Calculus I, Fall 2014

Midterm Exam 2

Dec 03, 2014, 10:10AM

1. (10 pts) Solve for y(x) on x < 0 from y′′(x) = x^{−2}, y(−1) = 1, y′(−1) = 2

2. (12 pts) Graph f(x) = x^{1/3}(x − 4). Indicate all critical points and points of inflection.

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) Evaluate

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

5. (10 pts) Evaluate

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

6. (12 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 \underline{one and only one} solution on \{x ≥ 0\}.

7. (12 pts) State both parts of Fundamental Theorem of Calculus, prove that part 1 implies part 2.

8. (12 pts) Suppose f(x) satisfies ∫_{0}^{x^{2}} e^{−t}f(t)dt = x for all x ≥ 0. Find f(4) and f′(4).