Unit #6 Basic Integration and Applications Homework Packet

Transcription

1 Unit #6 Basic Integration and Applications Homework Packet For problems, find the indefinite integrals below.. x 3 3. x 3x 3. x x 3x 4. 3 / x x 5. x 6. 3x x3 x 3 x w w 7. y 3 y dy 8. dw Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 47

2 9. 3 x 3 x. ( x 3)( x 3). cos d. x sin x For problems 3 and 4, find the indicated function based on the given information. 3. If f '( x) x sin x and f() = 4, find f(x). 4. If f ''( x) x, f '() 6, and f ( ) 3, find f(x). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 473

3 Given below is a table of function values of h(x). Approximate each of the following definite integrals using the indicated Riemann or Trapezoidal sum, using the indicated subintervals of equal length. x h(x) h ( x) using two subintervals and a Left 3 Hand Riemann sum h ( x) using three subintervals and a Right 3 Hand Riemann sum h ( x) using three subintervals and a 3 Midpoint Riemann sum h ( x) using three subintervals and a 3 Trapezoidal sum h ( x) using six subintervals and a Trapezoidal sum. 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 474

4 For questions and, approximate the definite integrals. Make a table of values showing your intervals that you used.. Approximate xsin x using four subintervals of equal length and a Right Hand Riemann sum.. Approximate e x using four subintervals of equal length and a Trapezoidal sum.. Given the table to the right, approximate 9 ( x) P using three subintervals and a Midpoint Riemann sum. x P(x) Given the table to the right, approximate 9 ( x) P using six subintervals and a Trapezoidal sum. x P(x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 475

5 For exercises 4 9, find the value of the definite integral. Show your algebraic work. 4. t t dt 5. 3 x 4 u 6. du u 7. x x 8. ( sin ) 3 x 9. x 5x 4 3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 476

6 Pictured to the right is the graph of a function f. In exercises 3 35, find the values of each of the following definite integrals f ( x) 3. 4 f ( x) 3. f ( x) 33. f '( x) 34. f '( x) 35. f 4 3 '( x) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 477

7 Given f ( x) and ( ) g x, find the values of each of the following definite integrals, if possible, by rewriting the given integral using the properties of integrals. 6 6 a. [ f ( x) g( x)] b. [ f ( x) 3g( x)] c. 6 ( x) 6 g d. 6 g( x) f ( x) Given f ( x) 6 and ( ) 4 g x, find the values of each of the following definite integrals. Rewrite the given integral using the properties of integrals. Then, find the value. 4 a. [ ( x) 4] 4 f b. g ( x) x 3 4 c. f x) 3x ( Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 478

8 Pictured below is the graph of f '( x), the first derivative of a function f(x). Use the graph to answer the following questions Graph of f '( x) 38. What is the value of 7 f '( x) 39. If f() = 3, what is the value of f(3)? 4. If f(3) =, what is the value of f(7)? The graph of f '( x), the derivative of a function, f(x), is pictured below on the interval [, 6]. Write and find the value of a definite integral to find each of the indicated values of f(x) below. Also, f( ) = Find the value of f(). 4. Find the value of f(6). Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 479

9 Problem #4 Let f be a function defined on the closed interval 3 < x < 4 with f() = 3. The graph of of f, consists of one line segment and a semicircle, as shown above. a. On what intervals, if any, is f increasing. Justify your reasoning. f ', the derivative b. Find the x coordinate of each point of inflection of the graph of f on the open interval 3 < x < 4. Justify your answer. c. Find an equation for the line tangent to the graph of f at the point (, 3). d. Find f( 3) and f(4). Show the work that leads to your answers. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 48

10 The graph to the right represents the velocity, v(t) in meters per second, of a particle that is moving along the x axis on the time interval < t <. The initial position of the particle at time t = is. 43. On what interval(s) of time is the particle moving to the left and to the right? Justify your answer. 44. What is the total distance that the particle has traveled on the time interval < t < 7. Leave your answer in terms of π. Indicate units of measure. 45. What is the net distance that the particle travels on the interval 5 < t <? Round your answer to the nearest thousandth. Indicate units of measure. 46. What is the acceleration of the particle at time t =? Indicate units of measure. 47. What is the position of the particle at time t = 5? Indicate units of measure. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 48

