AP Calculus Prep Session Handout
The AP Calculus Exams include multiple choice and free response questions in which the stem of the question includes a table of numerical information from which the students were asked questions about the function, its graph, its derivative, or its definite integral. The answers were usually approximations. Sometimes a function that modeled the function in the table was also given and students were asked similar questions based on the model. Explanations of what was found were also required in some questions. Thus, starting with a numerical prompt, numerical, graphical, analytic, and verbal replies were required. What Students Should Be Able to Do Here are the most common things that are asked on AP Calculus exam table problems. Approximate a derivative (slope, rate of change, average rate of change) using difference quotients. Use a Riemann sum or a Trapezoidal approximation to approximate a definite integral. Explain the meaning of a definite integral in the context of the problem. Calculate a tangent line approximation (local linear approximation). Give the units of the answer (unit analysis). Answer theory question usually related to the Mean Value Theorem (MVT), Rolle s Theorem, the Intermediate Value Theorem (IVT) or the Extreme Value Theorem (EVT). Give information about the graph of the function. One thing not to do: Do not use your graphing calculators to produce a regression equation and use that to answer the questions. Finding regression equations, while very good mathematics, is not one of the four things you are allowed to do on the exam with your calculator. Also, a regression model is not the given function, only an approximation of it. You are expected to work from the table using calculus techniques and approximations. Using a regression will not earn any points. Page 2
Free Response 2003 AB 3 The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice differentiable and strictly increasing function R of time t. The graph of R and a table of selected values of R(t), for the time interval 0 t 90 minutes, are shown above. (a) Use data from the table to find an approximation for R (45). Show the computations that lead to your answer. Indicate units of measure. (b) (c) The rate of fuel consumption is increasing fastest at time t = 45 minutes. What is the value of R (45)? Explain your reasoning. Approximate the value of 0 90 R() t dt using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approximation less than the value of 0 90 R() t dt? Explain your reasoning. b (d) For 0< b 90 minutes, explain the meaning of R () t dt in terms of fuel 0 1 b consumption for the plane. Explain the meaning of R () t dt b in terms of fuel 0 consumption for the plane. Indicate units of measure in both answers. Page 3
2005 AB 3 BC 3 Distance cm Temperature 0 1 5 6 8 100 93 70 62 55 A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the temperature, in degrees Celsius, of the wire cm from the heated end. The function is decreasing and twice differentiable. (a) Estimate 7. Show the work that leads to your answer. Indicate units of measure. (b) Write an integral expression in terms of for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indi cated by the data in the table. Indicate units of measure. (c) Find, and indicate units of measure. Explain the meaning of in terms of the temperature of the wire. (d) Are the data in the table consistent with the assertion that " 0 for every in the interval 0 8? Explain your answer. Page 4
2003 AB 3 and BC 3 Form B Distance x (mm) Diameter B( x ) (mm) 0 60 120 180 240 300 360 24 30 28 30 26 24 26 A blood vessel is 360 millimeters (mm) long with circular cross sections of varying diameter. The table above gives the measurements of the diameter of the blood vessel at selected points along the length of the blood vessel, where x represents the distance from one end of the blood vessel and B( x) is a twice differentiable function that represents the diameter at that point. (a) Write an integral expression in terms of B( x ) that represents the average radius, in mm, of the blood vessel between x = 0 and x = 360. (b) (c) Approximate the value of your answer from part (a) using the data from the table and a midpoint Riemann sum with three subintervals of equal length. Show the computations that lead to your answer. 275 Bx ( ) Using correct units, explain the meaning of π dx 125 2 in terms of the blood vessel. 2 (d) Explain why there must be at least one value x, for 0 < x < 360, such that B ( x) = 0. Page 5
1998 AB 3 The graph of the velocity vt (), in ft/sec, of a car traveling on a straight road, for 0 t 50, is shown above. A table of values for vt (), at 5 second intervals of time t, is shown to the right of the graph. (a) During what intervals of time is the acceleration of the car positive? Give a reason for your answer. 2 (b) Find the average acceleration of the car, in ft/sec, over the interval 0 t 50. 2 (c) Find one approximation for the acceleration of the car, in ft/sec, at t = 40. Show the computations you used to arrive at your answer. 50 (d) Approximate vt () dtwith a Riemann, sum, using the midpoints of five 0 subintervals of equal length. Using correct units, explain the meaning of this integral. Page 6
Multiple Choice Part A. No Calculator Allowed. 2 5 7 8 10 30 40 20 1. The function is continuous on the closed interval [2, 8] and has values that are given in the table above. Using the subintervals [2, 5], [5, 7], and [7, 8], what is the trapezoidal approximation of? a. 110 b. 130 c. 160 d. 190 e. 210-1 6 5 3-2 1 3-3 -1 2 3 1-2 2 3 2. The table above gives values of,,, and at selected values of. If, then 1 a. 5 b. 6 c. 9 d. 10 e. 12 Page 7
3. For all in the closed interval [2, 5], the function has a positive first derivative and a negative sec ond derivative. Which of the following could be a table of values for? 2 7 3 9 4 12 5 16 2 16 3 14 4 11 5 7 a. d. 2 7 3 11 4 14 5 16 2 16 3 13 4 10 5 7 b. e. 2 16 3 12 4 9 5 7 c. Page 8
Part B. Graphing Calculator Allowed. 0 0.5 1.0 1.5 2.0 3 3 5 8 13 4. A table of values for a continuous function is shown above. If four equal subintervals of [0, 2] are used, which of the following is the trapezoidal approximation of? a. 8 b. 12 c. 16 d. 24 e. 32 sec 0 2 4 6 ft/ 5 2 8 3 5. The data for the acceleration of a car from 0 to 6 seconds are given in the table above. If the velocity at 0 is 11 feet per second, the approximate value of the velocity at 6, computed using a left hand Riemann sum with three subintervals of equal length, is a. 26 ft/sec b. 30 ft/sec c. 37 ft/sec d. 39 ft/sec e. 41 ft/sec Page 9
2 5 10 14 12 28 34 30 6. The function is continuous on the closed interval [2, 14] and has values as shown in the table above. Using the subintervals [2, 5], [5, 10], and [10, 14], what is the approximation of a. 296 b. 312 c. 343 d. 374 e. 390 found by using a right Riemann sum? 0 1 2 1 2 7. The function is continuous on the closed interval [0, 2] and has values that are given in the table above. The equation must have at least two solutions in the interval [0, 2] if a. 0 b. c. 1 d. 2 e. 3 Page 10
-4-3 -2-1 0 1 2 3 4 2 3 0-3 -2-1 0 3 2 8. The derivative of a function is continuous and has exactly two zeros. Selected values of are given in the table above. If the domain of is the set of all real numbers, then is decreasing on which of the following intervals? a. 2 2 only b. 1 1 only c. 2 d. 2 only e. 2 or 2 0 1 2 3 4 2 3 4 3 2 9. The function is continuous and differentiable on the closed interval [0, 4]. The table gives selected values of on this interval. Which of the following statements must be true? a. The minimum value of on [0, 4] is 2. b. The maxim um value of on [0, 4] is 4. c. 0 for 0 4. d. 0 fo r 2 4 e. There exists, with 0 4, for which 0. Page 11