Calculus I, Fall 2015

Final Exam

Jan 12, 2016, 10:10AM

1. (12 pts) Find the solutions for

(a) \frac{dy}{dx} = 2x\sqrt{1 − y^{2}}, −1 < y < 1
(b) \frac{dy}{dx} − y = −y^{2}, Hint: let u = y^{−1}

2. (12 pts) Find the volume and surface area of the object obtained by rotating {(x − 3)^{2} + y^{2} ≤ 1, x ≥ 3} 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. (10 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. (50 pts)

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

(4) ∫_{1}^{2} \frac{1}{e^{2x} − e^{−x}} dx (5) ∫ \frac{1}{\sqrt{4x − x^{2}}} dx

6. (10 pts) Express ∫_{0}^{π} cos^{6} x sin^{4} x dx in terms of ∫_{0}^{π} cos^{4} x sin^{4} x dx.

7. (10 pts) Start with domain and range for sech and sech^{−1}, derive the formula for the derivative of sech^{−1}.

8. (8 pts) For what values of p > 0, is the function f(x) = |x|^{p} differentiable at x = 0? Explain.

9. (12 pts) State both parts of Fundamental Theorem of Calculus, prove that part 1 implies part 2.