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

Brief answer to selected problems in Homework 08

1. Section 4.5: problem 80: a = −2, b = −\frac{8}{3}. problem 84(c): Answer = 1. problem 86: Solve for \frac{d}{dx}x^{\frac{1}{x^{n}}} = 0, one gets x^{\frac{1}{x^{n}}} \frac{1−n ln x}{x^{n+1}} = 0. So the only critical point is x = e^{\frac{1}{n}}. Next show that x = e^{\frac{1}{n}} is indeed a maximum by first derivative test. Answer: Maximum = e^{\frac{1}{ne}}.

2. Section 4.6: problem 12: maximize V (y) = \frac{1}{3}π(9 − y^{2})(3 + y) on 0 ≤ y ≤ 3. It is easier to calculate than using V (x).

problem 48: Similar to Example 4. Minimize t = \frac{\sqrt{a^{2} + x^{2}}}{c} + \frac{\sqrt{b^{2} + (d − x)^{2}}}{c} with respect to x,