Fall 2009
David Perkinson
L316, ext. 7417
Office
hours
Course Description
Limits, continuity, derivatives, integration, fundamental theorem of
calculus, applications.
Prequisite: three years of high school mathematics.
Text
Calculus of a Single Variable, 8th Edition
by Larson, Hostetler, and Edwards, available at the bookstore.
Books on Reserve
- Calculus of a Single Variable, 8th Edition by Larson,
Hostetler, and Edwards.
Math Help Center
Student tutors are available to help you on SMTWTh evenings from 7:00
to 9:00 in L387 or L389.
Exams
- The midterm will be Friday, October 16. It will be in-class,
closed-book, closed-notes, etc. (Note: this is the Friday before fall break.)
Here is a review sheet.
- The final exam will be Tuesday, December 15, 9–12 noon, L314.
This Week
- Monday. Integration by parts. Trig. substitutions.
- Tuesday. Go over homework.
Trig substitutions.
- Wednesday. Quiz. Partial fractions.
- Friday. No class. Thanksgiving break.
Assignment: There is no assignment due this
week, but here are some practice exercises for the Thanksgiving holiday:
- Review exercises to Chapter 8: 2, 4, 6, 10, 12, 18, 20, 26, 50, 56.
Here is a handout with these
problems. Please try them over break.
Assignments
- No assignments due for the first week.
- HW 1, due September 11.
- Section 1.2: 20, 38, 42, 44, 60, 64, 66, 75.
- Reading: sections 1.1, 1.2, and 1.3.
- HW 2, due September 18.
- Section 1.3: 8, 16, 24, 30, 34, 38, 44 (parts (a) and (b),
plus a simpler function agreeing with the given one except at one
point), 50, 52, 54, 84, 86, 104, 116 (explain or give a specific
counterexample). [NOTE: None of these problems assigned from section
1.3 require epsilon-delta proofs.]
- Let f be a function defined at all real numbers. Prove, using
the definition of the limit, that if the limit of f(x) as x
approaches 0 is 1, then f(x) is positive for x in an open
interval about 0, except possibly at 0, itself.
- Let f(x) = 2x+3 for x less than or equal to 2, and
let f(x) = −3x+13 for x greater than 2. Guess the limit
of f(x) as x approaches 2 and give an epsilon-delta
proof that your guess is correct.
- Reading: sections 1.4–1.5.
- HW 3, due September 25.
- Section 1.4: 16, 18, 26, 84 (don't forget to mention
continuity), 92, 94, 104.
- Section 1.5: 34, 36, 42, 46 (remember this is the infinite
limits section), 76 (see the definition on p. 83).
- Section 2.1: 14, 18, 24, 38, 40, 66.
- Extra Credit (please turn in on a separate sheet of paper):
Section 1.4: 114.
- Reading: Sections 2.1–2.3.
- Notes:
- When applying the intermediate value theorem, don't forget
to mention continuity.
- For T-F questions, our in-house rule is that you either
explain why the statement is true or you give a concrete counter-example.
- HW 4, due October 2.
- Section 2.1: 20 (use the definition of the derivative directly).
- Section 2.2: 4, 8, 18, 42, 50, 54 (a), 74, 90, 100. [NOTE:
Please read p. 109, example 4, and pp. 113–114.]
- Section 2.3: 2, 6, 12, 44, 54, 74, 84, 87, 128, 136.
- Section 2.4: 8, 14, 42, 54, 98.
- Reading: Sections 2.4–2.6.
- HW 5, due October 9.
- Section 2.1: 22 (use the definition of the derivative directly).
- Section 2.2: 116 (use the definition of the derivative; see p. 112).
- Section 2.3: 118 (a) (which is distance, which is velocity, and which is
acceleration?).
- Section 2.4: 22, 26, 58.
- Section 2.5: 32 (we may not cover this section of the book in class;
please read it on your own).
- Section 2.6: 27 (a), 32, 36 (a,b). [NOTE: Make sure you do not
accidentally do problems from the review section instead of from section 2.6.
They are adjacent in the text.]
- Section 3.1: 40 (a,b,c,d), 44, 56 (a,b), 58 (a,b), 60, 64 (if T, explain;
if F, give a specific counter-example).
- Section 3.7: 4, 22.
- Extra Credit: Section 3.7: 65 (Please turn in on a separate sheet of
paper.)
- Reading: Sections 3.1–3.7.
- HW 6, due October 30.
- Section 2.5: 76.
