AP Calculus AB / Mr. Hansen 11/16/2005

Name: _________________________

Test on Chapter 4

Time limit: 26⅔ minutes (40 minutes for extended time)

Scoring	· Each question is worth 6 points, except for the proof at the end, which is 12 points. · Your name is also worth 6 points. That makes a total of 102 points possible (out of 100). · Simplification is not required. Work is not required either, in most cases. · If you make a mistake, partial credit is possible if you show some relevant work.
Pace	Allowing 1½ minutes per problem will leave you a few minutes at the end to double-check. If a problem looks too hard, skip it and try solving it at the end. You can omit 2 problems (or the proof, which counts double) and still earn an A.
Other	You may use your calculator wherever you wish, although most problems do not require it.
1.	Did you write your name above (Y/N)? _______________________________ .
2.	If y = x tan^–1 5x, then y¢ = __________________________________________ .
3.	If xy + 2y = cos xy, then y¢ = _______________________________________ .
	Work:
4.	If , then y¢ = _________________________________________ .
5.	Consider the circle defined by x = 5 cos t, y = 5 sin t, –2p £ t £ 2p. Convert this to a relation having a nonparametric (x^2 + y^2) equation. A little bit of work is required here.
	Work:
	Final equation: ___________________________________________________
6.	For the circle in #5, the slope of the tangent at the point (–4, 3) is ________ . Use the calculus if you wish, or a faster method if you happen to know of one.
	Work (optional):
7.	Sketch the relation defined by x = 2 cos (t + 2), y = sin^2 t, –p £ t £ p.
8.	For the relation you sketched above, dy/dx = _____________________________ .
9.	Justify (1 sentence) why #8 proves that the relation in #7 has a horizontal tangent whenever t is a multiple of p/2. If you cannot do this, then show some horizontal tangents on the sketch for partial credit.
	__________________________________________________________________
	__________________________________________________________________
10, 11.	On your sketch above, mark with a star (*) a place where dy/dx is undefined. The value of dy/dx is undefined at (*) because ___________________________ .
12, 13.	Let f be a one-to-one function. Whenever x is an integer, f satisfies f (x) = 2x – 8, f ¢(x) = 7x. (We have no information about the behavior of f elsewhere, except that f is one-to-one.) Let h(x) = f ^–1(x). Based on this information, h(4) = ______________ and h¢(4) = ______________ .
	Work (optional):
14.	Sketch the graph of a function f that has all of the following features. · f is continuous on all of  · f (–1) = 3 · f ¢(–1) = 1.5 · f ¢¢(–1) > 0 · f is not differentiable at x = 0

15.	Let f be defined piecewise as follows, where A is a constant:
	If f is to be continuous and differentiable at x = 0, then A = ___________ .
	Work (optional):
16. (double credit)	Let a > 0 and b > 0 be unequal constants. Let be the equation of an ellipse centered at the origin, and let (u, v) be any point on the ellipse that does not lie on the x-axis or on the y-axis. (Note: Since it is given that a ¹ b, the relation is not a circle.) Prove that the segment from the origin to (u, v) is not perpendicular to the tangent to the ellipse at (u, v).