.3 84) The equation gives the position s = f(t) of a bod moving on a coordinate line (s in meters, t in seconds). 8) s = + cos t Find the bod's jerk at time t = p/3 sec. A) - 3 m/sec3 B) - m/sec 3 C) m/sec 3 D) 3 m/sec3 8) 6

17 Find. 86) = ) A) C) B) D) Find the value of (f g) at the given value of. 87) f(u) = u, u = g() = 6 -, = 87) A) 4 B) - 4 C) 4 D) - 4 Use implicit differentiation to find d/d. 88) + + = A) B) C) + - D) ) Given = f(u) and u = g(), find d/d = f (g())g (). 89) = tan u, u = A) sec(- + 6) B) - sec(- + 6) C) - sec (- + 6) tan (- + 6) D) - sec(- + 6) 89) Use implicit differentiation to find d/d and d/d. 90) - + = 8 A) d d = - + ; d d = - ( + ) C) d d = ; d d = + ( + ) B) d d = ; d d = - ( + ) D) d d = + + ; d d = + ( + ) 90) Solve the problem. 9) The line that is normal to the curve - + = 9 at (3, 3) intersects the curve at what other point? A) (0, -3) B) (-6, -6) C) (-3, 0) D) (-3, -3) 9) 7

18 Parametric equations and and a parameter interval for the motion of a particle in the -plane are given. Identif the particle's path b finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced b the particle and the direction of motion. 9) = 4 sin t, = 3 cos t, 0 t p A) = ) Counterclockwise from (3, 0) to (3, 0), one rotation B) = Counterclockwise from (0, 4) to (0, 4), one rotation 8

19 C) = Counterclockwise from (4, 0) to (4, 0), one rotation D) = Counterclockwise from (0, 3) to (0, 3), one rotation 9

20 The figure shows the graph of a function. At the given value of, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? 93) = 93) A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable Find the derivative. 94) r = 3 s3-9 s 94) A) 3 s4-9 s B) - 9 s + 9 s C) - 9 s4 + 9 s D) 9 s4-9 s The graphs show the position s, velocit v = ds/dt, and acceleration a = ds/dt of a bod moving along a coordinate line as functions of time t. Which graph is which? 9) 9) t B A C A) A = position, B = velocit, C = acceleration B) A = position, C = velocit, B = acceleration C) B = position, A = velocit, C = acceleration D) C = position, A = velocit, B = acceleration Find the indicated derivative. 96) Find if = 8 sin. A) = - 6 cos + 8 sin B) = - 8 sin C) = 8 cos - 6 sin D) = 6 cos - 8 sin 96) 0

21 Given the graph of f, find an values of at which f is not defined. 97) 97) A) = B) = 0 C) = -, D) = -, 0, The figure shows the graph of a function. At the given value of, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? 98) = - 98) A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable Use implicit differentiation to find d/d. 99) + = A) - + B) - + C) + D) + 99) Find an equation for the line tangent to the curve at the point defined b the given value of t. 00) = sin t, = 3 sin t, t = p 3 00) A) = B) = C) = D) = 3 Find the linearization L() of f() at = a. 0) f() = sin, a = 0 A) L() = B) L() = - C) L() = 0 D) L() = 3 + 0)

22 Find the derivative of the function. 0) = A) d d = C) d d = B) d d = D) d d = ) The figure shows the velocit v or position s of a bod moving along a coordinate line as a function of time t. Use the figure to answer the question. 03) s (m) 03) 4 3 t (sec) When is the bod moving backward? A) < t < 3, 4 < t <, 7 < t < 9 B) 8 < t < 9 C) 8 < t 0 D) < t < 3, 4 < t <, 7 < t < 8 Use implicit differentiation to find d/d. 04) = cot A) - 4 csc cot B) - 4 csc C) csc 4 D) 4 csc 04) Find. 0) = tan(-8 - ) 0) A) sec(-8 - ) B) sec (-8 - ) tan(-8 - ) C) - 8 sec (-8 - ) D) 8 sec (-8 - ) tan(-8 - ) Find the derivative of the function. 06) = 8 A) d d = 8()7/8 C) d d = 8()7/8 B) d d = - 8()9/8 D) d d = ()7/8 06)

23 Suppose u and v are differentiable functions of. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 07) u() =, u () = -6, v() = 7, v () = -4. d (uv) at = d A) 0 B) -0 C) 34 D) -40 Use implicit differentiation to find d/d. 07) 08) cos = ) A) 4 sin - 4 B) 4 sin + cos - 4 C) 4 sin + cos - 4 D) 4 - sin cos - 4 Solve the problem. 09) At the two points where the curve + + = crosses the -ais, the tangents to the curve are parallel. What is the common slope of these tangents? A) 3 B) C) - D) - 09) Find the derivative. 0) s = t4 cos t - t sin t - cos t A) ds dt = - t 4 sin t + 4t3 cos t - t cos t - sin t B) ds dt = - 4t 3 sin t - cos t + sin t C) ds dt = t 4 sin t - 4t3 cos t + t cos t D) ds dt = - t 4 sin t + 4t3 cos t - t cos t 0) Find the derivative of the function. ) r = (sec q + tan q)-6 A) dr dq = -6 sec q (sec q + tan q)6 B) dr dq = -6(sec q + tan q) -7(tan q + sec q tan q) C) dr dq = -6(sec q tan q + sec q) -7 D) dr dq = -6(sec q + tan q) -7 ) 3

