Calculus I, Fall 2013

Midterm Exam 2

Dec 10, 2013, 10:10AM

1. (16 pts) Graph the function y = \frac{x^{3} + 2x − 2}{x + 1}. Indicate all critical points and points of inflection.

2. (10 pts) Let f be a real valued function defined on {x ≥ 0} satisfying (a): f(0) = −1, (b): f′(x) ≥ 1/2 for all x ≥ 0. Prove that f(x) = 0 has exactly one solution on {x ≥ 0}.

3. (16 pts) Find the limits of the following expressions:

(a) lim_{x→0^{+}} x^{x} (b) lim_{x→0} \frac{x^{2} cos \frac{1}{x}}{sin x}

4. (16 pts) State both parts of Fundamental Theorem of Calculus, prove that part 1

implies part 2, then evaluate

\frac{d}{dx} ∫_{sin x}^{1} e^{t^{2}}dt.

5. (16 pts) Evaluate

(a) ∫_{1}^{2} \frac{1}{x(1 + ln^{2} x)}dx (b) ∫_{0}^{4} x\sqrt{2x + 1} dx

6. (10 pts) Evaluate

lim_{n→∞} ∑_{k=n}^{2n} \frac{n}{k^{2}}

7. (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.