Honors AP Calculus / Mr. Hansen	Name: _______________________________________
3/1/2007
	Elapsed time (for informational purposes only): ____________

Test on Chapter 9 + Chapter 8 Rehash

Instructions

• Calculator is permitted.
• Show adequate work.
• Use space provided. Write “OVER” if you need more room.
• Point values are shown in parentheses in the left margin.

1.	Consider the ellipse having equation , where a and b are positive constants.
(a)	Solve for y as two functions of x.
(4)
(b) (8)	Use the fact that to prove that the ellipse has area . You may take advantage of ellipse symmetry if you wish.

(c) (3)	The ellipse can be parameterized as x = a cos q, y = b sin q. Therefore, the ellipse has what polar equation? (Raise your hand to buy a hint if you can’t get this, since you need it for part (d).)
(d) (8)	Use part (c) and the polar area formula to prove that the ellipse has area .
2.	The line segment y = 2 – 2x, for –1 £ x £ 1, is revolved about the axis x = –1 in order to create a solid of revolution bounded below by the x-axis.
(a) (4)	Sketch the solid.
(b) (3)	Compute the volume by a geometric formula.
(c) (8)	Compute the volume by the calculus.

3. (5 + 5)	Consider the curve for x Î [–4, 4]. Compute the perimeter and area of the “sideways D” region bounded by the curve and the lines y = 10, x = –4, and x = 4.
4. (8)	Compute .
5.(a) (6)	State three methods by which can be computed.
(b) (6)	Do it.

(c) (6)	Do it by another of the three methods.
6. (6)	Compute .
7. (6)	Compute . Hint: Think of arctangents.
8.(a) (8)	State the definitions of sinh x cosh x tanh x sech x
(b) (6)	Without using the identity cosh^2 x – sinh^2 x = 1 (i.e., simply by part (a) and QR), prove that .