Mathematics 133 – Intensive Calculus for Science 1 Sample Midterm Exam 3 November 19, 2001

Note: This is somewhat longer than the actual exam will be, to give you an idea of the range of types of problems I could ask

I. Compute the derivatives of the following:

A) f(x) = cos(3x^{4} + x)

B) g(x) = \frac{ln(x)}{x^{2}+3}

C) h(x) = arcsin(4x) + arctan(2^{x})

II. All parts of this question refer to the function

f(x) = e^{−x} cos(x).

A) Compute f′(x) and f′′(x).

B) Find the critical points p of f satisfying −π ≤ p ≤ π, and classify each one as a local maximum, local minimum, or neither.

C) What are the global maximum and minimum values of f(x) for −1 ≤ x ≤ 1

III. Because of security concerns, the Holy Cross Buildings and Grounds department wants to build a new enclosed parking lot for administrators. To save costs they decide to make the lot by fencing in a rectangular area and using the existing fence at the non-enclosed end of Fitton Field (the end behind the scoreboard) as one of the four sides. The area of the lot must be 500 square yards. What should the dimensions of the lot be to minimize the length of new fencing needed?

IV. For these questions, say whether the statement is true or false. If it is true, give a reason; if it is false, give an example of a function showing that the statement is not true. A) If f′(p) = 0 and f′ is increasing at p, then f has a local minimum at x = p.

B) If f′(p) = 0, and f′′(p) = 0, then f has neither a local maximum nor a local minimum at x = p.

V. Using the following information about f′, and f′′ give a qualitative sketch of the graph y = f(x).