Exam 3 - Part I 28 questions No Calculator Allowed - Solutions. cos3x ( ) = 2 3. f x. du D. 4 u du E. u du x dx = 1

Transcription

1 . If f = cos Eam - Part I 8 questions No Calculator Allowed - Solutions =, then f A. B. sin C. sin D. sin cos E. sin cos cos C. Chain rule. f [ ] = cos = f [ cos ( ) ] sin [ ] = sin cos. d is equivalent to A. 5 u du B. u du C. u du D. u du E. u du 5 C. d 6 8 u =,du = d =,u = 6, =,u = 8 d = 8 u du = u du Let f be the function defined below, where c and d are constants. If f is differentiable at =, what is the value of c d? f = + ( c +) d, e + + c + d, < A. - B. C. D. E. E. Continuity : lim + f = f = lim + c +, e + + c, < Differentiability : lim c d = += + ( c +) d = c d c + d = f = lim f f + c += + c c =,d = BC Eam Illegal to post on Internet

2 . An oil slick forms in a lake. The oil is feet deep. The slick is 8 feet wide and is divided into 6 sections of equal width having measurements as shown in the figure to the right. Tim uses a trapezoidal sum with si trapezoids to approimate the volume while Doug uses a midpoint sum using three equally spaced rectangles. What is the difference between their approimations of the volume of the slick in cubic feet? A. 9 B. 6 C. D. E. B. Tim : Area = ( )( ) = 5 Volume = ( 5) =8 Doug : Area = 6( 5 + +) = 5 Volume = ( 5) =6 Difference : 6 5. The graph of the piecewise linear function f is shown in the figure to right. If g A. g D. g = f ( t) dt, which of the following is the greatest? B. g( ) C. g E. g( 5) B. g - the area of the triangle = = f ( t) dt : g the area of the triangle = = f ( t) dt : ( ) = 6 ( ) = all others will be less. 6. A spider is walking up a vertical wall. At t =, the spider is 8 inches off the floor. The velocity v of the spider, measured in inches per minute at time t minutes, t is given by the function whose graph is shown to the right. How many inches off the floor is the spider at t =? A..5 B. 9.5 C. D..5 E. 7.5 D. y t = 8 + v( t) dt = = BC Eam Illegal to post on Internet

3 7. e d = A. e C B. e ( 8 + ) + C C. e + C D. e + + C E. e + C D. u = v = e du = d dv = e d e d = e e d u = v = e du = d dv = e d e d = e = e e + e = e + + C e e d sin π + h 8. lim h h sinπ = A. B. cos C. - D. π E. C. This is asking for the derivative of y = sin at = π. d d sin = cos and cosπ =. 9. A new robotic dog called the IPup went on sale at (9 AM) and sold out within 8 hours. The number of customers in line to purchase the IPup at time t is modeled by a differentiable function A where t 8. Values of A( t) are shown in the table below. For t 8, what is the fewest number of times at which A t =? t (hours) A( t) people A. B. C. D. E. 5 C. A is differentiable on,8 and for some t on ( 6,7). A t [ ] so the MVT implies that A ( t) > for some t on (,) < for some t on (,) and for some t on ( 7,8). Since f is continuous, the IVT implies that there must be at least one value of t on, where A ( t) = and the same must be true on ( 6,8). And there must be at least one one value of t on,5 where A ( t) =. So there must be at least values of t on (,8) where A ( t) = BC Eam Illegal to post on Internet

4 = + ( ) + ( ) ( ) + 6( ) be the fourth-degree Taylor polynomial for the. Let P function f about =. What is the value of f? A. - B. - C. - D. E. A. f! = f =.. A particle moving along a straight line that at any time t its velocity is given by v( t) = sint cost. For which values of t is the particle speeding up? I. t = π 6 II. t = π III. t = π A. I only B. II only C. III only D. I and II only E. I and III only cost cost sint sint E. a( t) = = ( cost) v π = 6 >,a π 6 = v π = <,a π ( ) + cost cost + sint sint ( cost) ( ) > v π = <,a π ( = )( ) + > ( ) ( ) ( = )( ) + ( )( ) < Speed increasing if v( t) and a( t) have the same signs.. Which of the following series converge? I. n n n = II. e n III. n = n = sin nπ n A. I only B. I and II only C. I and III only D. II and III only E. I, II and III n + C. lim n n n + n = <... Convergent e... the harmonic series - divergent n n = sin nπ = n +... alternating series - convergent n = BC Eam Illegal to post on Internet

