Honors AP Calculus / Mr. Hansen 11/24/2008

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

Test through §7.2 (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 except in Part IV. 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.
Part I (30 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. However, because a few of these are extremely difficult, you may miss one question in Part I without penalty.
___ 1.	A function h that is defined on , and for which there is at least one value such that does not exist, is continuous on .
___ 2.	A function f that is continuous on [a, b], where b > a, has an antiderivative A(x) that satisfies A(b) = 0.
___ 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 Simpson’s Rule estimate (using infinite-precision arithmetic) of the definite integral of a quadratic function equals the true exact value of the integral.
___ 6.	The general solution of , where k > 0 and x > 0, is y = Ce^kx.
___ 7.	If (a, b) is a point of inflection for a twice-differentiable function f, then .
___ 8.	If (a, b) is a point of inflection for a one-to-one function f, then (b, a) is a point of inflection for function f ^–1.
___ 9.	The trapezoid rule gives the same result that one would obtain by averaging the left and right endpoint rule estimates defined on the same partition.
___ 10.	For a function g(x) having domain and range , .
___ 11.	If (a, b) is a nonempty interval in the domain of f, and if f is a function for which , then f is continuous on [a, b] and differentiable on (a, b).
Part II	(16 points)
12.	Some textbooks, though not ours, make a big point of proving a result called the “Chain Rule for Integrals” (CRI for short), which is stated below: Let c be a constant. For any continuous function f and any differentiable function v such that the range of v is a subset of the domain of f, .
(a)	Use the regular Chain Rule and FTC to prove CRI.
(b)	Apply CRI to evaluate .
Part III	Fill-Ins (6 pts. per numbered problem, 18 pts. total).
13.	If f and g are differentiable functions such that , state precisely what L’Hôpital’s Rule says about , in the simple form that was presented in your textbook. (Do not consider the more sophisticated corollaries and extensions that we discussed in class.)
	Answer: = _________________________________________________ .
14.	For an investment that is compounded quarterly at an annual interest rate of 3 percent, the ______________________ states that V(t), the nominal value of the investment as a function of time t in years, will double in approximately ______ years. The true answer, found by solving the compound interest formula V(t) = V(0) · (1 + (.03/4))^4t for t, where we set V(t) = ___________ , is 23.191 years.
15.	In any situation where the instantaneous ____________________________ of an investment or a quantity is proportional to the ____________________________ at that instant, we have a case of exponential growth (or decay). In precalculus, you used the formula y = P(t) = P0e^rt, but in the calculus we usually use the equation ____________________________ to denote the general solution for y as a function of t.
Part IV (18 pts.)	Calculations (6 pts. each). Work is optional but must be shown if you desire partial credit in the event of a mistake.
16.	Hansenium is a radioactive material whose half-life is 13 days. In other words, a 22 g lump of Hansenium will be 11 g after 13 days. Felonious pranksters have installed a 22 g lump of Hansenium in Mr. Hansen’s office in order to delay the grading of a certain HappyCal test. Mr. Hansen can safely re-enter his office when the lump has decayed to 1 g. How long will Mr. Hansen have to avoid his office? Give answer to the nearest tenth of a day.
17.	Compute cot^–1(log8 41.844), correct to 3 decimal places.
18.
Part V (18 pts.)	Limits (6 pts. each). Here, work is required. (The answers by themselves are not worth anything, since you could easily use your calculator.) Show work in the manner that was required in homework.
19.
20.
21.