24 Assuming that the equations define and implicitl as differentiable functions = f(t), = g(t), find the slope of the curve = f(t), = g(t) at the given value of t. ) (t + ) - 4t = 36, + 43/ = t3 + t, t = 0 ) A) -4 B) - C) - D) - 4 Solve the problem. 3) A = pr, where r is the radius, in centimeters. B approimatel how much does the area of a circle decrease when the radius is decreased from.0 cm to 4.8 cm? (Use 3.4 for p.) A) 6. cm B) 3. cm C) 6. cm D) 6.3 cm 3) The figure shows the velocit v or position s of a bod moving along a coordinate line as a function of time t. Use the figure to answer the question. 4) v (ft/sec) 4) 4 3 t (sec) When is the bod's acceleration equal to zero? A) t =, t = 3, t =, t = 6 B) t = 0, t = 4, t = 7 C) < t < 3, < t < 6 D) 0 < t <, 6 < t < 7 4

25 Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given. ) ) (, ) = = A) Since lim + f () = while lim - f () =, f() is not differentiable at =. B) Since lim + f () = while lim - f () =, f() is not differentiable at =. C) Since lim + f () = while lim - f () =, f() is not differentiable at =. D) Since lim + f () = while lim - f () =, f() is differentiable at =. Find d. 6) = A) d B) 9-4 (9 + 7)3/ d C) (9 + 7)3/ d D) d 6) Calculate the derivative of the function. Then find the value of the derivative as specified. 8 7) f() = + ; f (0) 7) A) f () = 8; f (0) = 8 B) f () = - 8( + ); f (0) = - 3 C) f () = - 8 ( + ) ; f (0) = - D) f () = 8 ( + ) ; f (0) = Solve the problem. 8) If a and b are the lengths of the legs of a right triangle and c is the length of the hpotenuse, c = a + b. How is dc/dt related to da/dt and db/dt? A) dc dt = a da dt + b db B) dc dt dt = a da dt + bdb dt C) dc dt = ada db + b dt dt D) dc dt = c a da dt + bdb dt 8)

26 Solve the problem. Round our answer, if appropriate. 9) The volume of a rectangular bo with a square base remains constant at 400 cm3 as the area of the base increases at a rate of cm/sec. Find the rate at which the height of the bo is decreasing when each side of the base is 9 cm long. (Do not round our answer.) A) cm/sec B) cm/sec C) cm/sec D) cm/sec 9) Find an equation of the tangent line at the indicated point on the graph of the function. 0) = f() = -, (, ) = (-, -6) A) = B) = + 4 C) = D) = -3-4 Solve the problem. 0 ) Find an equation for the tangent to the curve = at the point (, ). + A) = B) = 0 C) = D) = + Find the derivative of the function. ) g() = (6 + 4) /3 A) g () = C) g () = (6 + 4) /3 B) g () = (6 + 4) /3 36 (6 + 4) /3 D) g () = (6 + 4) /3 0) ) ) Solve the problem. 3) Find the slope of the curve 3 - = -4 at (-, ). 3) A) B) 3 C) - 6 D) - 3 4) Find the tangent to = cot at = p 4. 4) A) = + B) = - + p + C) = - + p D) = - p + ) Find the points on the curve + = + where the tangent is parallel to the -ais. A) (, ), (, -) B) (, 0), (, ) C) (, + ), (, - ) D) (, + ), (, - ) Find an equation of the tangent line at the indicated point on the graph of the function. 6) = f() = 3, (, ) = (8, 6) A) = + 96 B) = 3 + C) = 3 - D) = 96 - ) 6) 6

27 Parametric equations and and a parameter interval for the motion of a particle in the -plane are given. Identif the particle's path b finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced b the particle and the direction of motion. 7) = 36t, = 6t, -«t « A) = ) Entire parabola, left to right (from second quadrant to origin to first quadrant) B) = Entire parabola, top to bottom (from first quadrant to origin to fourth quadrant) 7

28 C) = Entire parabola, right to left (from first quadrant to origin to second quadrant) D) = Entire parabola, bottom to top (from fourth quadrant to origin to first quadrant) At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 8) = +, slope at (0, ) A) B) 3 C) 3 D) - 3 8) Solve the problem. 9) The size of a population of lions after t months is P = 00 ( + 0.t + 0.0t). Find the growth rate when P = 00. A) 80 lions/month B) 60 lions/month C) 0,00 lions/month D) 40 lions/month 9) At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 30) 44 = 6, slope at (, ) 30) A) B) - 4 C) -8 D) - 8

