## Math 111: Calculus, Fall 2014

Section F01: MTWF 8-8:50am, Library 389Section F03: MTWF 11-11:50am, Eliot 207

Office Hours: T 9-11am, Wθ 1:30-2:30pm, Library 313

Math
Center: SuMTWθ 7-9pm, Library 387

Textbook: *Calculus of a single variable* by Larson, Hostetler,
and Edwards, 8th ed.

### Week 14

- Monday: Read §6.3. Separation of variables, logistic growth, and the Lotka-Voltera model.
- Tuesday:
*Pretend Thursday. No class.* - Wednesday: Review session. [FINAL REVIEW SHEET]
- Friday:
*Reading period. No class.* - Tuesday, Dec. 16: Final exam in Physics 123, 6-9pm.

#### Homework due Tuesday night.

- Email me one review topic and a question typifiying that topic which you'd like to cover in Wednesday's review session. Please include the words "final review" in your subject heading so I can sort the emails appropriately.

### Week 13

- Monday: Read §8.8. Improper integrals.
- Tuesday: Integration review. Evaluations.
- Wednesday: Quiz [SOLUTIONS]. Problem session.
- Friday: Read §§6.1, 6.2. Differential equations.

#### Homework due Friday

- §8.8: 2, 4, 8, 24, 66, 86, 88, 102.
- Ch. 8 Review (pp.589-590): 2, 10, 20, 28, 52, 54.
- Ch. 8 P.S. (p.592): 12, 16.

### Week 12

*Announcement*: Homework is due Monday, Dec. 1. A different
(and final) homework set will be due on Friday, Dec. 5.

- Monday: Read §§8.3, 8.4. Trig integrals and trig substitution.
- Tuesday: Read §8.5. Partial fractions.
- Wednesday: Mysteries of the universe explained.
- Friday:
*No class. Happy Thanksgiving!*

#### Homework due Monday, Dec. 1.

- §3.6: 24.
- §5.5: 50, 52.
- §8.2: 18, 26, 90. (For 90, also compute an explicit
formula, not involving integrals, when
*n*= 5.) - §8.3: 18, 70, 96. (For 96, also compute an explicit
formula, not involving integrals, when
*n*= 5.) - §8.4: 22, 30, 42, 62, 82, 86. (Read the start of §7.4 to learn about arclength.)
- §8.5: 10, 20, 48, 58.

### Week 11

*Announcement*: Our final exam takes place Tuesday, Dec. 16, 6-9pm
in Physics 123.

- Monday: Read §§5.4, 5.5. Exponentiation.
- Tuesday: Read §§5.6, 5.7. Inverse trig functions.
- Wednesday: Quiz [SOLUTIONS]. Problem session.
- Friday: Read §8.2. Integration by parts.

#### Homework due Friday

- Review exercises for Chapter 2 (p.160): 110.
- §3.6: 14.
- §3.7: 56.
- §5.2: 18, 30, 34.
- §5.4: 36, 38, 40, 46, 50.
- §5.5: 38, 62.
- §5.7; 2, 4, 6, 8.
- Let
*U*and*V*be subsets of the real numbers. Prove that a function*f*:*U→V*has an inverse*g*:*V→U*if and only if*f*is bijective. (Recall that*f*is*bijective*when for every*v*in*V*there is a unique*u*in*U*such that*f*(*u*) =*v*. If you use this definition properly, your argument should be very short.)

### Week 10

*Announcement*: Women in Math & Physics Brunch, Parker House,
10am-Noon, Saturday, Nov. 15. RSVP by Nov. 10 to Isabella Jorissen
(isjoriss@reed.edu).

- Monday: Read §5.1. The logarithm.
- Tuesday: Quiz [SOLUTIONS]. Properties of logarithms.
- Wednesday: Read §5.2. Integrals involving logs and trig functions.
- Friday: Read §5.3 and §5.4. The inverse function theorem and the exponential function.

#### Homework due Friday

- §4.4: 10, 12, 16, 18, 32, 38, 78, 86, 104. (Ignore any "graphics utility" instructions.)
- §4.5: 8, 10, 12, 14, 20, 24, 34, 44, 64, 66, 72, 80, 90. (Remember to check your answer by taking the derivative in problems 8-34.)
- §5.1: 46, 48, 50, 56.
- §5.2: 2, 6, 8, 10, 20, 26.

### Week 9

[INTEGRALS] (A terse pdf recounting the theory of integrals as done in class. Meant to bolster, not replace, your own notes.)

