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

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1 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 cos B) = 8 sin = - 8 sin D) = sin ) ) Find if = sin. A) = - cos + sin B) = - sin = cos - sin D) = cos - sin ) Find. ) = ( + 8)(7 - ) A) B) D) ) ) = ( - )( + ) A) 0-7 B) D) 0-7 ) The equation gives the position s = f(t) of a bod moving on a coordinate line (s in meters, t in seconds). ) s (m) ) t (sec) When is the bod moving forward? A) 0 < t <, < t <, < t < 7, 9 < t < B) 0 < t <, < t <, < t < 8 0 < t < 8 D) 0 < t <, < t <, < t < 7

2 ) s (m) ) t (sec) What is the bod's velocit when t =. sec? A) m/sec B) m/sec - m/sec D) m/sec 7) s = sin t - cos t Find the bod's velocit at time t = π/ sec. 7) A) - m/sec B) m/sec m/sec D) - m/sec 8) s (m) 8) t (sec) When is the bod standing still? A) 8 < t B) < t < < t <, < t <, 7 < t < 9 D) t =, t =, t = 8 Use implicit differentiation to find d/d and d/d. 9) - + = A) d d = ; d d = - ( + ) d d = - + ; d d = - ( + ) B) d d = ; d d = + ( + ) D) d d = + + ; d d = + ( + ) 9)

3 ) - = A) d d = ; d d = - d d = ; d d = - B) d d = ; d d = - D) d d = - ; d d = - ) Find an equation of the tangent line at = a. ) = - - ; a = A) = - B) = - = 8 - D) = 8 - ) Find the derivative of the function. ) f(t) = ( - t)( + t) - A) f (t) = t - 8t - ( + t) B) f (t) = t - 8t - + t f (t) = - t + 8t - ( + t) D) f (t) = - t + 8t - ( + t) ) ) = (9 + 9) ) A) 7 (9 + 9) - - (9 + 9) B) 9 (9 + 9) + - D) (9) ) = ) A) = (7 - ) = (7 - ) B) = (7 - ) D) = (7 - ) ) q = 0r - r7 A) -7r 0r - r7 B) 0-7r 0r - r7 0-7r D) 0r - r7 ) ) h() = cos + sin A) - sin cos - cos ( + sin ) cos + sin B) - sin cos D) cos + sin )

4 7) = ( + )( + )- A) ( + )( - - ) ( + ) B) ( + )( + - ) ( + ) 7) -( + )( - - ) ( + ) D) -( + )( + - ) ( + ) 8) s = sin 7πt A) 7π - 7π - cos 7πt cos 7πt cos 7πt + 7π - 7π sin 7πt sin 7πt B) cos 7πt D) 7π cos 7πt + sin 7πt - 7π sin 7πt 8) 9) = A) = = + 8 B) = / D) = + 8 / 9) 0) r = θ - 9 θ A) r = (θ + 9) θ - 8 r = 9 θ + 9 B) r = - D) r = 9 θ(θ + 9) 9 θ(θ + 9) 0) 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. ) u() = 9, u () =, v() = -, v () = -. ) d u d v at = A) B) D) Find an equation for the tangent to the curve at the given point. ) f() = - +, (0, ) ) A) = - + B) = = - + D) = -

5 Provide an appropriate response. ) The curve = a + b + c passes through the point (, 8) and is tangent to the line = at the origin. Find a, b, and c. A) a = 0, b =, c = B) a =, b =, c = 0 a =, b = 0, c = D) a =, b = 0, c = 0 7 ) Find an equation for the tangent to the curve = at the point (, 9). + A) = - + B) = - = - + D) = + ) ) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. ) Graph = - tan and its derivative together on - π, π = - tan ever positive? Eplain.. Is the slope of the graph of ) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ) Find all points (, ) on the graph of f() = - with tangent lines parallel to the line = A) (, 8) B) (, 9) (, 9) D) (0, 0), (, 9) ) 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? 7) = - 7) A) Differentiable B) Continuous but not differentiable Neither continuous nor differentiable Find the value(s) of for which the slope of the curve = f() is 0. 8) f() = - + A) = ± B) = - + = - ± D) = - - 8)

