The Orange Baron's Sample AP Calculus Exam

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Sample AP Calculus Examination

Part I: Multiple-Choice Questions: You will have approximately 2.34 minutes to complete this section. Do not spend too much time on any one problem. You are not expected to be able to answer every problem in this test. (Oh, by the way, I was too excited as I did this to even bother, in my joyful state, to make choices for every one of these questions. For most you'll just have to figure 'em out without the benefit of choices.)

1. The velocity v(t) of a particle is given by: v(t) = 5x^2 - 4x + 2. At what time t does the particle reach maximum acceleration?

2. The derivative of a function f is dy/dx = 4x^2 + 3x + 4y + 7r +24u. Differentiate implicitly with respect to x.

3. The second derivative f''' of a function f is given by f'''(x) = 28x^2. If f''(5) = 0, and f'(4) =12, what is the average value of the function f on the interval [3,4]?

4. The graph of f is shown below. Which of the following is true?

(a) f is not continuous at x = 2
(b) f does not equal 0 at x = 0
(c) f has a valley at x = 2
(d) f has a peak at x = 0
(e) The slope of the tangent line to f at x = 2 is 0.0000001

5. Suppose the number of annoying integration rules grows at an exponential rate proportional to the growing annoyance of the calculus reform movement, whose overall annoyance is given by A = ae^(rt), where t is the time in years and r is the amount of rage given to students. Supposing the constant factor of rage is r = 2000, how annoyed will students be twenty years from now if the initial annoyance is 22?

Part II: Sample Free-Response Questions

1. The tortoise and the hare decide to run a race. They are designated respectively by A and B. If runner A runs at a velocity graphed by y=t for t on the interval [0, 3] and y=3 thereafter on the interval [3,10] and runner B has a velocity given by v = (8t-2)/(2t-1) then answer the following questions:

a. Find the total distance run by each runner
b. If the hare overconfidently stops at five lemonade stands, spending 25 seconds at each, how much distance does he lose?
c. What is the speed of each runner?
d. What is the average rate of change of the acceleration of the tortoise and the hare?

2. A cylindrical tank, okay, a used laundry bucket in some jerk's backyard, is filled 3/4 of the way with water. Some jerk comes along and puts a hole in the bucket 12.45 inches away from the top of the bucket, with height h, and water flows out at a constant rate of 2.344 inches per second.

a. What is the rate of change of the height of the water?
b. If the jerk spits in the bucket a loogie of approx. 2 in^2, what is the total volume of the water in the bucket at the instant the hole is made?
c. If the bucket sits for 5 minutes before the jerk notices the hole, how much flows out?
d. Find the average flow of the water out of the bucket.

STOP! END OF EXAMINATION. BELIEVE IT OR NOT, THERE IS NO TEST MATERIAL ON THE BLANK PAGE THAT FOLLOWS. AND, SURPRISE: IF YOU FINISH EARLY, YOU MAY GO BACK AND CHECK YOUR ANSWERS, EINSTEIN!