Honors AP Calculus / Mr. Hansen 12/18/2008

Name: _________________________ Bonus (for Mr. Hansen’s use only): ________

Test through §8-5 (Calculator Required)

Rules	You may not write calculator notation anywhere unless you cross it out. For example, fnInt(X^2,X,1,2) is not allowed; write instead. Adequate justification is required for free-response questions. All final answers in free-response portions should be circled or boxed. Decimal approximations must be correct to at least 3 places after the decimal point. Questions in Part I are worth 3 points each. Questions in Part III are worth 5 points each. All other questions are worth 8 points each.
Part I (15 pts.)	Unlike the scoring on the AP exam, there is no penalty in this section for wrong guesses (other than the loss of the points you could have earned, of course). In the small blank provided, write a capital letter A if the statement is always true, S if it is sometimes true, or N if it is never true. There is no partial credit.
___ 1.	For a function f that is defined on , there is at least one value x = c such that f (x) is continuous at x = c.
___ 2.	A function g that is continuous on [a, b], where b > a, has a unique value g(c), where c [a, b], such that g(c) is a maximum function value on the interval. [Note: We are not asserting that the x-value, x = c, is unique, merely that the function value is unique.]
___ 3.	For function f, continuous on , there is at least one value such that exists.
___ 4.	For a continuous, even function f, and for a > 0,
___ 5.	The general solution of , where k > 0 and x > 0, is y = Ce^kx.

Part II	Consider the differential equation .
6.	Find the general solution. Use ample space below or on reverse side.
7.	Find the particular solution that passes through (–1.5, –0.75).
8.	Sketch your answer to #7 as an overlay on the slope field that uses lattice points on [–3, 3] [–3, 3].
9.	Use Euler’s Method to estimate y when x = 0. Use a step size of 0.5. Show all relevant work.

Part III	“Quickies”
10.
11.
12.	Let F(x) be a differentiable function whose derivative equals e^44x – sin x cos(7x^9). If F(2.2) = 3.3, find F(1.1). A decimal approximation is not required. Justify that your answer satisfies the conditions imposed.
Part IV	Consider the region R in Quadrant I bounded by the functions f (x) = x + 2 and g(x) = e^x.
13.	Compute the area of R by means of horizontal slicing. Show setup and answer below or on reverse side.
14.	Can the area of R also be computed by vertical slicing? If yes, show setup and answer. If not, explain why not.
15.	R is used as the base for a solid that is built such that all cross sections perpendicular to the x-axis are semicircles. The diameter of each such semicircle is the distance between functions f and g. Compute the volume of the solid so defined.
16.	If R is used as the starting point for a solid of revolution (i.e., if R is revolved about the x-axis, creating the figure sketched on the board), compute the volume.
17.	Find the x value for which the absolute difference between f and g is (a) maximized and (b) minimized if x is restricted to the domain [–1, 1]. Please note, this is different from the domain used to define R. Show all necessary justification; answers alone are not sufficient.