Calculus I, Fall 2013

Final Exam

Jan 14, 2014

1. (8 pts) Find the solutions for \frac{dy}{dx} = 3x^{2}e^{−y}.

2. (8 pts) Write down the form of partial fraction expansion for \frac{x^{7}}{(1 − x^{4})^{2}} . Need NOT find the undetermined coefficients.

3. (8 pts) Order e^{x}, x^{x}, (ln x)^{x} and x^{e} from slowest to fastest growing rate as x → ∞.

4. (64 pts) Evaluate

(1) ∫ \frac{1}{2 + sin x}dx (2) ∫ e^{x} sin x dx (3) ∫ \frac{1}{\sqrt{4x − x^{2}}} dx (4) ∫_{0}^{π/4} tan^{3} x sec^{3} x dx

(5) ∫_{1}^{2} \frac{1}{e^{x} − e^{−x}} dx (6) ∫_{0}^{∞} x^{2}e^{−x}dx (7) ∫_{0}^{1} \frac{1}{\sqrt{1 + e^{x}}} dx (8) ∫_{0}^{π} \sqrt{1 + sin x} dx

5. (12 pts) Does the improper integral ∫_{1}^{∞} \frac{1}{\sqrt{x^{3} − x}}dx converge or diverge? Explain (DO NOT try to integrate explicitly).

6. (8 pts) Express ∫_{1}^{2} sin x dx as a Riemann sum. That is, lim_{···} ∑_{···}^{···} · · ·.

7. (8 pts) Evaluate lim_{x→0^{+}} x^{x ln x}.

8. (16 pts) Find the volume and surface area of the object obtained by rotating the region {(x − 2)^{2} + y^{2} ≤ 1, x ≥ 2} around the y axis. Note the surface area consists of two parts, one generated by a half circle, the other generated by a line segment.

9. (.100 pts) Schedule recitation change for next semester.