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Derivatives and Integrals An Annotated Discourse

Robert Beezer, A. Second Author

Appendix B Solutions to Selected Exercises

This is an introduction, where you might explain that this division of the back matter contains various hints, answers, solutions of inline exercises, divisional exercises, project-like blocks, worksheet exercises, and/or reading questions. See the source to see just how this solutions division was built.
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4 An Interesting Corollary
4.2 A Pedagogical Note about Subsection 4.1
4.2.1 Symbolic and Numerical Integrals

Checkpoint 4.6. Essay Question: Compare and Contrast.

Hint.
Start writing!
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4.2.2 Advice

Checkpoint 4.9. An Inline Exercise.

Hint.
A good hint.
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Answer.
42.
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Solution.
If your exercise feels like proving a theorem, then you might want to make some comments, but also clearly delineate which part of the solution is a the complete proof.
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Proof.
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Proin lorem diam, convallis in nulla sed, accumsan fermentum urna. Pellentesque aliquet leo elit, ut consequat nunc dapibus ac. Sed lobortis leo tincidunt, vulputate nunc at, ultricies leo. Vivamus purus diam, tristique laoreet purus eget, mollis gravida sapien. Nunc vulputate nisl ac mauris hendrerit cursus. Sed vel molestie velit. Suspendisse sem sem, elementum at vehicula id, volutpat ac mi. Nullam ullamcorper fringilla purus in accumsan. Mauris at nunc accumsan orci dictum vulputate id id augue. Suspendisse at dignissim elit, non euismod nunc. Aliquam faucibus magna ac molestie semper. Aliquam hendrerit sem sit amet metus congue tempor. Donec laoreet laoreet metus, id interdum purus mattis vulputate. Proin condimentum vitae erat varius mollis. Donec venenatis libero sed turpis pretium tempor.
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Praesent rutrum scelerisque felis sit amet adipiscing. Phasellus in mollis velit. Nunc malesuada felis sit amet massa cursus, eget elementum neque viverra. Integer sagittis dictum turpis vel aliquet. Fusce ut suscipit dolor, nec tristique nisl. Aenean luctus, leo et ornare fermentum, nibh dui vulputate leo, nec tincidunt augue ipsum sed odio. Nunc non erat sollicitudin, iaculis eros consequat, dapibus eros.
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Activity 4.3. Hints, Answers, Solutions.

Answer.
The result, but no help in getting there.
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Checkpoint 4.12. A very structured exercise.

4.12.c A title of a task that has a subtask with an <answer> for the Solutions.
4.12.c.i A task with a title and an <answer> for the Solutions.
Answer.
With a proof.
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Proof.
In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
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And a bit more to say.
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4.12.c.ii
Answer.
Right! In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
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4.12.d
Hint.
One hint. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
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In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
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Answer 1.
First answer. In interdum suscipit ullamcorper.
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Answer 2.
Second answer. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
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In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
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In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
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Solution.
At last, the solution. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
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Project 4.4. A very structured project.

4.4.c
4.4.c.i
Answer.
With a proof.
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Proof.
In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
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And a bit more to say.
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4.4.c.ii
Answer.
Right! In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
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4.4.d
Answer 1.
First answer. In interdum suscipit ullamcorper.
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Answer 2.
Second answer. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
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In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
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In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
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Project 4.5.

Answer.
Here’s the answer we are looking for.
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4.2.3 Exercises

4.2.3.1.

Solution.
This solution will migrate to a list of solutions in the backmatter. We include a sidebyside as a test.
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This is a skinny paragraph which should be just 30% of the width.
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And another skinny paragraph which should also be just 30% of the width.
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4.6 Introductions and Conclusions
4.6.2 Test Two

Checkpoint 4.16.

Answer.
An answer so there is something to appear in a <solutions>.
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What Did You Learn?

