Honors AP Calculus / Mr. Hansen 10/27/2008

Name: _________________________ Start time: _______ End time: _______

Test through §5-5 (Calculator Required)

Part I	Each correct answer in Part I is worth 4 points. Unlike the scoring on the AP exam, there is no penalty today for wrong guesses (other than the loss of the 4 points, 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.	A continuous function on a closed interval is Riemann integrable.
___ 2.	A continuous function is differentiable.
___ 3.	For a continuous increasing function, upper Riemann sums give the same result that the right endpoint rule would give.
___ 4.	Adaptive quadrature over an interval [a, b] refers to the practice of letting n grow large, where by definition.
___ 5.	For a differentiable linear function y, dy = .
___ 6.	For a differentiable function y = f (x) with upward concavity, such as an exponential function with a base greater than 1, even if < 0.
___ 7.	If , then the point (c, f (c)) is either a local minimum or a local maximum.
___ 8.	The trapezoid rule gives the same result that one would obtain by averaging the upper and lower Riemann sums defined on the same partition.
___ 9.	For a Riemann sum formed from equal-width intervals for independent variable t, the norm of the partition equals .
___ 10.	When no initial conditions have been specified, there exists a unique solution to the differential equation .
Part II	Fill in each blank (1.5 points each) with the word, symbol, or phrase that best fits.
11.	The _____________ Riemann integral of a function f on a _____________ interval from a to b can be thought of as the _____________ under the curve from a to b. Technically, however, we define this type of Riemann integral as the _____________ of all possible Riemann sums as the norm of the partition approaches _____________ .
12.	The _____________ rule is an example of a _____________ sum, since the function whose definite integral is being _____________ is evaluated at the center point of each interval in the partition, and each such height is multiplied by _____________ in order to find the area of a thin rectangle.
13.	_____________ , also known as _____________ integration, is the process of finding a family of functions whose derivative equals the stated function. Although this task is sometimes difficult or even impossible (in closed form), checking an answer is always easy, since if the _____________ of a proposed answer equals the integrand, we know we have found the correct answer.
Part II	Evaluate each differential dy (4 pts. each).
14.	y = sec^2 (2 sin x^2)
15.	y = 74 + 312 cot^–1 5x^3
Part III	Evaluate (8 pts. each). For full credit, show u-substitution work even if you personally find the work to be a waste of time.
16.
17.	Hint: Factor x^0.5 out of the denominator.
Part IV	Answer all the questions posed (18 points). Show enough work to make it clear that you know what you are talking about.
18.	Consider the function y = f (x) = 1.08^exp(x).
(a)	Rewrite f using calculator notation. Note: This is never permitted on the AP exam. The only reason we are doing this today is to make sure you have the correct function. Raise your hand before proceeding. Y1 = ______________________________
(b)	Estimate by using lower Riemann sums with n = 5 subintervals. Show your work.
(c)	In part (b), how can you prove that your method is valid? In other words, how can you satisfy the AP graders that you really were using the lower sums? (Remember, finding minimums is not a permitted calculator operation.)
(d)	As in part (b), estimate the integral, but this time, use the trapezoid rule with n = 5. Show your work.

(e)	We know that f (0) = ________ by inspection. (Fill in the blank.)
(f)	Use the linear approximator that best fits f at the point (0, [answer you gave in (e)]) to estimate f (0.77).
(g)	Compute your error in part (f). Which direction is the error? Why is this not surprising? Explain briefly.