29 Find the second derivative of the function. 3) = ( - 9)( + 3) 3 3) A) d d = C) d d = B) d d = D) d d = Given the graph of f, find an values of at which f is not defined. 3) 3) A) = B) =, C) = D) Defined for all values of Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given. 33) 33) (-, -) = = - A) Since lim - + f () = - while lim - - f () = 0, f() is not differentiable at = -. B) Since lim - + f () = 0 while lim - - f () = -, f() is not differentiable at = -. C) Since lim - + f () = 0 while lim - - f () =, f() is not differentiable at = -. D) Since lim - + f () = 0 while lim - - f () = 0, f() is differentiable at = -. 9

30 Solve the problem. 34) A ball dropped from the top of a building has a height of s = 400-6t meters after t seconds. How long does it take the ball to reach the ground? What is the ball's velocit at the moment of impact? A) sec, -800 m/sec B) sec, 60 m/sec C) sec, -60 m/sec D) 0 sec, -80 m/sec 34) Find the derivative of the function. 3) = ( - 6)( + ) 3 A) d d = C) d 4 = 4 + d B) d d = D) d d = ) Solve the problem. 36) Use the following information to graph the function f over the closed interval [-, 6]. i) The graph of f is made of closed line segments joined end to end. ii) The graph starts at the point (-, ). iii) The derivative of f is the step function in the figure shown here. 36) A) B) (-3, ) (3, ) (-3, ) (3, 6) (-, ) (0, ) (-, ) (0, ) (6, -) (6, 0) 30

31 C) D) (-3, ) (3, ) (-3, 6) (3, ) (-, ) (0, ) (-, ) (0, ) (6, -) (6, 0) Find the derivative. 37) = A) B) C) D) ) Find d. 38) = cos(7 ) A) C) 7 sin(7 ) d B) -7 sin(7 ) d D) -7 sin(7 ) 7 sin(7 ) d d 38) Find the second derivative. 39) = A) 8-4 B) 84-4 C) 8-4 D) 84-4 Solve the problem. Round our answer, if appropriate. 40) Water is discharged from a pipeline at a velocit v (in ft/sec) given b v = 40p(/), where p is the pressure (in psi). If the water pressure is changing at a rate of psi/sec, find the acceleration (dv/dt) of the water when p = 33.0 psi. A) 08 ft/sec B) 40 ft/sec C) 3.6 ft/sec D) 43.8 ft/sec Find the second derivative of the function. 4) = ) 40) 4) A) d d = C) d d = B) d d = D) d d =

32 The graphs show the position s, velocit v = ds/dt, and acceleration a = ds/dt of a bod moving along a coordinate line as functions of time t. Which graph is which? 4) B 4) A C t A) B = position, A = velocit, C = acceleration B) A = position, B = velocit, C = acceleration C) C = position, A = velocit, B = acceleration D) A = position, C = velocit, B = acceleration Find the derivative of the function. 43) = 8 + A) d d = C) d d = + B) d d = - 4 ( + )3/ D) d d = ) Find. 44) = (3 + 6)(7-3) A) B) C) D) ) 3

33 The figure shows the velocit v or position s of a bod moving along a coordinate line as a function of time t. Use the figure to answer the question. 4) v (ft/sec) 4) t (sec) What is the bod's acceleration when t = 8 sec? A) - ft/sec B) ft/sec C).67 ft/sec D) 0 ft/sec Assuming that the equations define and implicitl as differentiable functions = f(t), = g(t), find the slope of the curve = f(t), = g(t) at the given value of t. 46) + 43/ = t3 + t, (t + ) - 4t =, t = 0 A) - B) 0 C) - D) -0 46) Find the linearization L() of f() at = a. 47) f() = - 4 +, a = 3 A) L() = B) L() = C) L() = D) L() = 6-44 Calculate the derivative of the function. Then find the value of the derivative as specified. 48) f() = + 9; f () Solve the problem. A) f () = 0; f () = 0 B) f () = 9; f () = 9 C) f () = ; f () = 0 D) f () = ; f () = 49) The elasticit e of a particular thermoplastic can be modeled approimatel b the relation e =.0, where T is the Kelvin temperature. If the thermometer used to measure T is accurate to T.3 %, and if the measured temperature is 480 K, how should the elasticit be reported? A) e = 0.70 ± B) e = 0.70 C) e = 0.70 ± 0.00 D) e = ± ) The range R of a projectile is related to the initial velocit v and projection angle q b the equation R = v sin q, where g is a constant. How is dr/dt related to dq/dt if v is constant? g A) dr dt = - v cos q dq g dt C) dr dt = v cos q dq g dt B) dr dt = v sin q dq g dt D) dr dt = v cos q dq g dt 47) 48) 49) 0) 33

34 Find the linearization L() of f() at = a. ) f() = +, a = 4 ) A) L() = 6 + B) L() = C) L() = 6 + D) L() = The graphs show the position s, velocit v = ds/dt, and acceleration a = ds/dt of a bod moving along a coordinate line as functions of time t. Which graph is which? ) ) A t C B A) C = position, B = velocit, A = acceleration B) B = position, C = velocit, A = acceleration C) A = position, C = velocit, B = acceleration D) B = position, A = velocit, C = acceleration Find the derivative of the function. 3) = / A) d d = / B) d d = 7/ C) d d = 7/ D) d d = -7/ 3) The graphs show the position s, velocit v = ds/dt, and acceleration a = ds/dt of a bod moving along a coordinate line as functions of time t. Which graph is which? 4) 4) B C A t A) C = position, A = velocit, B = acceleration B) A = position, B = velocit, C = acceleration C) A = position, C = velocit, B = acceleration D) B = position, A = velocit, C = acceleration 34