11 Pictured to the right is the graph of a function which represents a particle s velocity on the interval [, 4]. Answer the following questions. 48. For what values is the particle moving to the right? Justify your answer. 49. For what values is the particle moving to the left? Justify your answer. 5. For what values is the speed of the particle increasing? Justify your answer. 5. For what values is the speed of the particle decreasing? Justify your answer. 5. What is the net distance that the particle travels on the interval [, 4]? 53. What is the total distance that the particle travels on the interval [, 4]? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 48

12 A car travels on a straight track. During the time interval < t < 6 seconds, the car s velocity, v, measured in feet per second, and acceleration, a, measured in feet per second per second, are continuous functions. The table below shows selected values of these functions. t (sec) v(t) (ft/sec) a(t) (ft/sec ) Using appropriate units, explain the meaning of 6 v( t) dt in terms of the car s motion. Approximate this integral using a midpoint approximation with three subintervals as determined by the table. 55. Using appropriate units, explain the meaning of 5 a( t) dt in terms of the car s motion. Find the 5 exact value of the integral. 56. Is there a value of t such that a (t) =? If so, on what interval does such a value exist? Justify your reasoning. 57. Using appropriate units, approximate the value of v (3). What does this value say about the motion of the car at t = Using appropriate units, find the value and explain the meaning of 6 a ( t 35 5 ) dt. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 483

13 At time t =, there are pounds of sand in a conical tank. Sand is being added to the tank at the rate of sin t S ( t) e pounds per hour. Sand from the tank is used at a rate of R( t) 5sin t 3 t per hour. The tank can hold a maximum of pounds of sand. 59. Find the value of 4 S ( t) dt. Using correct units, what does this value represent? 6. Find the value of 3 R ( t) dt. Using correct units, what does this value represent? 6. Find the value of 4 4 S ( t) dt. Using correct units, what does this value represent? 6. Write a function, A(t), containing an integral expression that represents the amount of sand in the tank at any given time, t. 63. How many pounds of sand are in the tank at time t = 7? 64. After time t = 7, sand is not used any more. Sand is, however, added until the tank is full. If k represents the value of t at which the tank is at maximum capacity, write, but do not solve, an equation using an integral expression to find how many hours it will take before the tank is completely full of sand. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 484

14 Problem # Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 485

15 Problem #3 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 486

16 Problem # Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 487

17 65. Using a right Riemann sum over the given intervals, estimate 35 F ( t) dt 5 A. 73 B. 66 C. 564 D. 474 E For the first six seconds of driving, a car accelerates at a rate of t a( t) sin meters per second. Which one of the following expressions represents the velocity of the car when it first begins to decelerate? A B C.. 75 D E a ( t) dt a ( t) dt a ( t) dt a ( t) dt a ( t) dt 67. The rate at which gas is flowing through a large pipeline is given in thousands of gallons per month in the chart below. Use a midpoint Riemann sum with two equal subintervals to approximate the number of gallons that pass through the pipeline in a year. A. 594, B. 67, C. 73, D. 744, E.,68, Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 488

18 68. Let f be a continuous function on the closed interval [, ]. If the values of f are given below at three points, use a trapezoidal approximation to find f ( x) using two subintervals. A. 65 B. 7 C. 9.5 D. 4 E. 8 b 69. If f x) a 3b a (, then f x) 3 b a ( = A. a 3b + 3 B. 3b 3a C. a D. 5a 6b E. a 6b Use the table below to answer questions 6 and 7. Suppose the function f(x) is a continuous function and f is the derivative of F(x). 7. What is 3 f ( x)? A. 5 B. 8 C. 4 D. 9 E. Cannot be determined 7. If the area under the curve of f(x) on the interval < x < is equal to the area under the curve f(x) on the interval < x < 3, then what is the value of A? A. 4 B. C. 5.5 D. 6 E. Cannot be determined Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 489