AP Calculus AB / Mr. Hansen 2/27/2006

Name: _________________________

Re-Test on Chapter 7

Time limit: 30 minutes (45 minutes for extended time)

Scoring	· The two numbered problems are weighted equally (AP-style). · No calculator is permitted. · Algebraic simplification of answers is not required.
1.	The instantaneous rate of growth of a certain value y is inversely proportional to the amount of y present at time t.
(a)	Without doing any mathematics, explain briefly why this situation cannot involve exponential growth.
(b)	Write a general differential equation.
(c)	Sketch the slope field for lattice points in [–5, 5] ´ [–5, 5] if the constant of proportionality is given to be 2.

(d)	Find the function y = f (t) that solves the differential equation subject to initial conditions (0, –2) and (4, –4). Note that there is nothing that can be assumed this time concerning the constant of proportionality.

2.	Let y = f (x) satisfy the properties that f (0) = –3 and f ¢(x) = –2x for all real numbers x.
(a)	Use Euler’s Method with a step size of 0.5 to estimate f (1). Show your work by means of equations or a table, so that it is clear how you are obtaining your estimates.
(b)	What function is function f, in reality? State an answer and then prove that your function f satisfies the two given properties.

(c)	Analyze the concavity of f. Does f have the same type of concavity (i.e., upward or downward) for all values of x, or is there a place where the concavity changes sign? You may use your answer from part (b), or if you could not get (b), you may work from the properties given.
(d)	Would a correct answer to part (a) be greater than, less than, or equal to the true value of f (1)? Explain your reasoning.