Calculus I, Fall 2014 (http://www.math.nthu.edu.tw/~wangwc/)

Solutions in Quiz04

1. lim_{x→0}(\frac{1}{x} − \frac{1}{sin x}) = lim_{x→0} \frac{sin x−x}{x sin x} = lim_{x→0} \frac{cos x−1}{sin x+x cos x} = lim_{x→0} \frac{− sin x}{2 cos x−x sin x} = 0.

2. lim_{x→∞} \frac{e^{x^{2}}}{xe^{x}} = lim_{x→∞} \frac{e^{x^{2}−x}}{x} = lim_{x→∞} \frac{e^{x^{2}−x}(2x−1)}{1} = ∞.

3. Let D^{2} = (x − 2)^{2} + (y − 0)^{2} = (x − 2)^{2} + x on y = \sqrt{x} and x ≥ 0. Then \frac{dD^{2}}{dx} = 2x − 3, hence x = \frac{3}{2} is a critical point. Moreover \frac{d^{2}D^{2}}{dx^{2}} = 2 > 0 for all x ≥ 0. Hence (\frac{3}{2}, \sqrt{\frac{3}{2}}) is the point on y = \sqrt{x} which is closest to (2,0).

4. Let f(x) = x^{3} − 2, then given x_{0} = 1, compute x_{n+1} = x_{n} − \frac{x_{n}^{3}−2}{3x_{n}^{2}}.

5. ∫ x^{2} + 2^{x}dx = \frac{x^{3}}{3} + \frac{1}{ln 2}2^{x} + c for some constant c ∈ R.