Calculus I, Fall 2014

Final Exam

Jan 08, 2015

1. (14 pts) Find the solutions for \frac{dy}{dx} = e^{x−y} and x\frac{dy}{dx} + y = sin x, respectively.

2. (12 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.

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

4. (6 pts) Write down the form of partial fraction expansion for \frac{x^{7}}{(1 − x^{4})^{2}} . Need NOT

find the undetermined coefficients.

5. (64 pts) Evaluate

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

(5) ∫_{0}^{π/4} tan^{3} x sec^{3} x dx (6) ∫ x^{2}e^{−x}dx (7) ∫_{0}^{1} sechx dx (8) ∫_{0}^{1} \frac{1}{\sqrt{1 + e^{x}}} dx