Extracts from MATH 142 Final Exam (Spring, 1995) On this part of the exam no calculators or computers may be used. For full

credit you must show enough work so that your methods and reasoning are clear.

1. Compute the derivative.

a. \frac{d}{dr} sin(πr)e^{−r}

b. \frac{d}{dt} ln \sqrt[3]{te^{t}}

2. Compute each integral.

a. ∫ (cos(\frac{x}{4}) + \frac{6}{x^{3}} ) dx b. ∫ tan 3z dz c. ∫ x^{2} ln x dx d. ∫_{0}^{∞} r^{2}e^{−r^{3}} dr e. ∫ \frac{t+14}{t^{2}+3t−4} dt

3. Compute the average value of the function f(w) = \frac{1}{16 + w^{2}} on the interval 0 ≤ w ≤ 4 .

On the remainder of the exam calculators or Maple may be used. For full credit you must show enough work so that your methods and reasoning are clear. If you use Maple or a calculator, say what exactly you are having the machine do for you.

4. Suppose the graph of ln(g(x)) is the one shown below. Find the formula for g(x) itself, and sketch its graph.

5. Compute the area enclosed between the curves y = 2x^{3} and y = −x^{2} + 4x + 3 .