Chapter 29 BC Calculus Practice Test

Transcription

1 Chapter 9 BC Calculus Practice Test

2 The Eam AP Calculus BC Eam SECTION I: Multiple-Choice Questions DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time hour and 5 minutes Number of Questions 5 Percent of Total Grade 50% Writing Instrument Pencil required Instructions Section I of this eamination contains 5 multiple-choice questions. Fill in only the ovals for numbers through 5 on your answer sheet. CALCULATORS MAY NOT BE USED IN THIS PART OF THE EXAMINATION. Indicate all of your answers to the multiple-choice questions on the answer sheet. No credit will be given for anything written in this eam booklet, but you may use the booklet for notes or scratch work. After you have decided which of the suggested answers is best, completely fill in the corresponding oval on the answer sheet. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely. Here is a sample question and answer. Sample Question Sample Answer Chicago is a A B C D E (A) state (B) city country continent (E) village Use your time effectively, working as quickly as you can without losing accuracy. Do not spend too much time on any one question. Go on to other questions and come back to the ones you have not answered if you have time. It is not epected that everyone will know the answers to all the multiple-choice questions. About Guessing Many candidates wonder whether or not to guess the answers to questions about which they are not certain. Multiple choice scores are based on the number of questions answered correctly. Points are not deducted for incorrect answers, and no points are awarded for unanswered questions. Because points are not deducted for incorrect answers, you are encouraged to answer all multiple-choice questions. On any questions you do not know the answer to, you should eliminate as many choices as you can, and then select the best answer among the remaining choices.

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4 CALCULUS BC SECTION I, Part A Time 55 Minutes Number of questions 8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems, using the available space for scratchwork. After eamining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem. In this test: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f() is a real number.. Which of the following is a y-coordinate for the equation y = when the tangents to the curve equal zero? (A) 5 6 (B) 6 0 (E) What is the sum of the series (A) ln( + 5) (B) e 5 ln( 5 ) 5 (E) The series diverges. n n+ 5 n + +? AP Calculus AB & BC Practice Questions

5 + +. lim 0 (A) 0 (B) 6 8 (E) Undefined. Find d y d at = for y y = + 7. (A) 6 9 (B) (E) 6 9 BC Calculus Practice Test 9

6 + h + h 5. lim h 0 h (A) (B) (E) Undefined = d 6. = (A) tan 5 C (B) sin ( 5) + C tan ( 5) + C (E) tan sin ( 5) + C ( 5) + C 7. ( + 9) + d = (A) ln tan + ln + + C 5 (B) ln 9 tan ln C ln tan ln + + C ln tan + ln + + C (E) ln 9 tan + + ln C AP Calculus AB & BC Practice Questions

7 8. If dy = y and y(0) =, find an equation for y in terms of. d (A) y = e (B) y = e y = e y = e (E) y = e ( ) 9. Find the derivative of y = + (A) (B) (E) y + + y y y + + y 6 + BC Calculus Practice Test 9

8 0. dy = ( ) y f = d and. Find an equation for y in terms of. (A) y = (B) y = y = y = (E) y = +. Find the derivative of y = cos +. (A) (B) (E) AP Calculus AB & BC Practice Questions

9 =. lim 5 + (A) (B) 8 0 (E) 8. d ( csc sec ) = d (A) sec csc (B) sec csc csc sec sec + csc (E) csc + sec. d tan d + = (A) (B) (E) + sec sec + + sec + + sec sec + + BC Calculus Practice Test 9 5

10 lim = (A) 0 (B) 6 (E) The limit is undefined. 6. Where is the tangent line perpendicular to the y-ais for the curve y = + 7 located? (A) y = 5 (B) y = 7 = 5 y = (E) = 7 7. If f is continuous on the interval [,] and differentiable everywhere on (,), find = c, where f(c) is the mean value of = + f. (A) (B) 0 (E) AP Calculus AB & BC Practice Questions

11 8. A toy manufacturer has determined the total profit for a month can be determined by the equation P = , where is the number of thousands of toys sold. How many thousands of toys should be sold to maimize the profit that month? (A) (B) 5 (E) 6 9. Find the derivative of f =. ( (A) ) + ln (B) e ln ( + ln ) + ln + ln ln (E) e + ln = 0. If f, f () = (A) y + (B) + y + y ( ) (E) BC Calculus Practice Test 9 7

12 + 6. Find the equation of the line normal to the graph of y = + at (,6). (A) y = + 5 (B) y = 7 y = + 5 y = + 7 (E) y = +. Find the value of C that satisfies the MVTD for f 5 on the interval [0,]. = + (A) 6 9 (B) 9 (E) AP Calculus AB & BC Practice Questions

13 . If y = 7 cos, then dy d = ( cos cos sin ) ( ln )( cos + cos sin ) ( ln )( cos + cos sin ) ( ln )( cos cos sin ) cos (A) 7 cos + cos sin cos (B) 7 cos 7 7 cos 9 7 cos (E) 9 7. Find the derivative of the inverse of y = + 5 when y = 7. (A) 9 (B) 0 9 (E) = ( ) 5. Find the derivative of the inverse of f sin 6π at f π. = for 0 (A) (B) (E) BC Calculus Practice Test 9 9

14 = d (A) C (B) + C C (E) C C 7. Find the derivative of y = + +? (A) (B) y y y y y y y y (E) y AP Calculus AB & BC Practice Questions

15 8. Find lim. cos π (A) (B) (E) 0 End of Part a, Section I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY. DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO. BC Calculus Practice Test 0

16 CALCULUS BC SECTION I, Part B Time 50 Minutes Number of questions 7 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems, using the available space for scratchwork. After eamining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem. In this test:. The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approimates the eact numerical value.. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f() is a real number. 9. e + 7 d = (A) e C (B) e C e e C C (E) e C AP Calculus AB & BC Practice Questions

17 ( + ) d = 7 (A) ln ln C 0 5 (B) 7 ln ln C ln + ln C ln ln C 5 5 (E) 7 ln + ln C 5 5. Find b where y = a + b and y = + c have a common tangent at (,). (A) (B) 8 8 (E) 0. Car A and car B leave a town at the same time. Car A drives due north at a rate of 60 km/hr and car B goes east at a rate of 80 km/hr. How fast is the distance between them increasing after hours? (A) 0 km/hr (B) 60 km/hr 00 km/hr 00 km/hr (E) 70 km/hr BC Calculus Practice Test 0

18 . Approimate cos 9. (A) 79 π 80 (B) 79π 80 π 80 π 80 (E) 0. A 0-foot ladder leaning against a wall is pushed up the wall at a rate of ft/sec. How fast is the ladder sliding across the ground towards the wall when it is 8 feet up the wall from the ground? (A) 9 ft/sec (B) 9 ft/sec ft/sec ft/sec (E) ft/sec 5. Approimate the area under the curve y = + from = to = using four left-endpoint rectangles. (A). (B) (E) AP Calculus AB & BC Practice Questions

19 6. What is the mean value of f (A) 96 (B) 96 (E) There is no such value. = cos + over the interval (,)? 7. Find the second derivative of y = at =. (A) 0 (B) 6 (E) 0 8. What is dy d (A) (B) ln if y =? ln (E) ln( ) BC Calculus Practice Test 0 5

20 9. + d + ln (A) (B) (ln ) (ln ) ln ln (E) (ln ) 0. Find the derivative of the inverse of y = at y =. (A) (B) (E) AP Calculus AB & BC Practice Questions

21 . A frame is bought for a photo that is in. The artist would like to have a mat that is in on the top and bottom and 6 in on each side. Find the dimensions of the frame that will minimize its area. ( + ) ( + ) in ( + ) in in ( + ) in ( + ) ( + ) in ( + ) (A) + in 6 6 in (B) (E) 6 6 in in in. What is the length of the curve y = from = to = 6. (A) 7 (B) (E) Find the length of the curve defined by = t + 7 and y = t 8 ( + ) from t = 0 to t = 8. (A) 6 (B) (E) 00 BC Calculus Practice Test 0 7

