• See here for the syllabus.

Announcements

Problem sets

  • Week 12 (Due Friday 11/22) Section 5.2 #1, 5, 6, 12, 22, 25, 29, 31, 36, 42; Section 5.9 #1, 3, 12, 13 (for each, use 3 iterations of Newton’s method). Extra credit: write a compute program to compute Newton’s method.

  • Week 11 (Due Monday 11/18) Setion 4.5 #1, 2, 8, 10, 15, 16.

    Section 4.6: Choose three integrals from #1 to #20, with the corresponding choice of $\Delta x$. Numerically approximate the integrals. State which method you used, and at least one must be approximated using the trapezoidal rule and at least one must be approximated using Simpson’s rule. You are encouraged to use a calculator or write a computer program to calculate the integral.

    Section 5.1: #1, 2, 4, 6, 10, 12

  • Week 10 (Due Wednesday 11/6) Integrate the following:

    $\displaystyle \int_1^4 \frac{e^{\sqrt x}}{\sqrt x} \,\mathrm{d}x$

    $\displaystyle \int_0^1 (4t + 3t^2)(t^3 + 2t^2)^4 \,\mathrm{d}t$

    $\displaystyle \int_0^\pi \cos u e^{\sin u} \,\mathrm{d}u$

    $\displaystyle \int_0^1 \frac{\sqrt{\arctan x}}{1 + x^2} \,\mathrm{d}x$

    A common form where you use substitution is when $u = kx$ for some constant $k$. Work out the general rule for what to do in this situation:

    • If $u = kx$ determine $\mathrm{d}u$.

    • What is $\displaystyle \int e^{2x} \,\mathrm{d}x$?

    • What is $\displaystyle \int e^{kx} \,\mathrm{d}x$? ($k$ is a fixed constant)

    • What is $\displaystyle \int \sin(kx) \,\mathrm{d}x$?

    • What is the general rule to calculate $\displaystyle \int f(kx) \,\mathrm{d}x$, if $f$ is some function?

  • Week 9 (Due Monday 11/4) Section 4.2 #3, 5, 16, 20; Section 4.3 #1, 2, 12, 16, 18, 30; Section 4.4 #1, 2, 7, 27, 30, 56, 83, 84

  • Week 8 (Due Monday 10/28) Section 3.7 #1, 13, 39, do the same process as in this section’s problems to the functions $b(x) = (x^2 - x)e^x$ and $c(x) = \arctan x$; Section 4.1 #1, 2, 4, 10, 22.

  • Week 7 (Due Monday 10/21) Section 3.5 #1, 2, 4, 12, 24, 27; Section 3.6 your choice of four total from #2–20 or from the problems on the 10-18 worksheet.

  • Week 6 (Due Monday 10/14) Section 3.2 your choice of three from #5–12; Section 3.3 #2, 3, 4, 6, 8, 16; Section 3.4 #1, 2, 4, 9, 18, 26

  • Week 5 (Due Wednesday 9/25) Differentiate: $\displaystyle a(x) = 4x^2e^x + \frac{\cos(e^{\arctan x})}{\ln x}$

Differentiate: $\displaystyle b(x) = \sin(\cos(\tan(\sec(x)e^{x^2})))$

Differentiate: $\displaystyle c(x) = \sqrt{x^4 + e^4 + \sin(4x) + \tan(e^x - x^2) + \frac{\log_3 x}{x}}$

  • Week 4 (Due Monday 9/23) Section 2.8 #4, 8, 22, 27, 30, 36; Section 8.7 #1, 3, 4, 6, 7, 9; Section 2.4 #1, 2, 4, 14, 17, 18

  • Week 3 (Due Monday 9/16) See here, page 1.

  • Week 2 (Due Monday 9/9). Section 1.6 #1, 2, 4, 8, 25, 26, 30; Section 2.1 #3, 4, 6, 12, 26; Section 2.2 #11, 12, 22, 24, 27

  • Week 1 (Due Monday 9/2). Section 1.2 #2, 4, 10, 12, 16, 24; Section 1.3 #10, 34; Section 1.5 #2, 4, 6, 7, 18, 20, 24, 33

Writing assignments

  • Week 12 No writing assignment.

  • Week 11 (Due Monday 11/18) Write explanations for Section 4.6 #22 and 23. (Extra credit) Write a program to do numerical integration, implementing at least two kinds of Riemann sums and the trapezoidal rule. Write comments in your program explaining how it works.

  • Week 10 No writing assignment.

  • Week 9 (Due Monday 11/4) Work out formulas in terms of $k$, where $k$ is an arbitrary positive integer, which gives the values of the definite integrals $\displaystyle \int_0^1 x^k \, \mathrm{d} x$ and $\displaystyle \int_{-1}^1 x^k \, \mathrm{d} x$. Explain why these formulas work.

