Calculus with the Graphing Calculator Using a graphing calculator on the AP Calculus exam Students are expected to know how to use their graphing calculators on the AP Calculus exams proficiently to accomplish the following four capabilities: Plot the graph of a function within an arbitrary viewing window Find the zeros of functions (solve equations numerically) Numerically calculate the derivative of a function Numerically calculate the value of a definite integral AP Calculus exam format The AP exam has four parts: 1. 8 question multiple choice test without calculators (55 min). 17 question multiple choice test with calculators (50 min) 3. question free response test with calculators (30 min) 4. 4 question free response test without calculators (60 min) After the completion of the first free response portion, students will receive the "non-calculator free response" test, but will still be able to work on the original free response questions. However, they will no longer have access to their graphing calculators for those questions. A student's ability to use a graphing calculator effectively will be tested on the calculator active portions of the AP Exam. Students should know when and how to properly use their graphing calculators on the calculator active questions. One very common mistake that hurts a student's score is trying to work out a complex problem by hand when a calculator is available!!! IMPORTANT NOTE: Graphing calculators are a valuable tool for numeric calculations and to understand the behavior of a graph but CANNOT be used as justification on free response questions. An appropriate mathematical justification and/or explanation is necessary. Also, when using a graphing calculator on the free response questions, avoid writing down calculator syntax on the exam. Credit will not be awarded for simply writing down what is "typed" into the calculator. Proper calculus notation must be used.
Multiple Choice Questions (all are AP-like and calculator allowed) 1. f( x) x(cos x)(ln x)( e x ); f 1 (A) 0 (B) 1 (C) 1.31 (D) 1.406 (E) 1.469 Calculus with the Graphing Calculator. Let ln x f ( x) cosx for 1 x 6. On what intervals is f concave down? x e (A) (1.481, 4.76) (B) (1, 3.105) (C) (3.105, 6) (D) (1, 4.76) (E) (1, 1.481) (4.76, 6) 3. Given f ( x) sin x, use the tangent line at x 1 to estimate f (1.1). (A) 0 (B) 0.540 (C) 0.841 (D) 0.896 (E) 1
Calculus with the Graphing Calculator 4. Two particles start at the origin and move along the x-axis. For 0 t 4, their respective position functions are given to be x1 ln t and x ( t 5). For what value of t does the acceleration of x 1 equal the velocity of x? (A) 9.060 (B) 0.470 (C) 4.960 (D) 5.039 (E) None 5. The average value of the function f x ( ) cos( x ) 0, is on the closed interval (A) 0.31 (B) 0.461 (C) 0.780 (D) 0.977 (E) 1.53 t 6. Water is pumped out of a pond at the rate Rt () 6 cubic meters per minute, where t is t 1 measured in minutes. How much water is pumped from time t 0 to t 3? (A) 0.17 cubic meters (B).147 cubic meters (C) 5.196 cubic meters (D) 1.88 cubic meters (E) 5.456 cubic meters 7. If c satisfies the conclusion of the Mean Value Theorem for 0 x 1, then c is f ( x) sin 1 x on the interval (A) 0.500 (B) 0.771 (C) 0.785 (D) 1 (E) 1.186
Calculus with the Graphing Calculator 8. A particle moves along the x axis in a way such that the velocity of the particle for t 1 is sin t given by vt (). The total distance traveled by the particle from t to t 4 is t (A) 0.094 (B) 0.094 (C) 0.153 (D) 0.47 (E) 0.340 9. An object traveling in a straight line has position x() t at time t. If the initial position is x(0) 3 and the velocity of the object is time t 4? (A) 0.403 (B) 5.78 (C) 6.877 (D) 8.78 (E) 9.877 vt () 1 3 t, what is the position of the object at x 10. The area of the region bounded by the curves y e, y ln x, and the line x 1 is (A) 0.04 (B) 0.054 (C) 0.096 (D) 0.78 (E) 1.686
11. Let g( x ) be the function given by true? I. g is decreasing on (0,1) II. g is decreasing on (1,) III. g() 0 (A) I only (B) II only (C) IIIonly (D) I and III only (E) I, II, and III x t ( ) ( 1). 0 Calculus with the Graphing Calculator gx e t dt Which of the following must be 4 1. Let R be the region in the first quadrant enclosed by the graphs y x 1 and y x 16. The volume of the solid generated by revolving R about the x-axis is (A) 5.616 (B) 80.475 (C) 507.539 (D) 1594.480 (E) 3188.959 x t 13. Given f ( x) dt; f (1) 1 t ( e 1)(sin t) (A) 0 (B) 0.00 (C) 0.30 (D) 0.341 (E) does not exist
Free Response Questions (all are AP-like and calculator allowed) 14. y Calculus with the Graphing Calculator 5 feet 9 feet y = 9 x 4 65 0 An oil storage tank has the shape shown above, obtained by revolving the curve y 9 65 x 4 from x = 0 to x = 5 about the y-axis, where x and y are measured in feet. Oil flows into the tank at the constant rate of 8 cubic feet per minute. x (a) Find the volume of the tank. Indicate units of measure. (b) To the nearest minute, how long would it take to fill the tank if the tank was empty initially? (c) Let h be the depth, in feet, of oil in the tank. How fast is the depth of the oil in the tank increasing when h = 4? Indicate units of measure.
Calculus with the Graphing Calculator 15. A particle travels on the x axis so its velocity at time t is given by for 0t. The particle is at position x at time t 0. 1 vt () cost, (a) For what values of t is the particle moving to the left? (b) What is the particle's final position? (c) What is the total distance traveled by the particle over the interval 0t? (d) When t, is the speed of the particle increasing or decreasing? Justify your answer. 6
Calculus with the Graphing Calculator 16 16. Let R be the region bounded by the graphs of y 1 x and y 3. Set up and evaluate an integral expression for each of the following: (a) the area of R (b) the volume of the solid generated by revolving R about the x -axis (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are semicircles. Determine the volume of this solid.