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\newcommand{\myCourseNumber}{Math 142B}
\newcommand{\myAssignmentDueDate}{11:00pm Thursday, April 23, 2026}
\newcommand{\myAssignmentTitle}{Homework Assignment 2}
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\begin{questions}

%% Ross 29.14
\question Suppose $f$ is differentiable on $\mathbb{R}$, \ $1 \leq f'(x) \leq 2$ \ for all $x \in \mathbb{R}$, \ and \ $f(0) = 0$. \ Show that \ $x \leq f(x) \leq 2 x$ \ for all $x \geq 0$.

%% Ross 30.6
\question Let $f$ be differentiable on some interval $(c, \infty)$ such that $\displaystyle \lim_{x \to \infty} \left[ f(x) + f'(x) \right] = L$, with $L$ finite.

Show that $\displaystyle \lim_{x \to \infty} f(x) = L$ and $\displaystyle \lim_{x \to \infty} f'(x) = 0$.  \Big[Hint: Write $f(x) = \frac{f(x) \ e^x}{e^x}$.\Big]

%% Ross 6.32.2
\question Let \ $f(x) = \begin{cases} x & \text{if} \ x \in \mathbb{Q}, \\ 0 & \text{otherwise}. \end{cases}$
\begin{parts}
\part Compute the upper and lower Darboux integrals for $f$ on the interval $[0,b]$.
\part Is $f$ integrable on $[0,b]$?  Be sure to justify your answer.
\end{parts}

%% Ross 6.32.6
\question Let $f$ be a bounded function on $[a,b]$. \ Suppose there exist sequences $(L_n)$ and $(U_n)$ of upper and lower Darboux sums for $f$ such that \ $\lim (U_n - L_n) = 0$.

Show that $f$ is integrable on $[a,b]$ and that $\displaystyle \int_a^b f = \lim L_n = \lim U_n$.

%% Ross 6.32.7
\question Let $f$ be integrable on $[a,b]$, and suppose $g$ is a function on $[a,b]$ such that $g(x) = f(x)$ except for finitely many $x \in [a,b]$.

Show that $g$ is integrable on $[a,b]$ and that $\displaystyle \int_a^b g = \int_a^b f$.

%% Ross 6.33.1
\question Show that a decreasing function $f$ on $[a,b]$ is integrable.

%% Ross 6.33.7
\question Let $f$ be a bounded function on $[a,b]$ so that there is $B > 0$ for which $|f(x)| \leq B$ for all $x \in [a,b]$.
\begin{parts}
\part Show that
\begin{displaymath}
U(f^2,P) - L(f^2, P) \leq 2 B \left[ U(f,P) - L(f,P) \right]
\end{displaymath}
for all partitions $P$ of $[a,b]$.

\part Show that if $f$ is integrable on $[a,b]$, then $f^2$ is also integrable on $[a,b]$.
\end{parts}

%% Ross 6.33.8
\question Let $f$ and $g$ be integrable functions on $[a,b]$.
\begin{parts}
\part Show that $fg$ is integrable on $[a,b]$.

\part Show that $\max(f,g)$ and $\min(f,g)$ are integrable on $[a,b]$.
\end{parts}

%% Ross 6.33.13
\question Suppose $f$ and $g$ are continuous functions on $[a,b]$ such that $\displaystyle \int_a^b f = \int_a^b g$.  Prove that there exists $x \in (a,b)$ at which $f(x) = g(x)$.

%% Ross 6.33.14
\question
\begin{parts}
\part Prove that if $f$ and $g$ are continuous functions on $[a,b]$ with $g(t) \geq 0$ for all $t \in [a,b]$, then there exists $x \in (a,b)$ such that
\begin{displaymath} \int_a^b f(t) g(t) \, dt = f(x) \int_a^b g(t) \, dt. \end{displaymath}
	
\part Show that the \emph{Intermediate Value Theorem for Integrals} is a special case of part (a).

\part Does the conclusion in part (a) hold if $[a,b] = [-1,1]$ and $f(t) = g(t) = t$ for all $t \in [a,b]$?  Be sure to justify your answer.
\end{parts}

\end{questions}

\end{document}