35 Use implicit differentiation to find d/d and d/d. ) + = 9 A) d d = - ; d d = C) d d = - ; d d = B) d d = - ; d d = - + D) d d = ; d d = - ) Solve the problem. 6) The range R of a projectile is related to the initial velocit v and projection angle q b the equation R = v sin q, where g is a constant. How is dr/dt related to dv/dt and dq/dt if neither v nor q is g constant? A) dr dt = v dq dv v cos q + sin q g dt dt C) dr dt = dv dq v cos q + sin q g dt dt B) dr dt = dq dv 4v cos q g dt dt D) dr dt = v g v cos q dq dt + sin q dv dt 6) The function f() changes value when changes from 0 to 0 + d. Find the approimation error f - df. Round our answer, if appropriate. 7) f() = -, 0 =, d = ) A) 0.07 B) C) D) 0.44 Find an equation of the tangent line at the indicated point on the graph of the function. 8) w = g(z) = z - 4, (z, w) = (-3, ) A) w = -6z - 6 B) w = -6z - C) w = -6z - 3 D) w = -3z - 3 The equation gives the position s = f(t) of a bod moving on a coordinate line (s in meters, t in seconds). 9) s = 6 sin t - cos t Find the bod's velocit at time t = p/4 sec. A) - m/sec B) 7 m/sec C) m/sec D) - 7 m/sec 8) 9) Find. 60) = ( - + )( ) A) B) C) D) ) Calculate the derivative of the function. Then find the value of the derivative as specified. 6) f() = 8 ; f (-) 6) A) f () = - 8; f (-) = - 8 B) f () = - 8 ; f (-) = -8 C) f () = 8 ; f (-) = 8 D) f () = 8; f (-) = 8 3

36 At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 6) = + 0, normal at (0, ) A) = 3 + B) = - + C) = - 3 D) = + 6) The equation gives the position s = f(t) of a bod moving on a coordinate line (s in meters, t in seconds). 63) s = 8 + cos t Find the bod's acceleration at time t = p/3 sec. A) m/sec B) 3 m/sec C) - 3 m/sec D) - m/sec 63) Solve the problem. 64) V = 4 3 pr 3, where r is the radius, in centimeters. B approimatel how much does the volume of a 64) sphere increase when the radius is increased from.0 cm to. cm? (Use 3.4 for p.) A). cm3 B).0 cm3 C) 4.8 cm3 D) 0.3 cm3 Find the second derivative. 6) = A) 9 B) 9 C) 9-8 D) 9 6 6) Solve the problem. 66) The position (in centimeters) of an object oscillating up and down at the end of a spring is given b s = A sin k t at time t (in seconds). The value of A is the amplitude of the motion, k is a measure m of the stiffness of the spring, and m is the mass of the object. Find the object's acceleration at time t. A) a = - A sin C) a = - Ak m sin k m t cm/sec B) a = - A k m sin k m t cm/sec k m t cm/sec D) a = Ak m cos k m t cm/sec 66) Find the linearization L() of f() at = a. 67) f() = 6 + 8, a = 0 A) L() = 3-9 B) L() = C) L() = 3-9 D) L() = ) Solve the problem. 68) The curve = a + b + c passes through the point (, 8) and is tangent to the line = 4 at the origin. Find a, b, and c. A) a = 6, b = 0, c = 0 B) a = 0, b =, c = 4 C) a = 4, b = 0, c = D) a =, b = 4, c = 0 Find the derivative of the function. 69) = ( + sin 7t)-4 A) = - 8( + sin 7t)- cos 7t B) = - 4( + sin 7t)- cos 7t C) = - 8(cos 7t)- D) = - 4( + sin 7t)- 68) 69) 36

37 Find an equation for the line tangent to the curve at the point defined b the given value of t. 70) = 9t - 6, = t, t = 70) A) = B) = 9-3 C) = 9 + D) = 9-3 Find the derivative of the function. 7) s = 9 t-4 A) ds dt = t -3/9 B) ds dt = 4 9 t 3/9 C) ds dt = t 3/9 D) ds dt = t -3/9 7) Find the second derivative of the function. 7) r = + 3q (3 - q) 3q 7) A) d r dq = - q3 - B) d r dq = - q - C) d r dq = q - q D) d r dq = q3 Find the derivative. 73) s = t - csc t + 8 A) ds dt = t 4 + cott B) ds dt = t 4 + csc t cot t 73) C) ds dt = t 4 - csc t cot t D) ds dt = t 4 - cott ) w = z- - z 74) A) -z-6 + z B) z-6 + z C) -z-6 - z D) z-6 - z Find. 7) = ( - 9) -3 A) 6 ( - 9) - B) - 3 ( - 9)- C) 3 4 ( - 9) D) - 3 ( - 9) ) Given = f(u) and u = g(), find d/d = f (g())g (). 76) = sin u, u = cos Solve the problem. A) cos sin B) - cos sin C) - cos(cos ) sin D) sin(cos ) sin 77) The concentration of a certain drug in the bloodstream hr after being administered is 6 approimatel C() =. Use the differential to approimate the change in concentration as 3 + changes from to.. A) 0.04 B) 0.47 C) 0.6 D) ) 77) 37