5 . The graph of f, the derivative of f, is shown to the right for 8. The areas of the regions between the graph of f and the ais are 8, and respectively. If f on the interval 8? A. - B. - C. -8 D. E. D. Since f = f + so at that value, f ( c) = f + =, what is the minimum value of f f ( t ) dt, there is a value c where f ( t ) dt = c f ( t ) dt = =. c. To the right is a graph of f =. Find the value of f ( ) d. A. B. C. D. 6 E. Divergent d = E. = ( ) d + + d Both calculations are divergent. 5. Find lim e t dt A. B. e C. e 8 D. e 6 E. noneistent C. lim lim e t dt = e t e = e dt = BC Eam Illegal to post on Internet

6 6. Let g be a strictly decreasing function such that g which of the following is true? A. f has a relative minimum at = B. f has a relative maimum at = C. f will be a strictly decreasing function D. f will be a strictly increasing function E. It cannot be determined if f has a relative etrema E. g < and g < f = ( ) g + < for all real numbers. If f = ( ) g g = ( ) [( ) g + g ] = + [ + ( ) + ( ) ] = [ + ( ) ] which may be positive or negative. There is a critical point at =. If >, f If <, f = So f is increasing to the right of = but not enough information to the left of = = a The function f is given by f. The figure to the + b right shows a portion of the graph of f. Which of the following could be the values of the constants a and b? A. a = -, b = - B. a =, b = C. a =, b = - D. a = -, b = E. a = 6, b = - A. Since f has a vertical asymptote at =, b must equal. Since f has a horizontal asymptote at y =, a must equal., 8. What is the equation of the line tangent to the polar curve r = θ at θ = π? A. y = π D. y = π 6π 6π B. y = π 6π π E. y = 6π C. y = 6π D. = θ cosθ y = θ sinθ d = ( cosθ θ sinθ) dθ dθ d dθ = = ( sinθ + θ cosθ ) dθ = d θ = π + π = π y + 6π = π y = π 6π 9. A population is modeled by a function G that satisfies the logistic differential equation BC Eam Illegal to post on Internet

7 dg dt = G e G. If G e A. B. e C. e D. e E. e =, for what value of G is the population growing the fastest? C. dg dt = G e G e = G e G e d G dt = e G e dg dt = eg = e P = e half the carrying capacity. The function g is differentiable for all real numbers. The table below gives values of the function and its first derivatives at selected values of. If g is the inverse function of g, what is the equation for the line tangent to the graph of y = g at =? g g 5 A. y + = ( 5 ) B. y + = ( 5 ) C. y + = ( ) D. y + = ( ) E. y + = ( ) E. g( ) = g = ( g ) = g g ( ) = g ( ) = Tangent line : y + = ( ). If f = ( ) ( +) ( ), then f has which of the following relative etrema? I. A relative maimum at = II. A relative minimum at = III. A relative minimum at = A. I only B. III only C. I and II only D. I and III only E. I, II and III D. Critical values at =, =, =. What is the area under the curve described by the parametric equations = cost and y = sin t for BC Eam Illegal to post on Internet

8 t π? A. B. 8 C. D. E. B. cost = cos t = and sin t = y Since sin t + cos t = + y = y = When t =, = and when t = π, = A = d =. Let y = f be a solution to the differential equation = 6 ( 6 + ) = 8 d = y with initial condition f k a constant, k. If Euler s method with steps of equal size starting at = gives the approimation f, find the value of k. = k, A. B. C. D. - E. A. y d k k k k k k k k + k k + k = = k + k k =, k = = e cos. If the fourth. The function f has derivatives of all orders for all real numbers and f ( 5) degree Taylor polynomial for f about = is used to approimate f on the interval [, ], what is the Lagrange error bound for the maimum error on the interval [, ]? A. B. e C. e D. E. 7 B. The maimum value of cos = at = so the maimum value of e cos = e = e. So the Lagrange error for the th degree Taylor Polynomial is e 5! 5 = e BC Eam Illegal to post on Internet

9 5. An object moving in the y-plane has position function r( t) = the motion of the object. ( t +),t 6ln t +, t. Describe A. Left and up B. Left and down C. Right and up D. Right and down E. Depend on the value of t E. d dt = which is always negative when t so object moving left ( t +) = ( t + t ) t + ( t ) dt = t 6 t + = t + t 6 = t + t + t + changes sign at t = so y changes direction at t =. dt 6. The graph of f, the derivative of f is shown to the right for - < 6. At what values of does f have a horizontal tangent line? A. = only B. = -, =, = C. = only D. = -, =, = 6 E. = -, =, =, = 6 E. Horizontal tangent lines occur when f =. This occurs at =, =, =, = d = A. ln C B. ln C C. 5ln + ln + + C D. ln + 5ln + + C E. ln C + C. u - substitution doesn't work here. u = + +,du = ( + ) d d = + + ( +) = + + ( +) d d = 5ln + ln + + C BC Eam Illegal to post on Internet

