Skip to main content
\(\require{cancel}\newcommand{\definiteintegral}[4]{\int_{#1}^{#2}\,#3\,d#4} \newcommand{\myequation}[2]{#1\amp =#2} \newcommand{\indefiniteintegral}[2]{\int#1\,d#2} \newcommand{\testingescapedpercent}{ \% } \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

AppendixBSolutions to Selected Exercises

Exercises4.2.4Exercises

Exercise1

This is an exercise in an “Exercises” subdivision at the level of a subsubsection. There is no question other than if the numbering is appropriate. Here is a self-referential link: Exercise 4.2.4.1.

The subsubsection has no title in the source, so one is provided automatically, and will adjust according to the language of the document.

Solution

This solution will migrate to a list of solutions in the backmatter. We include a sidebyside as a test.

This is a skinny paragraph which should be just 30% of the width.

And another skinny paragraph which should also be just 30% of the width.

Exercises11.4More Exercises

Exercise6

\(3+4+5\)

Hint

Addition is associative.

Answer

\(12\)

Solution

First, add \(3\) and \(4\) to get \(7\text{,}\) then add \(5\) to arrive at \(12\text{.}\)

Exercises16Exercises

Exercise1An Exercise in a Section

Exercises can appear in a “section” of their own. You need to give the section a title, even if it seems obvious what to call it. Individual exercises may have titles, as you choose. Problem: How should we hide solutions?

Solution

Maybe a global switch should be used to suppress solutions, while a separate processing regime could use them as part of a solutions manual.
Exercise42aAn Exercise with a Hard-Coded Problem Number

Compute the definite integral \(\definiteintegral{2}{4}{x^2}{x}\text{,}\) not as an approximate value from a Riemann sum, but as an exact value based of the limit by using the Fundamental Theorem.

Solution

An antiderivative of \(x^2\) is \(F(x)=x^3/3\text{,}\) so by the FTC,

\begin{equation*} \definiteintegral{2}{4}{x^2}{x}=F(4)-F(2)=\frac{1}{3}\left(4^3-2^3\right)=\frac{56}{3}\text{!?!} \end{equation*}

This is indeed an exciting result, but we are mostly interested in seeing that the sentence-ending punctuation is absorbed properly into the displayed equation.
Exercise3

Can you prove Corollary 4.1 directly? If not consider that a problem could have several parts, which should be formatted as a second-level list, since the problems normally get numbered at the top level.

  1. Why is this result a Corollary?

  2. Could you interchange the Theorem and Corollary?

Hint1MVT

Consider the definite integral as an area function and employ the Mean Value Theorem.

Hint2Motivator

Think harder!

AnswerHelpful

  1. It follows easily.

  2. Yes.

Solution

We could prove either result first, then obtain the other as an easy consequence.