38 Find the value of (f g) at the given value of. 78) f(u) = sin pu + u, u = g() = -, = 6 Solve the problem. A) B) 0 C) - D) -6 79) Suppose that the radius r and the circumference C = pr of a circle are differentiable functions of t. Write an equation that relates dc/dt to dr/dt. A) dc dt = pr dr dt B) dc dt = dr dt C) dc dt = p dr dt D) dr dt = p dc dt 78) 79) The function s = f(t) gives the position of a bod moving on a coordinate line, with s in meters and t in seconds. 80) s = - t3 + 4t - 4t, 0 t 4 Find the bod's speed and acceleration at the end of the time interval. A) 4 m/sec, 0 m/sec B) 0 m/sec, -6 m/sec C) -0 m/sec, -6 m/sec D) 0 m/sec, -4 m/sec 80) Solve the problem. 8) Does the graph of the function = sin have an horizontal tangents in the interval 0 p? If so, where? A) Yes, at = p 3, = p B) No 3 C) Yes, at = p 3, = 4p 3 D) Yes, at = p 3 8) Find the value of d/d at the point defined b the given value of t. 8) = tan t, = 3 sec t, t = 3p 4 A) 3 B) - 3 C) 4 D) ) The function s = f(t) gives the position of a bod moving on a coordinate line, with s in meters and t in seconds. 83) s = 3t + 4t + 0, 0 t Find the bod's speed and acceleration at the end of the time interval. A) 6 m/sec, 6 m/sec B) 6 m/sec, m/sec C) 6 m/sec, 6 m/sec D) 0 m/sec, m/sec 83) Find the limit. 84) lim p/3 3 + sin(p sec ) 84) A) 0 B) 3 + C) D) 3 Find the value of d/d at the point defined b the given value of t. 8) = t, = t, t = 7 A) - B) - C) D) - 4 8) 38

39 Suppose u and v are differentiable functions of. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 86) u() = 9, u () = 3, v() = -, v () = -. d (3v - u) at = d A) -8 B) - C) 3 D) - 86) Find the derivative of the function. ( + )( + ) 87) = ( - )( - ) A) = C) = 4-40 ( - )( - ) ( - )( - ) B) = D) = 4-40 ( - )( - ) ( - )( - ) 87) Use implicit differentiation to find d/d and d/d. 88) + 3 =, at the point (4, -) A) d d = 3 ; d d = 9 C) d d = 3 ; d d = - 9 B) d d = - 3 ; d d = 0 D) d d = 3; d d = ) Find the derivative of the function. 89) = A) = / C) = + 8 B) = D) = + 8 3/ 89) Calculate the derivative of the function. Then find the value of the derivative as specified. 90) ds dt t =-3 if s =t - t 90) A) ds dt C) ds dt = - t; ds dt t =-3 = = t - ; ds dt t =-3 = -7 ds ds B) = t - ; dt dt t =-3 = -4 ds ds D) = t + ; dt dt t =-3 = - Find the value of (f g) at the given value of. 9) f(u) = tan pu, u = g() =, = 9) A) -4p B) -p C) p D) 4 39

40 Solve the problem. 9) Does the graph of the function = tan - have an horizontal tangents in the interval 0 p? If so, where? 9) A) Yes, at = p B) No C) Yes, at = 0, = p, = p D) Yes, at = p, = 3p Find the slope of the tangent line at the given value of the independent variable. 93) f() = 3 + 9, = 4 93) A) 39 4 B) 7 6 C) 7 4 D) 39 6 Solve the problem. 94) A heat engine is a device that converts thermal energ into other forms. The thermal efficienc, e, of a heat engine is defined b e = Q h - Qc, Qh where Qh is the heat absorbed in one ccle and Qc, the heat released into a reservoir in one ccle, is a constant. Find de dqh. 94) A) de dqh = Q c Qh 3 B) de dqh = Q c Qh C) de dqh = - Q c Qh D) de dqh = - Q c Qh 3 Solve the problem. Round our answer, if appropriate. 9) As the zoom lens in a camera moves in and out, the size of the rectangular image changes. Assume that the current image is 6 cm cm. Find the rate at which the area of the image is changing (da/df) if the length of the image is changing at 0. cm/s and the width of the image is changing at 0. cm/s. A) 6. cm/sec B) 3. cm/sec C) 3. cm/sec D) 7.0 cm/sec 9) Calculate the derivative of the function. Then find the value of the derivative as specified. 96) g() = - ; g (-) 96) A) g () = - ; g (- ) = - B) g () = ; g (-) = C) g () = - ; g (- ) = - 8 D) g () = - ; g (- ) = - Solve the problem. 97) The area A = pr of a circular oil spill changes with the radius. At what rate does the area change with respect to the radius when r = 3 ft? A) 6p ft/ft B) 3p ft/ft C) 6 ft/ft D) 9p ft/ft 97) 40

