College of the Holy Cross MATH 135, section 1 Friday, November 22

Your Name:

Instructions: For full credit, you must show all work on the test pages and place your final answer in the box provided for the problem. Use the back of the preceding page if you need more space for scratch work. The numbers next to each part of the questions are their point values.

Please do not write in the space below

Problem Points/Poss
I  / 35
II  / 20
III  / 20
IV  / 25
Total  /100

I. For each of the following functions find the derivative and simplify. A. (5) f(x) = e^{2x} cos(4x) f′(x) =

B. (5) g(x) = \frac{x^{2} + 2}{ln(x)} g′(x) =

C. (5) h(x) = tan^{−1}(2x + 1) h′(x) = D. (10) k(x) = x^{sin(x)} k′(x) = E. (10) Find y′ by implicit differentiation if x^{3}y^{2} + 2y = 3x.

y′ = II. (20) A rocket is launched vertically and is tracked by a ground station 4 miles from the launch pad. What is the vertical speed of the rocket when its height above the ground is 3 miles and its distance to the ground station is increasing at 3300 miles per hour?

Answer:

III. All parts of this question refer to the function f(x) = −2x^{3} + 24x + 11.

A. (10) Find the critical numbers of f(x). Critical numbers:

B. (10) What does the Second Derivative Test tell you about the behavior of f at each of these critical numbers?

Answer:

IV. All parts of this question refer to the function f(x) = \frac{x}{(2x + 1)^{2}} , for which the first two derivatives are, in simplified form:

f′(x) = \frac{1 − 2x}{(2x + 1)^{3}}
f′′(x) = \frac{8x − 8}{(2x + 1)^{4}}

A. (5) What is the domain of f(x)? Domain:

B. (10) Find all critical numbers and determine where f(x) is increasing and decreasing. Critical numbers: Increasing: Decreasing:

C. (5) What is the absolute minimum value of f(x) on the interval [0, 3]? Absolute minimum value:

D. (5) Determine the concavity of f(x), and find all inflection points. Concave up:

Concave down:

Inflection points: