( + ) 3. AP Calculus BC Chapter 6 AP Exam Problems. Antiderivatives. + + x + C. 2. If the second derivative of f is given by f ( x) = 2x cosx

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1 Chapter 6 AP Eam Problems Antiderivatives. ( ) + d = ( + ) ( + ) 6 ( + ). If the second derivative of f is given by f ( ) = cos, which of the following could be f( )? + cos + cos + + cos + sin + + sin +. Find sec d = tan csc cos sec sec tan+ C Evaluating Definite Integrals. If f is a linear function and < a < b, then f ( ) d = a ab b b a b a 5. Find d = If ( ) + k d = 6, then k =

2 Chapter 6 AP Eam Problems 7. Find d = 7 ln 8. Find e d = e e e e e e+ e e 9. Find ( ) d = Find d = 6 6 noneistent. Find d = + 5 ln k. If ( ) k d = 8, then k = What are all values of k for which k d =? and,, and

3 Chapter 6 AP Eam Problems. Which of the following is equal to sin d? cos d sin d cos d sin d sin d 5. Find d = Find d = If the function f has a continuous derivative on [, c], then c f ( ) d= f( c) f() f( c ) f ( c) f () f( c) f() f( ) 8. If for f( ) =, then for > e f( ) d= e e Find ( ) d ( ) + d = If p is a polynomial of degree n, n >, what is the degree of the polynomial Q ( ) = ptdt ( )? n n n +

4 Chapter 6 AP Eam Problems. (988 AB6) Let f be the differentiable function, defined for all real numbers, with the following properties. (i) f ( ) = a + b (ii) f () = 6 and f () = 8 (iii) f( ) d = 8 Find f( ). Show your work. U Substitutions. Find d + 5 = 9 5 C 5 C 5 C ( ) + ) ( ) + + ( ) ( ) + + ( ). Find cosθ sin d θ = + θ ( ) ( + ) ( ). Find d = ln( + ) C ln + 5. Find ( ) + d =

5 Chapter 6 AP Eam Problems d =, then y = 6. If cos( ) cos( ) sin( ) cos ( ) sin ( ) sin( ) 7. Find sin( ) d = 8. Find sin( ) + d = cos + cos + cos( + ) ( ) cos( + ) ( ) cos + ( ) 9. If sin cos d = and if y = when =, what is the value of y when =?. Find d = 6 ln ln. Which of the following is equal to d? 5 arcsin 5 arcsin 5 5 arcsin 5 5 5

6 . An antiderivative for + is AP Calculus BC Chapter 6 AP Eam Problems ( ) ln( ) ) ln + + sec ( ) tan ( ) e. An antiderivative for f ( ) = e + is + e e + e e + e e + ( ). Find e d = e e + e e + e e e e e e e e + e e + e 5. Find e d = ( ) e e 6. Find tan( ) d = e e ( e ) ln cos( ) ln cos( ) ln cos( ) sec( )tan( ) ln cos( ) 7. Find + d = ln ln ln ln ln5 6

7 Chapter 6 AP Eam Problems du 8. For >, d = u C ) ln ( ln ) (ln ) 8 ln( ) 9. Let F () be an antiderivative of ( ln ). If F () =, then F (9) = ,6.5. Find + + d = ln8 ln ln8 ln + ln8 ln ln. If f is a continuous function and if F ( ) = f( ) for all real numbers, then F() F() F(6) F() F(6) F() F() F() F(6) F() f( ) d =. If the substitution u = is made, the integral d = u du u u du u u du u u du u u u du 7

8 Chapter 6 AP Eam Problems Separation of Variables. If d = y and if y = when =, then when =, y =. If y d = and if y = when =, what is the value of for which y =? ln ln ln 5. At each point (, y ) on a certain curve, the slope of the curve is point (, 8), then its equation is y. If the curve contains the 6. If y y d = 8e = + 8 y ( ) sec y= e + 7 = ln = y and if y = 5 when =, then y = y = If tan e + tan 5 e + 5 tan + 5 d = y, then y could be tan e tan + 5e ln 7 e + e e + 8. (998 AB) Let f be a function with f () = such that for all points (, y) on the graph of f the + slope is given by. y (a) Find the slope of the graph of f at the point where =. (b) Write an equation for the line tangent to the graph of f at = and use it to approimate f (.). 8