    (Extra credit) A function $f(x)$ is odd if $f(-x) = - f(x)$ for all $x$ and $f(x)$ is even if $f(-x) = f(x)$ for all $x$. Explain why, for any $b$, $\displaystyle \int_{-b}^b f(x) \, \mathrm{d} x = 0$ for odd $f(x)$ and $\displaystyle \int_{-b}^b f(x) \, \mathrm{d} x = 2 \int_0^b f(x) \, \mathrm{d} x$ for even $f(x)$.

  • Week 8 (Due Monday 10/28) Read the textbook’s proof of the extreme value theorem on page 164. Summarize the key ideas of the argument in your own words, and draw a picture to illustrate it. (Extra credit) Do the same for the textbook’s proofs of Rolle’s theorem and the mean value theorem, on pages 165 and

  • Week 7 (Due Monday 10/21) Your friend insists that a function can have its maximum occur at two different places without having a minimum. Explain why they are correct by giving an example. Are they still correct if you insist the function be continuous everywhere on its domain? Justify your answer. (Extra credit 1) Consider the function $r(x)$ defined as $r(x) = x$ when $x$ is rational and $r(x) = 0$ when $x$ is irrational. Determine the set of points where $r(x)$ is continuous. (Extra credit 2) Consider a fixed finite set $S = \{a_1, \ldots, a_n\}$ of real numbers. Define the function $\chi_S(x)$ as $\chi_S(x) = 1$ if $x$ is an element of $S$ and $\chi_S(x) = 0$ otherwise. Determine the set of points where $\chi_S(x)$ is continuous.

  • Week 6 (Due Monday 10/14) Read pages 29–31 of Penelope Maddy, “How applied mathematics became pure” (2008), where she discusses how calculus is used to approximate a discrete reality. Pick one of the problems from section 3.2 you did. Following Maddy’s analysis, explain for your specific problem the idealizations and simplifications made to be able to apply calculus.

  • Week 5 No writing assignment.

  • Week 4 (Due Monday 9/23) Use logarithmic differentiation to explain why the quotient rule for derivatives works. (Extra credit) Use the inverse function rule to explain why the rules for arcsin and arccos work.

  • Week 3 (Due Monday 9/16) See here, page 2.

  • Week 2 (Due Monday 9/9). Consider the step function $u(x)$ defined as $u(x) = 0$ if $x \le 0$ and $u(x) = 1$ if $x > 0$. Determine where $u(x)$ is differentiable and justify your answer. (Extra credit) Find the derivative of $f(x) = \lvert x^2 - 1 \rvert$, identifying where $f(x)$ is not differentiable. Justify your answer.

  • Week 1 (Due Monday 9/2). Write up an explanation for problem #25 and (extra credit) #26 on page 42.

Readings

  • By Wednesday 8/28. Skim sections 1.1 through 1.3 and confirm that you know all the concepts therein.

Schedule

This course is organized into three units. Units 1 and 2 each end in a midterm, while unit 3 ends in an oral final. Homework consists of problem sets and writing exercises, and is on a weekly schedule due Mondays. (Exam weeks have an adjusted schedule.)

The schedule below is tentative; we might have small adjustments in the dates. For each week I’ve included which sections from the textbook we will be covering.

  • Unit 1 (8/26–9/27): The derivative

    • Week 1: Introduction, hyperreals [1.4, 1.5]

    • Week 2: The derivative [1.6, 2.1, 2.2]

    • Week 3: Computing derivatives [Chapter 2]

    • Week 4: More about differentiation [2.6, 2.8]

    • Week 5: Yet more about differentiation, review, exam [None]

    (Fall break 9/30–10/4.)

  • Unit 2 (10/7–11/8): Uses of the derivative and an introduction to the integral

    • Week 6: Related rates, limits, and continuity [3.2, 3.3, 3.4]

    • Week 7: Extreme points [3.5, 3.6]

    (No class Wednesday 10/16.)

    • Week 8: The shape of graphs and integrals [3.7, 3.8, 4.1]

    • Week 9: The fundamental theorem of calculus and computing integrals [4.2, 4.3, 4.4]

    • Week 10: More on integration by change of variables, review, exam [4.4]

  • Unit 3 (11/11–12/6): Miscellaneous applications

    • Week 11: Area between curves, numerical integration and infinite limits [4.5, 4.6, 5.1]

    • Week 12: L’Hôpital’s rule, Newton’s method, the epsilon-delta characterization of limits [5.2, 5.9, 5.8]

    (Thanksgiving break 11/25–11/29.)

    • Week 13: Increments, overspill time [5.10]

    • Week 14: Oral finals

Important dates:

  • Friday, Sept 27: Midterm 1

  • Friday, Nov 8: Midterm 2

  • Oral Final: Scheduled individually with me the week of 12/9. Exact times TBD.