AP Calculus AB / Mr. Hansen 9/30/2003, rev. 10/2/2003

Name: _________________________

Test #1: §§1-1 through 2-2 (alternate version)

	Instructions: If a blank is provided, simply fill it in (no work required). Otherwise, answer the question and justify your steps. In general, this means showing at least 3 things: a formula, the plug-ins, and the answer (circled, with units if appropriate). All numeric answers must be correct to 3 decimal places. You may use your calculator throughout. If a problem includes extraneous information, use only the information that you need in order answer the question(s) posed.
1.	If f is the derivative of g, i.e., if f (x) = g¢(x) for all x in the domain of g, then we say that g is __ ____________________________ of f. Is g the only possible function that can play such a role? _____ Explain briefly (1 sentence). ____________________________________________________________________
2.	In addition to simple calculator computations (addition, subtraction, etc.), the AP exam permits the following graphing calculator operations: function plotting, computation of a definite integral, evaluation of a _________________ ___ ____ _________, and finding roots.
3.	Sketch a function f having domain equal to all of  that is continuous except at the point x = 4, where the left-hand limit is infinite but the right-hand limit is finite.
4.	In question 3, does f ¢(4) exist? ______________ Why or why not? (Write a sentence or two.)
5.	Use the symmetric difference method with a step size of 0.1 on each side to estimate the derivative of the function y = sin x when x = –2.2. If you have forgotten what the symmetric difference method is, then simply use a step size of 0.1 on the left or right side (your choice) for most of the credit. Show your answer using correct notation for the derivative.
6.	Use your calculator to find the derivative requested in question 5: ______________

7.	As a rocket is launched from rest at time t = 0, it undergoes acceleration that changes as a function of time. Elapsed time, t, is measured in seconds, and acceleration is measured in meters per second per second, also known as m/sec^2. By using a device called an accelerometer, we can measure the acceleration at various instants. Use Dt = 0.5 seconds and the midpoint method to estimate the increase in velocity that occurs during the first 4 seconds of flight. Give answer using appropriate units.
	Elapsed time (sec.)	Acceleration (m/sec/sec)
	0	20
	0.25	16
	0.50	15
	0.75	14
	1.00	13.5
	1.25	14
	1.50	14.5
	1.75	15
	2.00	16
	2.25	17
	2.50	17.5
	2.75	18
	3.00	18.3
	3.25	18.4
	3.50	18.6
	3.75	18.7
	4.00	18.8
8.	In Lower School science and mathematics, you learned that velocity equals acceleration multiplied by time. In question 7, can we simply pick one of the acceleration values from the table (e.g., 16 m/sec^2) and multiply by the elapsed time (4 sec.) to compute the velocity at time t = 4? ____ Explain why or why not.
	_____________________________________________________________________
	_____________________________________________________________________

9.	Use correct notation to find the antiderivative of cos x – 3x^3 – 11x + 2p with respect to x. In other words, state the problem in correct mathematical notation, and then state the answer. No work is needed.
10.	The expression I = òa^b v dt represents the ___________ integral of _____ with respect to ______ over the interval ___________. If v (in miles per hour) represents velocity as a function of time (in hours), then the real-world interpretation of òa^b v dt is ____ _______ ___________ (in _____ ). Briefly explain, using moderately rigorous terminology that reveals knowledge of the underlying process, how the integral I is defined. Hint: You will need to use the word limit at least once.
	_____________________________________________________________________
	_____________________________________________________________________
	_____________________________________________________________________
11.	Make a sketch that shows the function y = f (x) = 7x^3 – 8x^2 on the domain [0, 1]. What is the limit of f (x) as x approaches 0.69? ______________ Suppose that we are trying to estimate d to the nearest hundredth. Let d = 0.04, and use your sketch to indicate whether or not every x in the restricted domain (0.69 – d, 0.69 + d) has a y value that is within 0.05 units of that limit. Please use dotted lines to show the “d and e bands” on your sketch. Conclusion: d is (circle one)   too large too small about right