9 Chapter 6 AP Eam Problems (c) Find f( ) by solving the separable differential equation with the initial condition f () =. (d) Use your solution from part (c) to find f (.). = d + y 9. (985 BC) Given the differential equation y =, y >. d ln y (a) (b) (c) Find the general solution of the differential equation. Find the solution that satisfies the condition that y= e when =. Epress your answer in the form y= f( ). Eplain why = is not in the domain of the solution found in part (b). 5. ( AB6) Consider the differential equation =. y d e (a) Find a solution y= f( ) to the differential equation satisfying (b) Find the domain and range of the function f founding part (a). f () =. 5. ( BC 5) Consider the differential equation =. d y (a) Let y= f( ) be the particular solution to the given differential equation for < < 5 such that the line y = is tangent to the graph of f. Find the coordinate of the point of tangency, and determine whether f has a local maimum, local minimum or neither at this point. (b) Let y= g( ) be the particular solution to the given differential equation for < < 8, with the initial condition g (6) =. Find y= g( ). 5. ( AB6) The function f is differentiable for all real numbers. The point, is on the graph of y= f( ), and the slope at each point (, y) on the graph is given by y (6 ) d =. (a) d y Find and evaluate it at the point, d. (b) Find y= f( ) by solving the differential equation condition f () = /. = y (6 ) with the initial d 9

10 Chapter 6 AP Eam Problems 5. ( BC5) Let f be the function satisfying f ( ) = f( ), for all real numbers, with f () = and lim f( ) =. (c) Write an epression for y= f( ) by solving the differential equation = y with d the initial condition f () =. (5 points) Integration by Parts 5. Which of the following is equal to ln? ln+ln ln8 ln t edt lntdt dt t 55. Find e d= e e e 56. Find cos d = 57. Find sec d= tan tan ln cos tan tan + ln cos sec + sec tan + C 58. Find f( ) d = f( ) f ( ) d f( ) f ( ) d f( ) ( ) f d f( ) d f( ) f( )

11 Chapter 6 AP Eam Problems 59. If f ( )sin d = f ( )cos + cos d, then ( ) f could be sin cos Integration by Partial Fractions d 6. Find ( )( + ) = ln + + ln ln ( )( + ) + ( ln )( ln ) ln ( )( + ) + C 6. Find ( )( ) d = ln 8 ln 5 ln 5 6. Find 6 8 d = + ln ln ln ( )( + ) ln ( )( ) ln ( )( )

12 Chapter 6 AP Eam Problems Slope Fields and Euler s Method 6. Shown above is a slope field for which of the following differential equations? d = ) y d = = d d y = + ln d = y d =. 6. ( BC6) Consider the differential equation given by ( y ) (a) On the aes provided, sketch a slope field for the given differential equation at the eleven points indicated. (b) Use the slope field for the given differential equation to eplain why a solution could not have the graph shown below.

13 Chapter 6 AP Eam Problems (c) Find the particular solution y= f( ) to the given differential equation with the initial condition f () =. (d) Find the range of the solution found in part (c). 65. (998 BC) Consider the differential equation given by y d =. (a) On the aes provided below, sketch a slope field for the given differential equation at the nine points indicated. (b) Let y= f( ) be the particular solution to the given differential equation with the initial condition f () =. Use Euler s method starting at =, with a step size of., to approimate f (.). Show the work that leads to your answer. (c) Find the particular solution y= f( ) to the given differential equation with the initial condition f () =. Use your solution to find f (.).

14 Chapter 6 AP Eam Problems 66. ( BC5) Consider the differential equation y d =. (a) The slope field for the given differential equation is provided. Sketch the solution curve that passes through the point (, ) and sketch the solution curve that passes through the point (, ). (b) (c) (d) Let f be the function that satisfies the given differential equation with the initial condition f () =. Use Euler s method, starting at = with a step size of. to approimate f (.). Show the work that leads to your answer. Find the value of b for which y = + b is a solution to the given differential equation. Justify your answer. Let g be the function that satisfies the given differential equation with the initial condition g () =. Does the graph of g have a local etreme at the point (, )? If so, is the point a local maimum or a local minimum? Justify your answer. 67. ( AB6) Consider the differential equation ( y ) d =. (a) On the aes provided, sketch a slope field for the given differential equation at the twelve points indicated.

15 Chapter 6 AP Eam Problems (b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the y plane. Describe all points in the y plane for which the slopes are positive. (c) Find the particular solution y= f( ) to the given differential equation with the initial condition f () =. Eponential Growth and Decay 68. A puppy weighs. pounds at birth and.5 pounds two months later. If the weight of the puppy during its first 6 months is increasing at a rate proportional to its weight, then how much will the puppy weigh when it is months old?. pounds.8 pounds 6.5 pounds.6 pounds 5.6 pounds 69. Population y grows according to the equation ky dt =, where k is a constant and t is measured in years. If the population doubles every years, then the value of k is Bacteria in a certain culture increase at a rate proportional to the number present. If the number of bacteria doubles in three hours, in how many hours will the number of bacteria triple? ln ln ln ln ln ln ln 7 9 ln 7. During a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. If, people are infected when the epidemic is first discovered, and, are infected 7 days later, how many people are infected days after the epidemic is first discovered?,,67,,57 dp P 7. The population Pt ( ) of a species satisfies the logistic differential equation = P dt 5, where the initial population P () =, and t is the time in years. What is lim Pt ( )?,5,, 5,, t 5