- Monday: Read §4.4. The fundamental theorem of calculus.
- Tuesday: Quiz [SOLUTIONS]. Basic properties of integrals.
- Wednesday: Read §4.1 and §4.5. Antiderivatives and substitution.
- Friday: MVT for integrals and FTC2.

#### Homework due Friday

- §4.1: 16, 18, 20, 22, 24, 26, 28.
- §4.2: 30.
- §4.3: 46.
- §4.4: 6, 8, 14.
- Use the fundamental theorem of calculus to turn the special
derivatives portion of the essential derivatives handout
into a list of integrals (excluding the log and exponential
portion). The result should be a list of 11 definite
integrals from
*a*to*b*along with restrictions on*a*and*b*that make the rules valid. (For example, your formula for the integral of*sec*^{2}should not permit,*e.g.*, π/2 to be between*a*and*b*. Be careful!)

### Week 8

- Monday: Read §4.2*. Definition of the integral, redux.
- Tuesday: Comparing lower and upper sums.
- Wednesday: Read §4.3*. Practice with infs, sups, upper sums, lower sums.
- Friday: Continuous functions are integrable.

*It's recommended that you read sections 4.2 and 4.3 above. Note, though, that the definition of the integral given in class is different from (and better than) the one in the book. We will use the class definition headed forward.

#### Homework due Friday

- §4.2: 18, 20, 24, 26, 80 [
*Hint*: The behavior of*x*sin(1/*x*) as*x*approaches ∞ is intimately linked to that of sin(*y*)/*y*as*y*approaches 0.] - §4.3: 14, 16, 18.
- The following problems will use suprema to define
*x*where^{r}*x*is real number greater than 1 and*r*is any real number. (Previously we only defined*x*for^{r}*r*rational.) We'll then verify that some familiar properties hold for this new function. In your arguments, you may use the fact that if*r*>*s*are both rational numbers and*x*> 1, then*x*>^{r}*x*.^{s}- If
*A*and*B*are subsets of the real numbers, define*AB*to be the set of numbers*ab*where*a*is in*A*and*b*is in*B*. Assume that*A*and*B*only consist of postive real numbers. If*S*= sup*A*and*T*= sup*B*, argue that*ST*= sup*AB*. (You must use the properties of*S*and*T*to show that the product*ST*is an upper bound of*AB*, and that it is the least such upper bound.) - For
*x*> 1 a real number, let*X*(*r*) denote the set of numbers*x*where^{t}*t*ranges over all*rational*numbers less than or equal to*r*. Use completeness of the real numbers to argue that sup*X*(*r*) exists. We define*x*= sup^{r}*X*(*r*). - If
*r*is rational and*x*> 1, prove that sup*X*(*r*) matches the "old"*x*.^{r} - For
*x*> 1, prove that*x*=^{r+s}*x*^{r}*x*for any real numbers^{s}*r*,*s*. (You should almost certainly use the fact you proved about sup*AB*.) - What about positive
*x*which are less than 1? Will the same definition work? If not, come up with a definition using sups or infs that will work. (You should make a persuasive argument about whether or not the old definition will work. If you make a new definition, you only need to write a couple of sentences about why it should work better.) - Think carefully about what you would have to do in order to
prove that
*d/dx*[*x*] =^{r}*rx*when^{r-1}*r*is a real number. (We know this is true when*r*is rational, but what if it's irrational?) Either prove this result, or come up with some reasonable lemmas that you think would help in its proof. (Feel free to look at some more advanced texts if you think it will help.) Explain your logic.

- If

### Week 7

- Monday: Infs, sups, and completeness.
- Tuesday: No quiz. Upper and lower sums, definition of the integral.
- Wednesday: In-class review. [REVIEW QUESTIONS]
- Friday: Midterm.

#### Homework due Friday

- None.
- Have a great fall break!

### Week 6

- Monday: Read §3.4. Concavity and the second derivative test.
- Tuesday: Quiz [SOLUTIONS]. Asymptotics.
- Wednesday: Read §§3.5, 3.6. Summary of curve sketching.
- Friday: Read §3.7. Optimization redux. [MIDTERM REVIEW HANDOUT]

#### Homework due Friday

- §2.5: 78.
- §2.6: 30(a,b).
- §3.3: 24(a,b,c), 36(a,b,c).
- §3.4: 2, 4, 6 [for each of 2, 4, 6, determine the intervals analytically (just referencing the picture is not enough)], 40, 60, 76.
- §3.5: 22, 28, 82.
- §3.6: 6, 16, 34, 42 [no need to use a graphing utility].
- §3.7: 44, 52 [you may need to do parts of problems 49-51 to complete this problem], 64.