6 Find the derivative. 9) = sin + cot A) = - csc cot + sec B) = csc cot - sec = cos - csc D) = csc cot - csc 9) 0) s = t - csc t + A) ds dt = t - csc t cot t ds dt = t + cott B) ds dt = t + csc t cot t D) ds dt = t - cott + 0) ) p = sec q + csc q csc q A) dp dq = sec q + dp = - csc q cot q dq B) dp dq = sec q dp D) = sec q tan q dq ) ) r = - θ7 cos θ A) dr dθ = 7θ sin θ dr dθ = 7θ cos θ - θ7 sin θ B) dr dθ = 7θ sin θ - θ7 cos θ D) dr dθ = - 7θ cos θ + θ7 sin θ ) ) s = t + 7t + A) t + 7 B) t + 7 t + 7 D) t + 7 ) ) = + sec A) = - sec tan B) = - - csc = - + tan D) = - + sec tan ) Solve the problem. Round our answer, if appropriate. ) The volume of a rectangular bo with a square base remains constant at 00 cm 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 cm long. (Do not round our answer.) 0 00 A) cm/sec B) 0 cm/sec 0 cm/sec D) 9 9 cm/sec ) ) Water is discharged from a pipeline at a velocit v (in ft/sec) given b v = 80p(/), where p is the pressure (in psi). If the water pressure is changing at a rate of 0.8 psi/sec, find the acceleration (dv/dt) of the water when p =.0 psi. A) 8 ft/sec B) 8.7 ft/sec 9. ft/sec D) 0 ft/sec )

7 7) The radius of a right circular clinder is increasing at the rate of 8 in./sec, while the height is decreasing at the rate of in./sec. At what rate is the volume of the clinder changing when the radius is in. and the height is 8 in.? A) 0π in./sec B) 0 in./sec 8π in./sec D) -8 in./sec 8) Water is being drained from a container which has the shape of an inverted right circular cone. The container has a radius of.00 inches at the top and a height of.00 inches. At the instant when the water in the container is.00 inches deep, the surface level is falling at a rate of 0.7 in./sec. Find the rate at which water is being drained from the container. A). in./s B).0 in.s 7.7 in./s D). in./s 9) A man ft tall walks at a rate of ft/sec awa from a lamppost that is ft high. At what rate is the length of his shadow changing when he is ft awa from the lamppost? (Do not round our answer) A) 8 ft/sec B) 7 ft/sec 9 8 ft/sec D) 9 9 ft/sec 7) 8) 9) Solve the problem. 0) A rock is thrown verticall upward from the surface of an airless planet. It reaches a height of s = t - t meters in t seconds. How high does the rock go? How long does it take the rock to reach its highest point? A) m, sec B) 70 m, sec 8 m, sec D) 70 m, sec 0) ) Find the tangent to = cos at = π. ) A) = - + π B) = = - - π D) = + π ) For a motorccle traveling at speed v (in mph) when the brakes are applied, the distance d (in feet) required to stop the motorccle ma be approimated b the formula d = 0.0v + v. Find the instantaneous rate of change of distance with respect to velocit when the speed is mph. A). mph B) mph. mph D). mph ) ) The area A = πr of a circular oil spill changes with the radius. At what rate does the area change with respect to the radius when r = 9 ft? A) 8π ft/ft B) 9π ft/ft 8 ft/ft D) 8π ft/ft ) ) Assume that the profit generated b a product is given b P() =, where is the number of units sold. If the profit keeps changing at a rate of $700 per month, then how fast are the sales changing when the number of units sold is 0? (Round our answer to the nearest dollar per month.) A) $,9/month B) $9,99/month $,/month D) $0/month ) 7

8 ) Does the graph of the function = tan - have an horizontal tangents in the interval 0 π? If so, where? ) A) No B) Yes, at = 0, = π, = π Yes, at = π, = π D) Yes, at = π ) A wheel with radius m rolls at rad/s. How fast is a point on the rim of the wheel rising when the point is π/ radians above the horizontal (and rising)? (Round our answer to one decimal place.) A) 8.0 m/s B).0 m/s.0 m/s D).0 m/s ) 7) Suppose that the dollar cost of producing radios is c() = Find the marginal cost when 0 radios are produced. A) -$880 B) $880 $ D) $ 7) 8) The graph of = f() in the accompaning figure is made of line segments joined end to end. Graph the derivative of f. 8) (, ) (, ) (-, ) (-, 0) (0, -) A) B)

