MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) = 2t + 1; D) = 2 - t;

Size: px

Start display at page:

Download "MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) = 2t + 1; D) = 2 - t;"

Winfred Merritt
6 years ago
Views:

1 Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculate the derivative of the function. Then find the value of the derivative as specified. ) f() = ; f (0) A) f () = + 7; f (0) = 7 B) f () = 2 + 7; f (0) = 7 C) f () = 2-2; f (0) = - 2 D) f () = 2; f (0) = 0 ) 2) ds dt t =- if s =t 2 - t 2) A) ds ds = t - ; dt dt t =- = -2 C) ds ds = 2t - ; dt dt t =- = -3 ds ds B) = 2t + ; dt dt t =- = - ds ds D) = 2 - t; dt dt t =- = 3 Find the slope of the tangent line at the given value of the independent variable. 7 3) g() = 4 +, = 8 A) B) 7 2 C) 7 44 D) ) Find an equation of the tangent line at the indicated point on the graph of the function. 4) = f() = - 2, (, ) = (, 0) A) = + B) = 3 - C) = 3 + D) = - + 4)

2 The graph of a function is given. Choose the answer that represents the graph of its derivative. 5) 5 5) A) B) C) D) Given the graph of f, find an values of at which f is not defined. 6) 6) A) = 0,, 2 B) = 2 C) = 0 D) = 2

3 7) 7) A) = 2, 5 B) = 2 C) = 5 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. 8) 8) (-, -) = = - 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 = -. 3

4 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? 9) = - 9) A) Differentiable B) Continuous but not differentiable C) Neither continuous nor differentiable SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 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. 0) ) Does the curve = have a tangent whose slope is -2? If so, find an equation for the line and the point of tangenc. If not, wh not? ) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the derivative. 2) w = z-3 - z 2) A) z-4 + z2 B) -3z-4 + z2 C) -3z-4 - z2 D) 3z-4 - z2 Find the second derivative. 3) = A) 42-6 B) 6-28 C) 6-42 D) ) Find. 4) = ( )( ) A) B) C) D) ) 4

5 Find the derivative of the function. 3 5) = - A) = ( - )2 C) = ( - )2 B) = ( - )2 D) = ( - )2 5) Find. 6) = + - 6) A) B) C) D) 2-2 Find the derivative of the function. 7) r = θ - 6 θ + 6 A) r = 6 θ C) r = (θ + 6) θ2-36 B) r = D) r = - 6 θ(θ + 6)2 6 θ(θ + 6)2 7) Find the derivative. 8et 8) s = 2et + 8) A) 8et (2et + )2 B) et (2et + )2 C) 8et (2et + )3 D) 8et (2et + ) 9) w = z9 - e A) (8 - e)z9 - e B) z9 - e C) z 0 - e 0 - e D) (9 - e)z8 - e 9) 20) w = 3 z e3. A) 2.2z e2. B) 2.2z e2. C) 2.2z e2. D) 2.2z e3. 20) Find the derivative of the function. 2) r = + 3θ (3 - θ) 3θ 2) A) dr dθ = θ 2 - B) dr dθ = - θ2 - dr C) dθ = θ2 + 3 dr D) dθ = θ2 + 5

6 22) z = 32e A) dz d = 3e + 62e B) dz d = 3e + 32e 22) C) dz d = 6e + 32e D) dz d = 6e - 32e Find the second derivative of the function. 23) r = + 7θ (7 - θ) 7θ 23) A) d 2r dθ2 = θ - θ B) d 2r dθ2 = - 2 θ3 - C) d 2r dθ2 = 2 θ3 D) d 2r dθ2 = - θ2 - 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. 24) u() = 5, u () = -7, v() = 6, v () = ) d u d v at = A) B) C) - 8 D) Solve the problem. 25) The curve = a2 + b + c passes through the point (2, 0) and is tangent to the line = 3 at the origin. Find a, b, and c. A) a = 2, b = 0, c = 0 B) a = 0, b =, c = 3 C) a = 3, b = 0, c = D) a =, b = 3, c = 0 25) The function s = f(t) gives the position of a bod moving on a coordinate line, with s in meters and t in seconds. 26) s = 8t - t2, 0 t 8 26) Find the bodʹs displacement and average velocit for the given time interval. A) 0 m, 0 m/sec B) -28 m, -6 m/sec C) 28 m, -8 m/sec D) 28 m, 6 m/sec Solve the problem. 27) At time t, the position of a bod moving along the s-ais is s = t3-27t t m. Find the bodʹs acceleration each time the velocit is zero. A) a(20) = 20 m/sec2, a(6) = 20 m/sec2 B) a(0) = 6 m/sec2, a(8) = -6 m/sec2 C) a(0) = -6 m/sec2, a(8) = 6 m/sec2 D) a(0) = 0 m/sec2, a(8) = 0 m/sec2 27) 6