41 98) The position of a particle moving along a coordinate line is s = + t, with s in meters and t in seconds. Find the particle's velocit at t = sec. 98) A) m/sec B) 4 m/sec C) - m/sec D) m/sec Find. 99) = A) B) C) D) ) Solve the problem. 00) The size of a population of mice after t months is P = 00( + 0.t + 0.0t). Find the growth rate at t = 4 months. A) mice/month B) 76 mice/month C) 38 mice/month D) 76 mice/month 00) Calculate the derivative of the function. Then find the value of the derivative as specified. 0) dr dq q = if r = 6 - q 0) A) dr dq = (6 - q)3/ ; dr dq q = = 4 C) dr dq = - (6 - q)3/ ; dr dq q = = - 8 B) dr dq = (6 - q)3/ ; dr dq q = = 8 D) dr dq = - (6 - q)3/ ; dr dq q = = - 4 The function f() changes value when changes from 0 to 0 + d. Find the approimation error f - df. Round our answer, if appropriate. 0) f() = 3, 0 =, d = 0.0 0) A) B) C) D) At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 03) 66 = 64, normal at (, ) A) = 3 B) = - + C) = - 3 D) = ) Find d. 04) = csc( - ) Solve the problem. A) -0 csc(0) cot(0) d B) 0 csc( - ) cot( - )d C) -0 csc( - ) cot( - ) d D) - csc( - ) cot( - ) d 0) At time t, the position of a bod moving along the s-ais is s = t3-8t + 60t m. Find the total distance traveled b the bod from t = 0 to t = 3. A) 0 m B) 49 m C) 0 m D) 4 m 04) 0) 4

42 06) The position (in centimeters) of an object oscillating up and down at the end of a spring is given b s = A sin k t at time t (in seconds). The value of A is the amplitude of the motion, k is a measure m of the stiffness of the spring, and m is the mass of the object. How fast is the object accelerating when it is accelerating the fastest? A) A cm/sec B) Ak m cm/sec C) A cm/sec D) A k m cm/sec 06) Find the derivative. 07) s = t3 tan t - t A) ds dt = 3t sec t - t C) ds dt = - t 3 sec t + 3t tan t + t B) ds dt = t 3 sec t tan t + 3t tan t - D) ds dt = t 3 sec t + 3t tan t - t t 07) Use implicit differentiation to find d/d and d/d. 08) + =, at the point (, ) A) d d = - ; d d = - C) d d = ; d d = d B) d = - ; d d = d D) d = - ; d d = 0 08) 09) 4 - = A) d d = - ; d d = 4 ( - ) 3 B) d d = - ; d d = - 09) C) d d = - ; d d = - 4 D) d d = - ; d d = - ( - ) Find the limit. p 0) lim cos sin + p cot -p/ 4 csc + 0) A) 0 B) C) - D) Solve the problem. ) A piece of land is shaped like a right triangle. Two people start at the right angle of the triangle at the same time, and walk at the same speed along different legs of the triangle. If the area formed b the positions of the two people and their starting point (the right angle) is changing at m/s, then how fast are the people moving when the are 3 m from the right angle? (Round our answer to two decimal places.) A) 0.83 m/s B) 3.33 m/s C).80 m/s D).67 m/s ) 4

43 Find the derivative of the function. ) s = sin 7pt - cos 7pt A) ds dt = 7p 7pt cos C) ds dt = 7p 7pt cos - 7p + 7p sin 7pt sin 7pt B) ds 7pt = cos dt D) ds dt = - 7p 7pt cos + sin 7pt - 7p sin 7pt ) Find the value of (f g) at the given value of. 3) f(u) = - u, u = g() = p, = cos3 u A) -p B) 3 - p C) p D) -p Find the derivative of the function. 4) f(t) = (4 - t)(4 + t3) - A) f (t) = t 3 - t t3 B) f (t) = - t 3 + t - 4 (4 + t3) C) f (t) = t 3 - t - 4 (4 + t3) D) f (t) = - 4t 3 + t - 4 (4 + t3) 3) 4) ) p = q q q7 + 6 q A) dp dq = q 4 + 8q8 + 4q7-4 q3 C) dp dq = q 0 + 0q4 + q3-4 q3 B) dp dq = q 0 + q4 + 3q3 + 4 q3 D) dp dq = q 0-4 q3 ) Find the slope of the tangent line at the given value of the independent variable. 6) s = 4t4 + 3t3, t = - A) - B) 7 C) -7 D) 6) Use implicit differentiation to find d/d and d/d. 7) - = 3 A) d d = ; d d = - 3 C) d d = ; d d = - B) d d = - ; d d = - 3 D) d d = ; d d = - 7) Solve the problem. 8) Suppose that the radius r and volume V = 4 3 pr 3 of a sphere are differentiable functions of t. Write 8) an equation that relates dv/dt to dr/dt. A) dv dt = 4 3 pr dr B) dv dt dt = 4pr dr dt C) dv dt = 3r dr dt D) dv dt = 4p dr dt 43

