Calculus I, Fall 2015 (http://www.math.nthu.edu.tw/˜ wangwc/)

Brief answer to selected problems in HW10

1. Section 5.3:

Problem 87: We take for granted from the problem that a continuous function on a closed interval [a, b] must be uniformly continuous (ie. Assume this statement is correct. This is an advanced calculus Theorem).

Therefore given ϵ > 0, one can find δ > 0 such that |x_{1} − x_{2}| < δ implies |f(x_{1}) − f(x_{2})| < ϵ. It is not difficult to see that for this δ, ∥P ∥ < δ implies U − L < ϵ(∑_{k} Δx_{k}) = ϵ(b − a).

2. Section 5.4:

Problems 84: Use L’Hôpital’s rule to get the limit. Answer = 2.

Problems 89: F (x) = ∫_{1}^{x^{2}} \sqrt{1 − t^{2}}dt. F′(x) = 2x\sqrt{1 − (x^{2})^{2}}. F′′(x) = \frac{2(1−3x^{4})}{\sqrt{1−x^{4}}}.