- Section 2.6: 30 (a), (b).
Note:
- There are two parts to both (a) and (b).
- Section 3.3: 24, 36.
Note:
- For each of these do parts (a)
through (c) and graph the function. You don't need to worry about
part (d).
- Section 3.7: 6, 24.
- Section 4.2: 10, 14, 16, 28, 48, 80.
Notes:
- In 16, you shouldn't need to add the terms one-by-one. Read
the first few pages of Section 4.2.
- For 48, see p. 266.
- For the limit in 80, see Theorem 8.4, p. 568, and example 4,
p. 570. An alternative is to modify example 10 on p. 66. At any
rate, show your work.]
- Extra Credit:
- Section 4.1: 97.
- Let X be a bounded subset of the real numbers with
sup(X) = s and inf(X) = l. Suppose that
x - y < k for all x and y in X. (i) Show
that s - l ≤ k. (ii) Give an example of a specific set
X for which equality is attained, i.e., s - l = k even
though for each pair of elements of X, you have x - y < k.
- NOTE: Please turn extra credit problems in on separate sheets of
paper, clearly marked as extra credit.
- Reading: Sections 4.2–4.4. Note that I expect you to
know the definition of the integral given in class (which is
somewhat different from that given in the book).
-
HW 7, due November 6.
- Section 2.3: 78.
- Section 2.4: 44, 52. (Careful, the secant is not squared in number 44, only its argument.)
- Section 2.6: 48.
- Section 3.4: 12, 32.
- I am leaving the reading of section 3.4 to you.
- Section 3.7: 54.
- Read Section 4.1.
- Section 4.1: 16, 18, 20, 22, 24, 26 (remember to include "+c" in
your solutions).
- Section 4.2: 30.
- Section 4.3: 46.
- Section 4.4: 6, 8, 14.
- HW 8, due November 13.
- Section 3.6: 10.
- Please read section 3.6 to see how to do problem 10.
- Section 4.4: 10, 12, 16, 18, 32, 38, 78, 86, 104.
- As usual, don't worry about the "graphics utility" stuff.
- Section 4.5: 8, 10, 12, 14, 20, 24, 34, 44, 64, 66, 72, 80,
90.
- Note that problems 8-34 require checking your answer by
taking the derivative.
- Section 5.1: 46, 48, 50, 56.
- Section 5.2: 2, 6, 8, 10, 20, 26.
- Extra Credit: (Please turn in on a separate sheet of paper.)
Section 4.5: 136.
- NOTE: Please double-check that you have not accidentally skipped any problems
on this assignment.
- HW 9, due November 20.
- Review exercises to chapter 2 (p. 160): 110.
- Section 3.6: 14.
- Section 3.7: 56.
- Section 5.2: 18, 30, 34.
- Section 5.4: 36, 38, 40, 46, 50.
- Section 5.5: 38, 62.
- Section 5.7: 2, 4, 6, 8.
- Extra Credit: (Please turn in on a separate sheet of paper.)
Find the least number A such that for any two squares of combined
area 1, a rectangle of area A exists such that the two squares can
be packed in the rectangle (without interior overlap). You may assume
that the sides of the squares are parallel to the sides of the
rectangle.
- Practice exercises for the Thanksgiving holiday.
- Review exercises to Chapter 8: 2, 4, 6, 10, 12, 18, 20, 26, 50, 56.
Here is a handout with these problems. Please try them over
break!
Quizzes
- Quiz 1, Wednesday, September 2. Definition of the limit of a function. See p. 52 of
the text.
- Quiz 2, Wednesday, September 16. Limit of a linear function (ε-δ proof).
- Quiz 3, September 23. Evaluating limits.
- Quiz 4, September 30. Definition of the derivative. Explain
what the derivative is measuring. Use the definition to compute a
derivative.
- Quiz 5, October 7.
- Take derivatives using the sum/product/quotient/chain rule.
- Know the derivatives of the six trig functions.
- Know how to calculate the derivatives of the six trig functions from
the derivatives of cosine and sine.
- Quiz 6, October 14.
- State the mean value theorem (including the hypotheses).
- Using the mean value theorem, prove that if f'(x)>0 on some
open interval, then f is strictly increasing on that interval.
- Quiz 7, November 4. State the definition of integrability and
the integral. The definition we are using is here.
- Quiz 8, November 11. State and prove the fundamental theorem of
calculus. See the handout.
- Quiz 9, November 18. Quiz on last week's HW. Make
up quiz on the definition of the integral.