7 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. 28) v (ft/sec) 28) t (sec) When is the bodʹs acceleration equal to zero? A) 2 < t < 3, 5 < t < 6 B) 0 < t < 2, 6 < t < 7 C) t = 0, t = 4, t = 7 D) t = 2, t = 3, t = 5, t = 6 The graphs show the position s, velocit v = ds/dt, and acceleration a = d2s/dt2 of a bod moving along a coordinate line as functions of time t. Which graph is which? 29) 29) A t C B A) C = position, B = velocit, A = acceleration B) B = position, A = velocit, C = acceleration C) A = position, C = velocit, B = acceleration D) B = position, C = velocit, A = acceleration Solve the problem. 30) The area A = πr2 of a circular oil spill changes with the radius. At what rate does the area change with respect to the radius when r = 4 ft? A) 8π ft2/ft B) 6π ft2/ft C) 4π ft2/ft D) 8 ft2/ft 3) The number of gallons of water in a swimming pool t minutes after the pool has started to drain is Q(t) = 50(20 - )2. How fast is the water running out at the end of 3 minutes? A) 225 gal/min B) 2450 gal/min C) 700 gal/min D) 350 gal/min 30) 3) 7

8 32) The driver of a car traveling at 42 ft/sec suddenl applies the brakes. The position of the car is s = 42t - 3t2, 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) 4 sec B) 7 sec C) 42 sec D) 2 sec 32) 33) The driver of a car traveling at 42 ft/sec suddenl applies the brakes. The position of the car is s = 42t - 3t2, t seconds after the driver applies the brakes. How far does the car go before coming to a stop? A) 7 ft B) 47 ft C) 294 ft D) 588 ft 33) Find the derivative. 34) s = t3 tan t - t A) ds dt = - t 3 sec2 t + 3t2 tan t + 2 t C) ds dt = 3t 2 sec2 t - 2 t B) ds dt = t 3 sec2 t + 3t2 tan t - 2 t D) ds dt = t 3 sec t tan t + 3t2 tan t - 2 t 34) 35) p = 9 + sec q 9 - sec q A) dp dq = - 2 sec 2 q tan q (9 - sec q)2 C) dp dq = 8 tan2 q (9 - sec q)2 B) dp dq = - 8 sin q (9 cos q - )2 D) dp dq = 8 sin q (9 cos q - )2 35) 36) = sin sin A) d d = cos cos C) d cos + sin 7 sin + 7 cos = + d 72 sin2 B) d sin - cos 7 cos - 7 sin = + d 492 sin2 D) d cos - sin 7 sin - 7 cos = + d 72 sin2 36) 37) r = 7 - θ7 cos θ A) dr dθ = 7θ 6 sin θ C) dr dθ = - 7θ 6 cos θ + θ7 sin θ B) dr dθ = 7θ 6 sin θ - θ7 cos θ D) dr dθ = 7θ 6 cos θ - θ7 sin θ 37) Given = f(u) and u = g(), find d/d = f (g())g (). 38) = u(u - ), u = 2 + A) B) C) D) ) 8

9 Write the function in the form = f(u) and u = g(). Then find d/d as a function of. 39) = ) A) = u8; u = 62-2 B) = u8; u = 62-2 C) = u8; u = ; d d = ; d d = ; d d = D) = 6u2-2 u - u; u = 8; d d = ) = cot(6-0) A) = 6u - 0; u = cot ; d d = - 6 cot csc 2 B) = cot u; u = 6-0; d d = - 6 csc 2(6-0) C) = cot u; u = 6-0; d = - 6 cot(6-0) csc(6-0) d D) = cot u; u = 6-0; d d = - csc 2(6-0) 40) Find the derivative of the function. 4) = 6 (8 + 7) ) A) = 2 (8) C) = 4 3 (8 + 7) B) = 4(8 + 7) D) = 2 (8 + 7) ) h() = cos 5 + sin A) h () = 5 cos 4 + sin C) h () = - 4 sin cos cos 4 + sin B) h () = -5 sin 4 cos D) h () = - 5 cos 4 ( + sin )5 42) 43) h(θ) = 7 + sin(2θ) 6 cos(2θ) A) h (θ) = 7 + sin(2θ) C) h (θ) = cos(2θ) (7 + sin(2θ))3/2 cos(2θ) B) h (θ) = sin(2θ) D) h (θ) = sin(2θ) 43) 9

10 Find the value of (f g) at the given value of. u 44) f(u) = u2 -, u = g() = , = 0 44) A) 5 9 B) 9 C) D) 3 Use implicit differentiation to find d/d and d2/d2. 45) - + = 6 A) d d = - + ; d 2 d2 = 2-2 ( + )2 C) d d = ; d 2 d2 = 2-2 ( + )2 B) d d = ; d 2 d2 = + ( + )2 D) d d = + + ; d 2 d2 = ( + )2 45) 46) 2-3 = 8 A) d d = 2 32 ; d 2 d2 = C) d d = 2 32 ; d 2 d2 = B) d d = 2 32 ; d 2 d2 = D) d d = 2 32 ; d 2 d2 = ) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 47) 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. 47) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 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. 48) = 2 + 9, slope at (0, ) 48) A) - 2 B) 3 C) 9 5 D) 9 7 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. 49) /2 = t3 + t, (t + ) - 4t = 25, t = 0 49) A) -5 B) -0 C) 0 D) ) cos t + = 2t, = t sin t + t, t = π 2 50) A) + π B) 2 + π C) 2 D) π 0

