Harrison Chapman

Math 317: Homework 12

Due: Friday, December 13, 2019

Note:

This homework assignment is just an example, for practice with our end-of-semester material. It will not be collected or graded.

Exercises:

1. Let $f$ be integrable on $[a,b]$. Prove that, for any subset $S \subseteq [a, b]$ we have

Hint. For $x_0, y_0 \in S$, we have $\vert f(x_0)\vert - \vert f(y_0)\vert \le \vert f(x_0) - f(y_0)\vert \le M(f, S) - m(f, S)$.

Note: This is the primary step in proving that $$|f|$$ is integrable whenever $$f$$ is.

Proof. The first part of the hint follows from the triangle inequality, as

The second part of the hint comes from how $M(f, S) \ge f(x_0)$ and $m(f, S) \le f(y_0)$ for all $x_0, y_0 \in S$.

On the other hand, we have that for any $\epsilon > 0$, properties of the supremum and infimum guarantee that there exists $x_0, y_0 \in S$ so that,

So we have that,

This holds for all $\epsilon > 0$, so we conclude that the desired inequality holds. $\Box$

2. Show that $\left\vert \int_{-2\pi}^{2\pi} x^2 \sin^8(e^x)\; dx \right\vert \le \frac{16\pi^3}{3}$.

Proof. Using Theorems 33.4 and 33.5, together with the FTC I (for example), we have that (as $\left\vert \sin^8(e^x )\right\vert \le 1$)

3. Let $f$ be the function,

Prove that $f$ is integrable on $[-1, 1]$. Hint. See Excercise 33.11(c) and its solution in the textbook. (Why can we not apply the Dominated Convergence Theorem to prove that $f$ is integrable?)

Proof. Let $\epsilon > 0$. Notice that $f$ is continuous on each of $[-1, -\epsilon/8]$ and $[\epsilon/8, 1]$, and is hence integrable. This means that there exist partitions $Q_1$ of $[-1, -\epsilon/8]$ and $Q_2$ of $[\epsilon/8, 1]$ so that

Notice that $P = Q_1 \cup Q_2$ is a partition of $[-1, 1]$ and furthermore that

and,

(Why? The only terms in the two sums are those contributed by the interval $[-\epsilon/8, \epsilon/8]$ but we know that the supremum and infimum of $f$ on any such interval are $1$ and $-1$ respectively and that the width of this interval is $\epsilon/4$) So,

We can find any such $P$ for any $\epsilon > 0$, so we conclude that $f$ is integrable on $[-1, 1]$.

4. Let $f$ be a continuous function on $\mathbb R$ and define

Show that $F$ is differentiable on $\mathbb R$ and compute $F'$.

Let $G(x) = \int_0^x{f(t)\; dt}$. Then $G(x)$ is differentiable and $G'(x) = f(x)$ by the FTC II as $f$ is continuous. Notice that $F(x) = G(\sin x)$. So by the chain rule, $F$ is differentiable and $F'(x) = G'(\sin x)\cos x = f(\sin x)\cos x$.

5. Let $f$ be the function,

1. Determine the function $F(x) = \int_0^x f(t)\; dt$.

2. Sketch $F$. Where is $F$ continuous?

3. Where is $F$ differentiable? Calculate $F'$ at the points of differentiability.

1. By the FTC II, $F(x)$ is continuous everywhere.

2. By the FTC II, $F(x)$ is differentiable everywhere that $f(x)$ is continuous, hence at all $x$ except possibly $x=0$ and $x=2$.

We can prove that $F(x)$ is not differentiable at $x=0$ and $x=2$ by showing that the left-hand and right-hand difference quotient limits disagree.