Honors AP Calculus / Mr. Hansen 10/15/2010

Name: _________________________

Test #2 Through Chapter 3

	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 decimal approximations in final answers must be correct to 3 decimal places; do not round early. You may use your calculator throughout. In this test, you may take as given that all polynomials are continuous on and that the limit properties (sum of a limit, limit of a constant, etc.) are all valid.
1.	The ______________________ of a function at a point , which may be informally defined as the “limit of the nearby slopes,” can be more formally defined using mathematical symbols as follows:
2.	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). ____________________________________________________________________
3.	In addition to simple calculator computations (addition, subtraction, etc.), the AP exam permits the following graphing calculator operations: function values/tables/graphs, computation of a definite integral, evaluation of a _________________ ___ ____ _________, and finding roots.
4.	Sketch a function f having domain equal to all of  such that f is continuous everywhere except at the point x = 4, where the left-hand limit is infinite but the right-hand limit is finite.
5.	In question 4, does f ¢(4) exist? ______________ Why or why not? (Write a sentence or two.)
6.	Use a step size of −0.1 (i.e., a step to the left) to estimate the derivative of the function y = sin x when x = –2.2. Show your work, and give your approximate answer using the correct notation for the derivative.

7.	Write an equation (any form acceptable) of the line that is tangent to the graph of the function given in #6, at the point where x = −2.2. Exact values are preferred, but if you use your calculator, a minimum of 3 decimal places of accuracy after the decimal point are required for credit.
8.	The _____________ was invented approximately _____ years ago by Newton and _________ . The subject divides into 2 parts: the ______________________ calculus, which is concerned with derivatives, and the ______________________ calculus, which is concerned with definite and ______________________ integrals.
9.	About ______ years ago, the metamathematician Kurt ______________________ shocked the world with his famous theorem that proved that any consistent system of mathematics must be ______________________ , by which we mean that there will be formally undecidable propositions in the system.
10.	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 any clear, coherent method to estimate the increase in velocity that occurs during the first 4 seconds of flight. Explain your computations briefly (1 sentence or so), and give answer using appropriate units. Circle your answer.
	Elapsed time (sec.)	Acceleration (m/sec^2)
	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

11.	In Lower School science and mathematics, you were taught that distance equals rate times time. In a similar way, you were probably also taught that velocity equals acceleration multiplied by time. In question 10, 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.
	_____________________________________________________________________
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12.	Find the unique antiderivative, g(x), of the function y = cos x – 3x^3 – 11x + 2p with respect to x, such that g(2) = 4.2.
13.	For the situation in #12, find g(3.8). Note that it is possible to answer this question even if you failed to answer question #12.
14.	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 _____ ).
15	What is the limit of y = f (x) = 7x^3 – 8x^2 as x approaches 0.6? ______________ Find the largest 3-decimal-place approximation of d such that every x-value in a punctured d-neighborhood of 0.6 keeps y within 0.04 units of the answer you gave for the limit. ___________ No work is required to be shown here.
16.	Write an EQUATION for the derivative of each given function. Please do not simplify. If you do not write an equation, points will be deducted.
(a)
(b)
(c)
17.	Consider the function y = f (x) = x^4 − 15x^2 − 10x + 26.
(a)	Compute the function and its first two derivatives at x = 0 and x = 2. Give proper notation for all 6 answers.
(b)	State the Intermediate Value Theorem (IVT).
(c)	Use the IVT to prove, rigorously, that f has a root on the interval (0, 2).