Calculus I, Fall 2015

Midterm Exam 2

Dec 08, 2015, 10:10AM

1. True or False? If true, prove it. If false, give a counter example.

If |f(x) − (3x + 2)| ≤ |x|^{1.5} for all x ∈ R, then f is differentiable at x = 0.

2. (a) Graph f(x) = \frac{x}{\sqrt{x^{2} + 1}}. Give all details including possible asymptotes.

(b) The function y = f(x) is odd (f(−x) = −f(x)) and the root x^{∗} to the equation f(x) = 0 is x^{∗} = 0. Give formula of Newton’s method for finding this root.

(c) The Newton’s method does not always converge. There is an a > 0 such that Newton’s method converges if and only if −a < x_{0} < a. Take this fact for granted and find a (show how to find a, but need NOT prove that Newton’s method converge if and only if −a < x_{0} < a).

3. Let f be a differentiable function defined on {x ≥ 0} satisfying (a): f(0) = −1, (b): f′(x) ≥ 1/2 for all x ≥ 0. Show that f(x) = 0 has \underline{one and only one} solution on \{x ≥ 0\}.

4. Find the limits of the following expressions:

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

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

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

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

8. Evaluate

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

9. True or False? If true, prove it. If false, give a counter example.

(a) If y = f(x) is differentiable at x = c then it is continuous at x = c.

(b) If y = f(x) is continuous at x = c then it is differentiable at x = c.

10. Start with domain and range for csc and csc^{−1}, derive the formula for the derivative of csc^{−1}.