11 Find the formula for df-/d. 5) f() = 643 A) 922 B) /3 4 C) 22/3 D) 22 5) Find the derivative of with respect to, t, or θ, as appropriate. 52) = ln ( - 2) A) B) C) D) ) 53) = ln - ( + 3)5 53) A) 4-8 ( + 3)6 B) 4-8 ( + 3)( - ) C) ( + 3) 5 - D) ln 6-8 ( + 3)6 54) = ln (cos (ln θ)) A) tan (ln θ) B) tan (ln θ) θ C) - tan (ln θ) D) - tan (ln θ) θ 54) Use logarithmic differentiation to find the derivative of. 55) = A) C) B) (ln - ln( + 3)) D) ) 56) = ( + 5)( - 2) A) ( + 5)( - 2)(ln + ln( + 5) + ln( - 2)) B) C) D) ( + 5)( - 2) ) 57) = sin + 5 A) 2 + sin B) sin cot C) sin + 5 ln + lnsin - 2 ln( + 5) D) + cot ) Find the derivative of with respect to, t, or θ, as appropriate. 58) = 6e - 6e A) 6e B) 6e C) 6e + 2e D) 6 58)

12 59) = sin e-θ5 A) 5θ4cos e-θ5 C) (-5θ4 e-θ5 ) cos e -θ5 B) cos e-θ5 D) cos (-5θ4 e-θ5 ) 59) 60) = esin t (ln t2 + 3) A) esin t (cos t)(ln t2 + 3) + 2 t C) ecos t (cos t)(ln t2 + 3) + 2e sin t t B) 2e sin t cos t t D) esin t ln t t 60) Find d d. 6) e5 = sin ( + 2) A) C) 5e5 - B) ln sin ( + 2) 2 cos ( + 2) e5 2 cos ( + 2) D) 5e 5 - cos ( + 2) 2 cos ( + 2) 6) 62) e = sin A) sin - e e B) cos - e e C) cos - e e D) cos e 62) Find the derivative of with respect to the independent variable. 63) = 9 A) 9 B) ln 9 C) 9 ln D) 9 ln 9 63) 64) = (cos θ) 5 A) - 5 cos θ sin θ B) - 5(cos θ) 5- sin θ C) -(cos θ) 5- sin θ D) 5(cos θ) 5-64) Use logarithmic differentiation to find the derivative of with respect to the independent variable. 65) = ( + 4) A) ( + 4) ln( + 4) + B) + (4)- + 4 C) ln( + 4) D) ln( + 4) ) 66) = (ln )ln A) ln ln (ln ) B) ln (ln ) + 66) C) (ln ) ln D) ln (ln ) + (ln ) ln 2

13 Solve the problem. 67) The kinetic energ K of an object with mass m and velocit v is K = 2 mv 2. How is dm/dt related 67) to dv/dt if K is constant? A) dm = m dv dt v dt C) dm = - 2m dv dt v dt B) dv dt = - 2m v D) dm dt dm dt = - 2mv3 dv dt 68) Suppose that the radius r and volume V = 4 3 πr 3 of a sphere are differentiable functions of t. Write 68) an equation that relates dv/dt to dr/dt. A) dv dt = 4 3 πr 2 dr B) dv dr = 4π dt dt dt C) dv dt = 4πr 2 dr dt D) dv dt = 3r 2 dr dt 3

14 Answer Ke Testname: CAL DIFF PROBSET ) B 2) C 3) D 4) D 5) C 6) C 7) B 8) C 9) B 0) Yes, a tangent line to a graph can intersect the graph at more than one point. For eample, the graph = 3-22 has a horizontal tangent at = 0. It intersects the graph at both (0, 0) and (2, 0). ) The curve has no tangent whose slope is -2. The derivative of the curve, = , is alwas positive and thus never equals -2. 2) B 3) A 4) D 5) C 6) C 7) B 8) A 9) D 20) B 2) B 22) C 23) C 24) B 25) D 26) A 27) B 28) A 29) D 30) A 3) C 32) B 33) B 34) B 35) D 36) D 37) C 38) C 39) A 40) B 4) B 42) D 43) A 44) C 45) A 46) A 4

15 Answer Ke Testname: CAL DIFF PROBSET 47) There is no difference at all. At = a, the slope of the tangent = lim a 48) B 49) B 50) D 5) C 52) D 53) B 54) D 55) A 56) D 57) B 58) B 59) C 60) A 6) D 62) C 63) D 64) B 65) A 66) D 67) C 68) C f() - f(a) - a = f (). 5

MATH 150/GRACEY EXAM 2 PRACTICE/CHAPTER 2. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MATH 150/GRACEY EXAM 2 PRACTICE/CHAPTER 2. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. MATH 0/GRACEY EXAM PRACTICE/CHAPTER Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated derivative. ) Find if = 8 sin. A) = 8