16 Chapter 6 AP Eam Problems 7. (989 AB6) Oil is being pumped continuously from a certain oil well at a rate proportional to the amount of oil left in the well; that is, ky dt =, where y is the amount of oil left in the well at any time t. Initially there were,, gallons of oil in the well and 6 years later there were 5, gallons remaining. It will no longer be profitable to pump oil when there are fewer than 5, gallons remaining. (a) Write an equation for y, the amount of oil remaining in the well at any time t. (b) At what rate is the amount of oil in the well decreasing when there are 6, gallons of oil remaining? (c) In order not to lose money, at what time t should oil no longer be pumped from the well? kt 7. (996 B The rate of consumption of cola in the United States is given by St ( ) = Ce, where S is measured in billions of gallons per year and t is measured in years from the beginning of 98. (a) The consumption rate doubles every 5 years and the consumption rate at the beginning of 98 was 6 billion gallons per year. Find C and k. (b) Find the average rate of consumption of cola over the year time period beginning January, 98. Indicate units of measure. (c) Use the trapezoidal rule with four equal subdivisions to estimate 7 5 Stdt ( ). (d) 7 Using correct units, eplain the meaning of Stdt ( ) in terms of cola consumption. 75. (97 AB7) The rate of change in the number of bacteria in a culture is proportional to the number present. In a certain laboratory eperiment, a culture had, bacteria initially,, bacteria at time t minutes, and, bacteria at t + minutes. 5 (a) (b) (c) In terms of t only, find the number of bacteria in the culture at any time t minutes, t. How many bacteria were there after minutes? How many minutes had elapsed when the, bacteria were observed? 76. (987 B At any timet, in days, the rate of growth of a bacteria population is given by y = ky, where k is a constant and y is the number of bacteria present. The initial population is, and the population triples during the first 5 days. (a) Write an epression for y at any time t. (b) By what factor will the population have increased in the first days? (c) At what time t, in days, will the population have increased by a factor of 6? 6

17 Chapter 6 AP Eam Problems 77. ( BC5) A population is modeled by a function P that satisfies the logistic differential dp P P equation = dt 5. (a) If P () =, what is lim Pt ( )? If P () =, what is lim Pt ( )? t (b) If P () =, for what value of P is the population growing the fastest? (d) A different population is modeled by a function Y that satisfies the separable differential dy Y t equation = dt 5. Find Yt ( ) if () (d) For the function Y found in part (c), what is lim Yt ( )? t t Antiderivatives. E 99 AB #7 65%. A 99 AB #8 8%. A 988 AB #5 89% Evaluating Definite Integrals. A 998 AB # % 5. D 985 AB # 89% 6. D 985 AB # 7% 7. C 998 AB # 7% 8. E 998 AB #7 % 9. D 988 AB #7 7%. E 985 BC #6 6%. A 985 AB # 66%. C 988 AB # 8%. A 998 AB # 69%. A 99 BC # 7% 5. D 985 AB #7 5% 6. C 988 AB #8 % 7. A 988 AB # 7% 8. B 99 BC #7 7% 9. B 99 AB #8 %. E 99 BC # 8%. 988 AB #6 U Substitutions. D 988 AB #7 79%. D 988 AB # 67%. A 99 AB # 69% 5. D 988 BC # 89% 6. C 985 AB # 8% 7. D 985 AB # 56% 8. C 985 BC #8 89% 9. B 998 BC #8 55%. A 99 AB # 9%. A 985 BC #7 57%. E 99 AB # %. E 985 BC #8 5%. A 988 BC #6 8% 5. A 99 BC #7 8% 6. B 985 AB # 55% 7. B 988 AB #9 58% 8. E 988 AB #8 7% 9. C 998 AB #88 55%. B 985 BC # 9%. E 998 AB #8 %. A 985 BC # 7% Separation of Variables. B 99 AB # %. C 985 BC # 8% 5. A 985 BC # 5% 6. C 988 BC #9 % 7. C 99 BC # % AB # BC # 5. AB #6 5. BC #5 FormB 5. AB #6 5. BC #5 Integration by Parts 5. E 985 AB #7 6% 55. C 985 AB #7 % 56. E 988 AB #6 59% 57. E 99 BC #9 6% 58. B 99 AB # 6% 59. B 985 BC # 6% Integration by Partial Fractions 6. A 985 BC # 6% 6. D 988 BC #7 7% 6. A 998 BC # 6% Slope Fields and Euler s Method 6. C 998 BC # 8% 6. BC # BC # 66. BC #5 67. AB #6 Eponential Growth and Decay 68. B 99 AB # % 69. A 998 AB #8 % 7. A 988 BC # % 7. C 99 BC #8 6% 7. E 998 BC #6 % AB #6 FRQ BC # FRQ AB #7 FRQ BC # FRQ 77. BC #5 FRQ 7