44 Find the second derivative of the function. 9) s = t 7 + 6t + 6 t A) d s dt = 0t + t + 36 t C) d s dt = t 4-6 t - t3 B) d s dt = t 3-6 t3 - t4 D) d s dt = 0t 3 + t t4 9) The function s = f(t) gives the position of a bod moving on a coordinate line, with s in meters and t in seconds. 0) s = t - t, 0 t Find the bod's speed and acceleration at the end of the time interval. A) m/sec, -4 m/sec B) - m/sec, - m/sec C) 6 m/sec, -4 m/sec D) m/sec, - m/sec 0) Solve the problem. ) Suppose that the dollar cost of producing radios is c() = Find the average cost per radio of producing the first 4 radios. A) $9.00 B) $3.00 C) $43.00 D) $39.00 ) At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. ) - p cos = 6p, normal at (, p) A) = p - p + p B) = - p + p + p C) = p - + p D) = -p + 3p p ) The figure shows the velocit v or position s of a bod moving along a coordinate line as a function of time t. Use the figure to answer the question. 3) v (ft/sec) 3) 4 3 t (sec) When is the bod moving backward? A) 0 < t < 4 B) 0 < t <, 6 < t < 7 C) 7 < t < 0 D) 4 < t < 7 44

45 Find. 4) = A) B) - 9 4(3 + 3)3/ C) D) - 4(3 + 3)3/ 4) Find the value of (f g) at the given value of. ) f(u) = u, u = g() = +, = - Solve the problem. A) B) C) 0 D) -30 6) A cube 7 inches on an edge is given a protective coating 0.3 inches thick. About how much coating should a production manager order for 00 cubes? A) About 730 in. B) About,40 in.3 C) About,00 in. D) About 44,00 in.3 ) 6) Suppose u and v are differentiable functions of. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 7) u() =, u () = -, v() = 7, v () = -3. d (u - 4v) at = d A) -8 B) 38 C) - D) The function f() changes value when changes from 0 to 0 + d. Find the approimation error f - df. Round our answer, if appropriate. 8) f() =, 0 = 9, d = 0.0 A) 0.00 B) 0.00 C) 0.8 D) 0.00 At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 9) 33 = 8, tangent at (, ) A) = B) = 4 - C) = - + D) = 4 + 9) 7) 8) Solve the problem. Round our answer, if appropriate. 30) Bole's law states that if the temperature of a gas remains constant, then PV = c, where P = pressure, V = volume, and c is a constant. Given a quantit of gas at constant temperature, if V is decreasing at a rate of 9 in. 3/sec, at what rate is P increasing when P = 0 lb/in. and V = 70 in.3? (Do not round our answer.) A) 49 lb/in. per sec B) 300 lb/in. per sec 9 C) 4 7 lb/in. per sec D) 63 lb/in. per sec 30) 4

46 Solve the problem. 3) Find all points on the curve = sin, 0 p, where the tangent line is parallel to the line =. 3) A) p 3,, p 3, B) p 3, 3, p 3, - 3 C) p 3, 3, p 3, 3 D) p 6,, p 6, - 3) About how accuratel must the interior diameter of a clindrical storage tank that is 4 m high be measured in order to calculate the tank's volume within 0.% of its true value? A) Within 0.% B) Within 0. meters C) Within 0. meters D) Within 0.% 3) Suppose that the functions f and g and their derivatives with respect to have the following values at the given values of. Find the derivative with respect to of the given combination at the given value of. f() g() f () g () 33) ) g(), = 3 A) - 3 B) 8 C) 3 8 D) 3 Find the derivative of the function. 34) f() = cos (0 + 3)-/ A) f () = - sin - (0 + 3)3/ B) f () = sin (0 + 3) -/ (0 + 3)3/ C) f () = - sin (0 + 3)-/ D) f () = - sin (0 + 3) -/ (0 + 3)3/ 34) 3) q = cos 6t + A) dq dq = - sin 6t + B) dt dt = - sin 3 6t + C) dq dt = - dq sin 6t + D) 6t + dt = - 3 sin 6t + 6t + 3) Find. 36) = sin( + ) Solve the problem. A) 0 cos( + ) B) - 0 cos( + ) C) - 0 sin( + ) D) - 0 sin( + ) 37) At time t 0, the velocit of a bod moving along the s-ais is v = t - 7t + 6. When is the bod moving backward? A) < t < 6 B) 0 t < C) 0 t < 6 D) t > 6 36) 37) 46

47 Find the second derivative of the function. 38) = ) A) d d = B) d d = C) d d = D) d 4 = - d 3 Find the linearization L() of f() at = a. 39) f() = 9-9, a = 0 39) A) L() = B) L() = C) L() = 9-9 D) L() = Find. 40) = ( - )(3 - + ) A) B) C) D) Write a differential formula that estimates the given change in volume or surface area. 4) The change in the volume V = prh of a right circular clinder when the height changes from h0 to h0 + dh and the radius does not change A) dv = pr0 dr B) dv = pr dh C) dv = prh0 dh D) dv = prh0 dh Find the indicated derivative. 4) Find (4) if = -8 cos. A) (4) = 8 sin B) (4) = -8 sin C) (4) = -8 cos D) (4) = 8 cos 40) 4) 4) The figure shows the velocit v or position s of a bod moving along a coordinate line as a function of time t. Use the figure to answer the question. 43) v (ft/sec) 43) 4 3 t (sec) What is the bod's greatest velocit? A) 4 ft/sec B) 3 ft/sec C) ft/sec D) ft/sec 47