10 8. The Maclaurin series for for +? + n is n =. Which of the following is a power series epansion A B C D E C. f = + = f ( ) = + = = f = BC Eam Illegal to post on Internet

11 Eam - Part II 7 questions Calculators Allowed - Solutions 9. Which of the following are convergent? I. d II. d III. d A. I only B. II only C. III only D. II and III only E. I, II and III D. d = which is divergent d = = d = =. The yellow bird in the popular game Angry Birds flies along the path y = + when. When = (the point on the figure to the right), the player touches the screen and the bird leaves the path and travels along the line tangent to the path at that point. If the bird crashes into the -ais, find the total distance the bird flies. A. 6.8 B.. C.. D. 5.6 E. 8. E. y = y Tangent line : y 8 = =. y = + 8 = 8 y = Length : L = + ( ) d + + ( ) d = 8.. A tube is in the shape of a right circular cylinder. The height is increasing at the rate of.5 inches/sec while the radius is decreasing that the rate of.5 inches/sec. At the time that the radius is inches and the length is inches, how is the lateral surface area of the tube changing? (The lateral surface area of a right circular cylinder with radius r and height h is πrh). A. Decreasing at in sec B. Increasing at in sec C. Decreasing at π in sec D. Increasing at 6π in sec E. Increasing at 8π in sec C. da dt dh = π r dt + h dh = π.5 dt [ +(.5) ] = π( ) = π in sec BC Eam Illegal to post on Internet

12 . Cellophane-wrapped Tastycakes come off the assembly line at the rate of C t are boed at the rate of B( t) = t = 5 + sin πt. They 6 and B( t) are measured in hundreds of cakes per hour, and t + t. C t is measured in hours for t 5. There are 5 cakes waiting to be boed at the start of the 5-hour shift. How long (in hours) does it take the number of cakes waiting to be boed to go from a minimum to a maimum during the 5-hour shift? A..56 B..869 C.. D..69 E. 5 C. The first screen is of C t we are interested in where B t and B( t). Since C( t) and B( t) are rates of change, C( t) = (nd screen). The area between the curve and the ais represents the number of cakes still to be boed ( plus the original 5). At the st intersection, some additional cakes have gathered. At the nd intersection, more cakes have been boed than are coming off the line, but by 5 hours, many more cakes are coming off the line than are being boed. So the minimum occurs at t =.869 and the maimum occurs at t = =... Consider the differential equation d = y +. If y = equation, at what point does f have a relative minimum? f is the solution to the differential A. (, ) B. (-, ) C. (, 9) D., E. (5, ) d y E. = d d To have a relative minimum, d = and d y d > A. d = D. d =, d y d = B. d =, d y d = E. d =, d y d = C. d =. If + y = a, where a is a constant, find d y d. A. y y B. y y C. y a D. y E. a D. + y d = d = y d y y + y d = d = y y = y y y = a y BC Eam Illegal to post on Internet

13 5. Let f be the function given by f =. For what positive value(s) of c is e e f c =? A.. B..8 and.6 C..58 and.6 D..6 E..5 and.7 A. f c ( e + e ) = ( e e ) = 8 e e or easier 6. Find the epression which represents the length L of the path described by the parametric equations = sin ( t) and y = cos( t ) for t π. π A. L = sint cost sint dt B. L = sin t + 9sin 9t dt π C. L = 6sin t cos t + 9sin 9t dt D. L = sin t cos t + 9sin t dt π E. L = 6sin t cos t + 9sin t dt E. d dt L = = sint cost π dt = sint π π 6sin t cos t + 9sin t dt 7. The polar graph in the figure to the right is r = θ + sinθ for θ 5π. Find the greatest distance of the curve from the origin. A..9 B..57 C..5 D..86 E. 5.6 D. dr = + cosθ = cosθ = dθ θ = π, π π π 5π θ The greatest distance is.86 when θ = π r BC Eam Illegal to post on Internet

