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

Section16Exercises

Exercises in an “exercises” section are numbered automatically. However, what if you have a busted problem and remove it? Then a bunch of problem numbers change and your list of homework for your students changes as well. What a mess. Use auto-numbering while writing and refining. Once stable, go ahead and hard-code problem numbers as you delete/add. Notice the oddly numbered problem below. Once you go down this road, you can't stop. But instructors will thank you for it.

More precisely, once you hard-code a number for a problem, you will likely need to hard-code every subsequent problem number in that section of exercises. This is because the automatic numbering is unlikely to be what you really want, or what you had before.

1An 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.
42aAn 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.
3

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.