4.6.2.1.
Answer.
An answer so there is something to appear in a <solutions>.
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Exercises

4.6.2.1.
Answer.
An answer so there is something to appear in a <solutions>.
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11 Further Reading
11.3 More Exercises

11.3.6.

Hint.
Addition is associative.
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Answer.
\(12\)
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Solution.
First, add \(3\) and \(4\) to get \(7\text{,}\) then add \(5\) to arrive at \(12\text{.}\)
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11.3.11.

Hint.
Addition is associative.
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Answer.
\(12\)
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Solution.
First, add \(3\) and \(4\) to get \(7\text{,}\) then add \(5\) to arrive at \(12\text{.}\)
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Proof.
And a bit more.
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11.3.16.

Hint.
Addition is associative.
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Answer.
\(12\)
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Solution.
First, add \(3\) and \(4\) to get \(7\text{,}\) then add \(5\) to arrive at \(12\text{.}\)
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11.3.21.

Hint.
Addition is associative.
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Answer.
\(12\)
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Solution.
First, add \(3\) and \(4\) to get \(7\text{,}\) then add \(5\) to arrive at \(12\text{.}\)
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11.3.25.

Answer.
\(x\) \(f(x)\)
\(0\) \(0\)
\(1\) \(1\)
\(2\) \(4\)
\(3\) \(9\)
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16 Interactive Exercises
16.1 True/False

16.1.1. True/False.

Hint.
\(P_n\text{,}\) the vector space of polynomials with degree at most \(n\text{,}\) has dimension \(n+1\) by 2.1. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter \(n\text{?}\)
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16.2 Multiple-Choice

16.2.1. Multiple-Choice, Not Randomized, One Answer.

Hint 1.
What did you see last time you went driving?
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Hint 2.
Maybe go out for a drive?
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16.3 Parsons Problem, Math Proof

16.3.1. Parsons Problem, Mathematical Proof.

Hint.
Dorothy will not be much help with this proof.
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17 Dynamic Exercises

Dynamic Fill-In

17.3. Fill-In Formula (Dynamic).
17.4. Decompose the Function.

31 Advanced Numbering
31.7 A Title with ] a Right Bracket

Checkpoint 31.19. A Right Brace } and a Right Bracket].

Hint.
A faux hint to get this exercise to migrate into a <solutions>.
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31.10 Just an Exercise

Checkpoint 31.20. An Extraneous Exercise.

Hint.
A hint, so this exercise looks identical in structure to the one in the previous subsection.
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35 Worksheets
35.1 A Geometric Prelude

37 Exercise Section, Structured

Easy Exercises

37.1. Arithmetic.
Answer.
\(7\)
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With an Exercise Group

38 Exercises, Multiple Subsections
38.5 Layout challenge

38.5.4.

Hint. OPEN ME!
\begin{alignat*}{5} 3u \amp {}+{} 1v \amp {}+{} 3w \amp {}+{} 2x \amp {}+{} \amp y \amp {}+{} \amp 3z \amp {}={} \amp 10\\ 2u \amp {}+{} 2v \amp {}-{} 2w \amp {}+{} x \amp \amp \amp {}+{} \amp z \amp {}={} \amp 6\\ 1u \amp {}-{} 3v \amp {}+{} 1w \amp {}+{} x \amp {}+{} \amp 3y \amp {}+{} \amp 2z \amp {}={} \amp 13 \end{alignat*}
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39 Exercises, Top-Level

39.10. An Exercise in a Section.

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.
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39.42a. An Exercise with a Hard-Coded Problem Number.

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.
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39.12.

Hint 1. MVT
Consider the definite integral as an area function and employ the Mean Value Theorem.
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Hint 2. Motivator
Think harder!
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Answer. Helpful
  1. It follows easily.
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  2. Yes.
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Solution.
We could prove either result first, then obtain the other as an easy consequence.
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And a conclusion to this solutions division, which may not be readily apparent as distinct from the final division’s worth of solutions, but since it is not prefixed with a number, it may be different enough.
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