22 y. Use Euler s method with h = 0. to estimate y = if y = and y(0) = 0. (A) (B) (E) Which of the following series converges? (A) (B) (E) n n= n= n= n n n= n= n n n STOP End of Part b, Section I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART B ONLY. DO NOT GO ON TO SECTION II UNTIL YOU ARE TOLD TO DO SO AP Calculus AB & BC Practice Questions

23 I SECTION II GENERAL INSTRUCTIONS You may wish to look over the problems before starting to work on them, since it is not epected that everyone will be able to complete all parts of all problems. All problems are given equal weight, but the parts of a particular problem are not necessarily given equal weight. A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS ON THIS SECTION OF THE EXAMINATION. You should write all work for each part of each problem in the space provided for that part in the booklet. Be sure to write clearly and legibly. If you make an error, you may save time by crossing it out rather than trying to erase it. Erased or crossed-out work will not be graded. Show all your work. You will be graded on the correctness and completeness of your methods as well as your answers. Correct answers without supporting work may not receive credit. Justifications require that you give mathematical (noncalculator) reasons and that you clearly identify functions, graphs, tables, or other objects you use. You are permitted to use your calculator to solve an equation, find the derivative of a function at a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your problem, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results. Your work must be epressed in standard mathematical notation rather than calculator synta. For eample, 5 ò d may not be written as fnint (X, X,, 5). Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approimation, it should be correct to three places after the decimal point. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f() is a real number. BC Calculus Practice Test 0 9

24 I SECTION II, PART A Time 0 minutes Number of problems A graphing calculator is required for some problems or parts of problems. During the timed portion for Part A, you may work only on the problems in Part A. On Part A, you are permitted to use your calculator to solve an equation, find the derivative of a function at a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your problem, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results dy d The equation for y is thrice differentiable for > 0 with y = at =, the second derivative is equal to at =, and the third derivative is at =. Values of the first derivative are given for select values of above. (a) Write an equation for the tangent line of y at =. Use this line to approimate y at =. (b) Use a right endpoint Riemann sum with five subintervals of equal length and values from the table to approimate dy d d. Use this approimation to estimate y at =. Show your work. (c) Use Euler s Method, starting at = with five steps of equal size, to approimate y at =. Show your work. (d) Write a third degree Taylor polynomial for y about =. Use it to approimate y at =.. Let R be the region bound by y = 8 and y = (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the -ais for 0. (c) Find the volume of the solid generated when R is revolved about the line = when AP Calculus AB & BC Practice Questions

25 I SECTION II, PART B Time hour Number of problems No calculator is allowed for these problems. During the timed portion for Part B, you may continue to work on the problems in Part A without the use of any calculator.. Consider the graph of the polar curve r = + sinθ for 0 θ π. Let S be the region bound between the inner and outer loops. (a) Write an integral epression for the area of S. (b) Write epressions for d dy and in terms of θ. dθ dθ π (c) Write an equation in terms of and y for the line normal to the graph of the polar curve at the point where θ =. Show your work.. For t 0, a particle is moving along a curve so that its position at time t is ((t), y(t)). At time t =, the particle is at position (,). It is known that d t dy = e ( t + ) and = cos t. dt dt (a) Is the horizontal movement to the left or to the right at time t =? Find the slope of the particle s path at t =. (b) Find the y-coordinate of the particle s position at time t = 6. (c) Find the speed and acceleration of the particle at t = 6. (d) Find the distance traveled by the particle from time t = to t = 6. BC Calculus Practice Test

26 I 5 5. A particle s position in the y-plane at any time t is given by = t and y = t t. Find: (a) The and y components of the particle s velocity. dy (b) at t =. d (c) The acceleration of the particle at t =. (d) The time(s) when the particle is changing direction. 6. Let y = y and y = y y. (a) If = 0 and y = 6, find y. (b) If = and y = e, find y. (c) Use Euler s method to approimate y when =. Start at = 0 using three steps. Check your answer against the real value of y at =. Is this a reasonable approimation? stop End of Eam 550 AP Calculus AB & BC Practice Questions