### Week 5

- Monday: Read §3.1. Extrema.
- Tuesday: Quiz [SOLUTIONS]. Related rates and extrema practice problems. [PROBLEMS, SOLUTIONS (now with diagrams!)]
- Wednesday: Read §3.2. The mean value theorem.
- Friday: Read §3.3. The first derivative test.

#### Homework due Friday

- §2.1: 22 (use the limit definition of the derivative directly).
- §2.2: 116.
- Use the quotient rule to compute the derivatives of
sec(
*x*), csc(*x*), and cot(*x*). - §2.3: 110.
- §2.4: 26, 56, 58.
- §2.5: 32.
- §2.6: 26, 28, 32, 36.
- §3.1: 40, 44, 56, 58, 60, 64 (if true, explain; if false, give a counterexample).
- §3.2: 2, 22, 38(a,b,c), 66, 78.

### Week 4

- Monday: Read pp.112-114 and pp.119-124 in advance. Linear approximation, derivatives of trig functions, the product rule.
- Tuesday: Quiz [SOLUTIONS]. Derive and conquer.
- Wednesday: Read §2.4 in advance. The chain and quotient rules.
- Friday: Read §§2.5, 2.6 in advance. Related rates. [ESSENTIAL DERIVATIVES HANDOUT]

#### Homework due Friday

- §2.2: 4, 8, 18, 42, 50, 54(a), 74, 76, 100.
- Use the concept of linear approximation and your newfound
knowledge of derivatives to approximate the value of
sin(3.1)
^{2}by hand. - On a stroll through the Canyon, you chance upon an oracle which
will tell you the instantaneous rate of change of Apple Inc. stock
(NASDAQ: AAPL). (That is, if
*f*(*t*) = the value of AAPL*t*seconds after NASDAQ opening on September 19, 2014, then the oracle will tell you the value of*df/dt*at the present moment.) Ignoring any moral qualms with capitalism (or at least relegating them to a footnote), write several sentences on the potential uses and pitfalls of such an oracle in your day-trading investment strategy. - §2.3: 2, 6, 12, 44, 54, 74, 84, 122, 136.
- §2.4: 8, 14, 42, 54, 98.

[SOLUTIONS (updated 28.IX.14)]

### Week 3

- Monday: Read §1.5 in advance. The intermediate value theorem, infinite limits, limits at infinity.
- Tuesday: Quiz [SOLUTIONS]. Sums of (in)finite geometric series.
- Wednesday: Read §2.1 in advance. The derivative and tangent lines.
- Friday: Read pp.107-111 in advance. Basic properties of derivatives.

#### Homework due Friday

- §1.3: 120. [HINT]
- §1.4: 16, 18, 26, 84, 92 (if true, prove it; if false, provide a counterexample), 100, 104.
- §1.5: 34, 42, 46, 62, 76 (
*N*-δ proof). - §2.1: 18, 24, 38, 40, 66.

[SOLUTIONS (updated 22.IX.14)]

### Week 2

- Monday: Read pp.59-66 and A2-A5 in advance. Limits and composition, functions that agree in a neighborhood.
- Tuesday: Quiz. Problem session. The epsilon-deltinator [Mathematica demonstration].
- Wednesday: Continuity of trig functions, the Squeeze Theorem.
- Friday: Read pp.70-76 in advance. Continuity, one-sided limits, the arithmetic of continuity.

#### Homework due Friday

**Important note:** For **all** true/false questions, provide a
proof if the statement is true, and provide a counterexample if it's false.

- §1.2: 46, 64, 66
- §1.3: 16, 24, 30, 38, 50, 54, 84, 104, 110, 112, 119
~~§1.4: 26, 28~~

[SOLUTIONS (updated 14.IX.14)]

### Week 1

- Tuesday: Introduction to the course, overview of calculus.
- Wednesday: Read §1.2 in advance. Definition and examples of limits.
- Friday: Read the limits handout and do its exercises in advance. Working with ε's and δ's.

#### Homework due Friday

- Fill out and return the pre-course survey.
- Exercises from the limits handout. [SOLUTIONS]

### The LaTeX document preparation system

Poor handwriting? Love escape characters? Too much free time? Try LaTeX!

- LaTeX at Reed.
- A short guide [pdf] to writing mathematics with LaTeX.
- Sample LaTeX input/output: sample.tex/sample.pdf.

Kyle M. Ormsby