- Quiz 10, November 25.
- State the definition of the natural logarithm, the exponential function,
and exponentiation by arbitrary real numbers.
- State the second version of the fundamental theorem of calculus. A
handout with the statement appears here.
- Calculate the derivative of the inverse
sine using the fact that sin(arcsin(x))=x.
Handouts
Class Summary
Week 1: Aug. 31–Sept. 4
- Monday. Overview of calculus: derivatives, integrals,
fundamental theorem. For tomorrow and Wednesday, read sections 1.1
and 1.2 and look at exercises 7 and 9 in 1.1 and 1, 3, 9, 11, 13, 15,
17 in 1.2 (do not turn in).
- Tuesday. Finish introduction: plausibility argument for the
fundamental theorem of calculus. Definition of the limit, discussion.
- Wednesday. Definition of the limit. Examples. For
Friday, look at section 1.2, exercises 23, 31, 33, 39, 41.
- Friday. Limits of some nonlinear functions. Relevant
reading: p. 54.
Week 2: Sept. 7–11
- Tuesday. Limit theorems.
- Wednesday. Limit theorems: proofs. Relevant reading:
pp. 59—60, Appendix A2–A3.
- Friday. Evaluating limits; a function with no limits. Practice problems in
section 1.3:
any odd-numbered problem 5–21; 27, 29, 31, 37, 41, 43, 49, 51,
53, 103, 105, 111.
Week 3: Sept. 14–18
- Monday. Continuity. [Section 1.4]
- Tuesday. Discuss HW. Practice problems (Hints.)
- Wednesday. Quiz. Variations on the definition of a
limit. Intermediate value theorem. [Section 1.4–1.5]
- Friday. Definition of the derivative. Practice problems in section 2.1: 3,
7, 11–23 (odds), 37, 39, 55, 93, 99, 101.
Week 4: Sept. 21–25
- Monday. Instantaneous change, tangent lines; first properties of
derivatives.
- Tuesday. Discuss HW. Trig. review.
- Wednesday. Quiz. First properties of derivatives.
- Friday. First properties of
derivatives. Derivative of sine and cosine.
Week 5: Sept. 29–Oct. 3
Week 6: Oct. 5–9
Week 7: Oct. 12–16.
Week 8: Oct. 26–30.
- Monday. Introduction to the integral.
- Tuesday. Return midterm. Properties
of lower and upper sums.
- Wednesday. No quiz this week. Examples.
- Friday. Continuous functions are integrable.
Week 9: Nov. 2–6.
Week 10: Nov. 10–14.
- Monday. The logarithm.
- Tuesday. Go over homework. Properties of the log.
- Wednesday. Quiz (see the handout). Integrals involving logs and trig functions.
- Friday. The inverse function
theorem and the exponential function.
Week 11: Nov. 17–21.
- Monday. Exponentiation.
- Tuesday. Go over homework. Inverse trig functions.
- Wednesday. "Definition of the integral" quiz retake. Quiz.
Group quiz. Solutions are here.
- Friday. Integration by parts. Trig substitutions.
Miscellaneous Links
Sage
Sage is a free open-source mathematical software system. It can plot
functions, take derivatives and limits, integrate, and solve equations (among
many other things). Check
out the Sage website. You can use it from your web
browser or download it for free onto your own computer (Linux, Mac, or
Windows). For tips on using Sage, click here.
Check out the Sage calculus
tutorial.
Homework and Grades
Your grade will be based
on the weekly homework and quizzes, class participation, an
in-class midterm, and a final exam. The homework is weighted
almost as highly as the midterm and final. It is important to not
miss any of the homework assignments or quizzes! When I return
your homework, I will put numbers next to each problem according to
the following scheme:
| 5 |
4 |
3 |
2 |
1 |
0 |
| perfect |
minor mistakes |
major mistake, right idea |
wrong but contains a significant idea |
wrong but contains a relevant idea |
none of the above |
NOTES:
- Homework is due each Friday at the beginning of class.
- Late assigments (i.e., turned in after class on Friday), if
complete, may be given half-credit, but it is unlikely you will get written comments on
your work.
- If you miss an assignment or quiz due to an illness,
please inform me as soon as possible.
- Note our final exam date before making airline reservations
to leave at the end of the semester.
- Deadline to add classes or change sections: Friday, September 15.
- Deadline to drop fall semester classes without the grade of "W"
(withdraw): Monday, October 6.
- Deadline to withdraw from fall semester classes: Monday, November 10.