14 8. What is the coefficient of in the Taylor series for f = + about =? A. B. 6 C. D. E. C. f + = + f ( ) = = ( +) f + = = ( +) f + = 5 = ( +) 5 f = Coefficient of =! = 6 = 9. The Cheesesteak Factory at a Philadelphia stadium has 5 steak sandwiches rea to sell when it opens for business. It cooks cheesesteaks at the rate of steaks per minute and sells cheesesteaks at the rate of + 6sin πt steaks per minute. For t, at what time is the number of steaks rea to sell at a minimum (nearest tenth of a minute)? A. B.. C.. D. 7. E. D. C t t = sin π dt C t = + 6sin πt = + 6sin πt Critical values : t =.7,t = 7.8 t C t = = sin π( 9) where t. Barstow, California has the daily summer temperature modeled by T t is measured in hours and t = corresponds to : midnight. What is the average temperature in Barstow during the period from : PM to 6 PM? A. 6 B. 89 C. D. E. 5 D. T avg = sin π 9 dt BC Eam Illegal to post on Internet

15 = ( +) + ( +) ( +) +... for all real. The function f is defined by the power series f numbers for which the series converges. What is the interval of convergence for f? A. (, ) B. (, ] C. [, ] D. (-, ) E. (-, ] D. f + = ( ) n n. This is a geometric series with ratio +. n = This converges when + < so < <. ( ) n + + OR lim n ( ) n + = : ( ) n + n = = : ( ) n + n = n + <= + < < <. n n = divergent n = divergent The interval of convergence is (, ).. + d = + + tan A. ln + D. ln + + tan + C B. tan + C C. ln + + C E. ln ( + ) + tan + C + D. d = d + d : u = +,du = d d = du + + u = ln + du d : u =,a = = + + a + u = tan + + d = + ln ( + ) + tan + C + C BC Eam Illegal to post on Internet

16 . If the region bounded by the -ais, the line = π, and the curve y = sin is revolved about the line = π, the volume of the solid generated would be A.. B..68 C..79 D.. E If f = + 5 +!! +! 5 5! E. R = π = π sin y π V = π sin y =.π =.586 :V = π by shells not in curriculum ( ) n n +... n! if n, find f π π sin d =.586 d A B. 5. C D E. 7.8 D. e = + + f = ! +! +! + 5!! +! 5 5! ( e ) d = ! +... n n! +... e = = !! +! 5 5!!! +! 5 5! 5. The Athabasca Glacier in Canada is one of the few glaciers to which you can drive. Like many glaciers in the world, it is receding (in this case its toe is moving away from the road). In the year, its toe was 5 feet from the road and in the year, its toe was 6 feet from the road. The rate the distance that the toe increases from the road is proportional to that distance. Assuming the glacier recedes at the same rate, what will be the average distance in feet per year (nearest integer) the glacier recedes from to the year 5? +... n n n! A. B. C. 5 D. 6 E. 8 C. s is the distance from the road : ds dt = ks s = Cekt. 5 = Ce s = 5e kt 6 = 5e k k = ln 6 5 The year 5 : s = 5e 5k. s Average rate of change : s = = e BC Eam Illegal to post on Internet

17 BC Eam Answers and Description # Ans. Topic (* Denotes BC Problem) # Ans. Topic (* Denotes BC Problem) C Derivatives of Basic Functions 9 D Improper Integrals * C Fundamental Thm of Calculus E Arc Length * E Continuity/Differentiability C Related Rates B Riemann Sums C Absolute Etrema 5 B Derivative of Accumulation Function E Second Derivative Test 6 D Straight-Line Motion - Integrals D Implicit Differentiation 7 D Integration by Parts * 5 A Tangent Lines and Local Linear Appro. 8 C Definition of Derivative 6 E Parametric Equations * 9 C Intermediate/Mean Value Theorem 7 D Polar Curves * A Taylor Polynomial Approimations * 8 C Taylor Series * E Straight-Line Motion - Derivatives 9 D Derivative as a Rate of Change C Series Convergence Intervals * D Average Value of a Function D Absolute Etrema D Series Convergence Intervals * E Improper Integrals * D Integration Techniques 5 C Limits - L'Hopital's Rule * E Area/Volume 6 E Function Analysis D Taylor Series * 7 A Limits/Horizontal Asymptotes 5 C Differential Equations 8 D Polar Curves * 9 C Logistic Growth * E Derivative of Inverse Functions D Function Analysis B Parametric Equations * A Euler's Method * B Error Bounds * 5 E Vector Functions * 6 E Horizontal/Vertical Tangent Lines 7 C Integration w/partial Fractions * 8 C Manipulation of Series * BC Eam Illegal to post on Internet