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\newcommand{\myCourseNumber}{Math 142B}
\newcommand{\myAssignmentDueDate}{11:00pm Thursday, May 9, 2024}
\newcommand{\myAssignmentTitle}{Homework Assignment 3}
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\begin{questions}

%%Ross 34.3
\question Define \ $f : \mathbb{R} \to \mathbb{R}$ \ by \ $f(t) = \begin{cases} 0 & \text{if} \enspace t < 0, \\ t & \text{if} \enspace 0 \leq t \leq 1, \\ 4 & \text{if} \enspace t > 1. \end{cases}$
\begin{parts}
\part Determine the function \ $\displaystyle F(x) = \int_0^x f(t) \, dt$.
\part Where is \ $F$ \ continuous?
\part Where is \ $F$ \ differentiable?  Compute \ $F'(x)$ \ at the points $x$ where \ $F$ \ is differentiable.
\end{parts}

%%Ross 34.5
\question Let $f$ be a continuous function on $\mathbb{R}$ and define \ $\displaystyle F(x) = \displaystyle \int_{x-1}^{x+1} f(t) \, dt$ \ for $x \in \mathbb{R}$.

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

%%Ross 34.6
\question Let \ $f$ \ be a continuous function on \ $\mathbb{R}$ \ and define \ $\displaystyle G(x) = \int_{0}^{\sin(x)} f(t) \, dt$ \ for \ $x \in \mathbb{R}$.

Show that \ $G$ \ is differentiable on $\mathbb{R}$ and compute \ $G'$.

%%Ross 34.11
\question Suppose \ $f$ \ is a continuous function on \ $[a,b]$. \ Show that if \ $\displaystyle \int_a^b f(x)^2 \, dx = 0$, \ then \ $f(x) = 0$ \ for all $x \in [a,b]$.

%Ross 34.12
\question Show that if \ $f$ \ is a continuous real-valued function on \ $[a,b]$ \ satisfying \ $\int\limits_a^b f(x) g(x) \, dx = 0$ \ for every continuous function \ $g$ \ on \ $[a,b]$, \ then \ $f(x) = 0$ \ for all $x \in [a,b]$.

%%Ross 36.3
\question
\begin{parts}
\part Show that
\begin{subparts}
\subpart $\displaystyle \int_0^1 x^{-p} \, dx = \frac{1}{1-p}$ \ if \ $0 < p < 1$.
\subpart $\displaystyle \int_0^1 x^{-p} \, dx = +\infty$ \ if \ $p > 1$.
\end{subparts}
\part Show that \ $\displaystyle \int_0^\infty x^{-p} \, dx = +\infty$ for all $p > 0$.
\end{parts}

%%Ross 36.4
\question Compute
\begin{parts}
\part $\displaystyle \int_0^1 \log(x) \, dx$
\part $\displaystyle \int_2^\infty \frac{\log(x)}{x} \, dx$
\part $\displaystyle \int_0^\infty \frac{1}{1+x^2} \, dx$
\end{parts}

%%Ross 36.6
\question Prove the following \emph{comparison tests}.  Let $f$ and $g$ be continuous functions on $(a,b)$ such that $0 \leq f(x) \leq g(x)$ for all $x \in (a,b)$ and where $a$ could be $-\infty$ and $b$ could be $+\infty$.
\begin{parts}
\part If \ $\displaystyle \int_a^b g(x) \, dx < \infty$, then $\displaystyle \int_a^b f(x) \, dx < \infty$.
\part If \ $\displaystyle \int_a^b f(x) \, dx = +\infty$, then $\displaystyle \int_a^b g(x) \, dx = +\infty$.
\end{parts}

%%Ross 36.7
\question \begin{parts}
\part Using a comparison test, show that $\displaystyle \int_{-\infty}^\infty e^{-x^2} \, dx < \infty$.
\part Show that $\displaystyle \int_{-\infty}^\infty e^{-x^2} \, dx = \sqrt{\pi}$.
\end{parts}

%%Ross 36.8
\question Suppose $f$ is continuous on $(a,b)$, where $a$ could be $-\infty$ and $b$ could be $+\infty$.  Show that if \ $\displaystyle \int_a^b |f(x)| \, dx < \infty$, then the integral \ $\displaystyle \int_a^b f(x) \, dx$ \ exists and is finite.


\end{questions}

\end{document}