48 Find the value of (f g) at the given value of. u 44) f(u) = u -, u = g() = , = 0 44) A) B) - 7 C) 47 D) 7 The figure shows the graph of a function. At the given value of, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? 4) = 0 4) A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable Solve the problem. 46) The position(in feet) of an object oscillating up and down at the end of a spring is given b s = A sin k t at time t (in seconds). The value of A is the amplitude of the motion, k is a measure m of the stiffness of the spring, and m is the mass of the object. Find the object's velocit at time t. A) v = A cos k m t ft/sec B) v = A k m cos k m t ft/sec C) v = - A k m cos k m t ft/sec D) v = A m k cos k m t ft/sec 46) The function s = f(t) gives the position of a bod moving on a coordinate line, with s in meters and t in seconds. 47) s = 6t + 4t + 4, 0 t Find the bod's displacement and average velocit for the given time interval. A) 40 m, 0 m/sec B) 3 m, 6 m/sec C) 3 m, 3 m/sec D) 0 m, 8 m/sec 47) 48

49 Given the graph of f, find an values of at which f is not defined. 48) 48) A) = -3, 3 B) = -, 0, C) = -3, 0, 3 D) = -, Find the derivative of the function. 49) r = + 6q (6 - q) 6q 49) A) dr dq = q + 6 dr B) dq = q - C) dr dq = q + dr D) dq = - q - The function f() changes value when changes from 0 to 0 + d. Find the approimation error f - df. Round our answer, if appropriate. 0) f() = +, 0 = 3, d = 0.0 0) A) B) C) D) Solve the problem. ) A manufacturer contracts to mint coins for the federal government. How much variation dr in the radius of the coins can be tolerated if the coins are to weigh within /0 of their ideal weight? Assume that the thickness does not var. A) 0.00% B).0% C) 0.00% D).0% ) Find d. ) = ) A) 7-9 d B) 7-9 d C) d D) d Given the graph of f, find an values of at which f is not defined. 3) 3) A) =, 3 B) = C) =,, 3 D) Defined for all values of Solve the problem. 4) The driver of a car traveling at 48 ft/sec suddenl applies the brakes. The position of the car is s = 48t - 3t, t seconds after the driver applies the brakes. How far does the car go before coming to a stop? A) 8 ft B) 768 ft C) 384 ft D) 9 ft 4) 49

50 Find the derivative. ) = A) B) C) D) + -4 Find the derivative of the function. 6) = (7 + 0) ) 6) A) = 3 (7) C) = (7 + 0) B) = 7 (7 + 0) D) = 3 (7 + 0) Solve the problem. 7) The number of gallons of water in a swimming pool t minutes after the pool has started to drain is Q(t) = 0(0 - ). How fast is the water running out at the end of minutes? A) 400 gal/min B) 40 gal/min C) 900 gal/min D) 0 gal/min 7) Find an equation for the line tangent to the curve at the point defined b the given value of t. 8) = csc t, = cot t, t = p 3 8) A) = B) = C) = D) = 4-3 Suppose u and v are differentiable functions of. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 9) u() =, u () = -6, v() = 7, v () = -4. d v d u at = A) B) - C) - 6 D) 9) Given = f(u) and u = g(), find d/d = f (g())g (). 60) = u(u - ), u = + Solve the problem. A) B) C) + 4 D) ) The driver of a car traveling at 60 ft/sec suddenl applies the brakes. The position of the car is s = 60t - 3t, t seconds after the driver applies the brakes. How man seconds after the driver applies the brakes does the car come to a stop? A) 60 sec B) 0 sec C) 30 sec D) 0 sec Use the linear approimation ( + )k + k, as specified. 6) Estimate (.0003)0. A).003 B).03 C).006 D).0 60) 6) 6) 0

51 Find the derivative of the function. 63) = (sin )-/ cos A) = - (sin )3/ C) = (cos )3/ B) = D) = - cos (sin )3/ (sin )3/ 63) Find the derivative. 64) = 6-9 A) -8 B) 6-8 C) -8 D) 6-9 Solve the problem. Round our answer, if appropriate. 6) One airplane is approaching an airport from the north at 4 km/hr. A second airplane approaches from the east at 48 km/hr. Find the rate at which the distance between the planes changes when the southbound plane is 3 km awa from the airport and the westbound plane is km from the airport. A) -60 km/hr B) -3 km/hr C) -48 km/hr D) -64 km/hr Find the slope of the tangent line at the given value of the independent variable. 8 66) g() = 9 +, = 3 64) 6) 66) A) 8 B) - 8 C) 3 D) - 3 Find d. 67) 6/ - + = 0 - A) 3-/ - d B) - 3-/ - d C) - 3-/ + d D) d 67) Use implicit differentiation to find d/d. 68) - = A) - B) - C) - D) - 68)