9 D) ) 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) 0 gal/min B) 0 gal/min 00 gal/min D) gal/min 0) 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 m from the right angle? (Round our answer to two decimal places.) A) 0.80 m/s B).0 m/s 0.0 m/s D). m/s ) Find the tangent to = - sin at = π. A) = - π + B) = - = - + D) = - + π - 9) 0) ) Use implicit differentiation to find d/d. ) = sec() A) cos() cot() B) sec() tan() ) cos() cot() D) sec() tan() ) + - = + ) A) ( - ) - + ( - ) B) ( - ) - - ( - ) ( - ) + + ( - ) D) ( - ) + - ( - ) ) - = A) - B) - - D) - ) 9

10 ) = cot A) - csc cot B) - csc csc D) csc ) ) + + = 8 A) + + B) D) + + ) 7) cos + = A) - sin + sin B) + sin + sin - sin D) - sin 7) Find the second derivative. 8) = A) + 9 B) 0 D) 7 8) 9) s = 7t + 7 A) t B) t + 7 7t D) 7t 9) Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given. 0) 0) (0, 0) = = A) Since lim 0 + f () = - while lim 0- f () = -, f() is not differentiable at = 0. B) Since lim 0 + f () = while lim 0- f () =, f() is not differentiable at = 0. Since lim 0 + f () = while lim 0 - f () =, f() is not differentiable at = 0. D) Since lim 0 + f () = while lim 0 - f () =, f() is differentiable at = 0.

11 Estimate the slope of the curve at the indicated point. ) ) A) - B) 0 D) Undefined The graph of a function is given. Choose the answer that represents the graph of its derivative. ) ) A) B) D)

12 Find d/dt. ) = cos( 8t + ) A) 8t + sin( 8t + ) B) -sin 8t + -sin( 8t + ) D) - sin( 8t + ) 8t + ) ) = t(t - 9) A) t(t - 9) (9t - ) B) 0t8(t - 9) t(t - 9) (t - 9) D) t(t - 9) (9t - ) ) ) = cos7(πt - ) A) -7π sin(πt - ) B) -7 cos(πt - ) sin(πt - ) 7 cos(πt - ) D) -7π cos(πt - ) sin(πt - ) ) Find the slope of the line tangent tangent to the graph a the given point using the limit process. ) = +, = 7 ) A) m = 0 B) m = - 0 m = D) m = - At the given point, find the slope of the curve or the line that is tangent to the curve, as requested. 7) =, tangent at (, ) A) = -8 + B) = - + = 8 - D) = 7) 8) + = +, slope at (0, ) 8) A) B) - D) 9) - π cos = π, slope at (, π) A) 0 B) π - π D) -π 9) Graph the equation and its tangent. 70) Graph = and the tangent to the curve at the point whose -coordinate is. 70) - -

13 A) B) D) ) Graph = + and the tangent to the curve at the point whose -coordinate is )

14 A) B) D) Find d for the given function. d 7) = + 7) A) B) + D) - + 7) = sin( + 7) A) - sin( + 7) B) -0 sin( + 7) -0 cos( + 7) D) cos( + 7) 7) Find the second derivative of the function. 7) = + 7 7) A) d d = + B) d d = - d d = + D) d = - d Graph the curve over the given interval together with its tangent at the given value of. Graph the tangent with a dashed line.

15 7) = cos(), -π π, = π 7) A) - B) D) At the given point, find the line that is normal to the curve at the given point. 7) - π cos = π, normal at (, π) A) = π - π + π B) = -π + π 7) = - π + π + π D) = π - π + π

16 Answer Ke Testname: M0_E_PRAC ) C ) D ) A ) D ) A ) C 7) B 8) B 9) C ) C ) D ) A ) A ) D ) B ) C 7) D 8) A 9) B 0) D ) D ) A ) B ) C ) No, the slope of the graph of = - tan is never positive. The slope at an point is equal to the derivative, which is = - sec. Since sec is never negative, = - sec is never positive, and the slope of the graph is never positive. ) B 7) B 8) C 9) A 0) B ) B ) D ) B

17 Answer Ke Testname: M0_E_PRAC ) D ) A ) B 7) C 8) A 9) B 0) D ) A ) C ) A ) A ) B ) C 7) D 8) B 9) C 0) A ) A ) C ) D ) B ) B ) B 7) A 8) C 9) A 0) C ) C ) D ) A ) D ) D ) B 7) B 8) A 9) D 70) A 7) B 7) C 7) B 7) C 7) A 7) A 7

Review Test 2. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) ds dt = 4t3 sec 2 t -

Review Test 2. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) ds dt = 4t3 sec 2 t - Review Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the derivative. ) = 7 + 0 sec ) A) = - 7 + 0 tan B) = - 7-0 csc C) = 7-0 sec tan