Math 19 — Stanford, Summer 2011

This page is maintained by the instructors, Jonathan Lee and Xiannan Li, and pertains to both sections of the class.

This class has a CourseWork page; it is for checking your grades and little else.

Who, what, when, where

Lecture sections

  • Lecture 1 (Xiannan Li) — 1:15–2:05pm on Monday, Tuesday, Wednesday and Thursday
  • Lecture 2 (Jonathan Lee) — 2:15pm–3:05pm on Monday, Tuesday, Wednesday and Thursday
Both sections are held in room 380-380Y, located in the basement of the Main Quadrangle's Math corner.

Office hours (tentative)

  • Xiannan's office hours: from 2:45 - 4:15pm on Thursday in 380H. Alternate arrangements can be made to meet after lecture.
  • Jonathan's offices hours: Monday–Wednesday — informally, after lecture, depending on availability and Thursdays — 3:30–5:00pm (room 380-380X)
  • Xin's office hours: Friday — 2:00pm–5:00pm (room 380-381B)

Assignment submission

This quarter, Math 19 assignments are due at 5:00pm on Friday. The Math 19 CA, Xin Zhou, will be collecting them at this time; no late assignments are accepted! You may hand in your assignments to Xin at 380-381B directly or leave them in his mailbox; those of you who wish to submit assignments earlier may do so at the end of Thursday lectures.

Course Components and Policies

Should you take this class?

Tentative topics for this class are:
  • tangents, velocities
  • limits, the squeeze theorem, continuous functions and the intermediate value theorem
  • derivatives (both as numbers and as functions), higher-order derivatives
  • the product rule, the quotient rule, the chain rule
  • sketching functions, finding minima and maxima based on the derivative
  • implicit differentiation, inverse trigonometric functions, logarithmic differentiation
  • linear approximation, Taylor polynomials
  • (optionally) related rates
If you are already extremely comfortable with these topics, you are still welcome to take this class, say, if you simply desire an easy summer class. Otherwise, if you are already comfortable with harder topics, such as integration, consider taking a more advanced class, such as Math 51. The Stanford mathematics department offers this advice for choosing classes; additionally, you're encouraged to consult with a mathematics faculty advisor for advice.


Students in this class need a solid understanding of algebra and trigonometry at the high school level. This includes, but is not limited to:
  • understanding mathematical vocabulary and notation
  • mastery of the concept of a function
  • fluency working with lines (slope, various ways to write the equation of a line)
  • fluency manipulating algebraic expressions.
If you are concerned about your background, please see one of us (Xiannan, Jon or Xin).

We will have a background review quiz on the second Monday of class (June 27).


Single Variable Calculus: Concepts and Contexts, 4th Edition by James Stewart. Written homework assignments and your independent reading will come primarily from this book. The book should be available from the campus bookstore; it is also the textbook used by Math 20.

If you already have a different calculus textbook, it's not strictly necessary for you to acquire this one if you're confident in your abilities. (Most of the homework exercises will be drawn from the textbook.)


Unlike some of the other entry-level math classes at Stanford, such as the Math 40 and 50 series, Math 19 has no TA-led discussion sections. As such, we expect all students to be active participants in lectures themselves: ask questions when you have them, and work to answer questions posed by the lecturer.

Remember that if you are stuck on something, it is likely that others are as well and will benefit from your question!

Weekly written homework

The written homework will be a chance to work on problems that need extra thought. Be sure that your final write-up is clean and clear and effectively communicates your reasoning and your final answer to the grader. The grade on your homework will depend on your clarity of explanation, neatness, and correctness of intermediate steps, in addition to having the correct conclusion in the end.

Note that due to limited resources, we may not grade all the problems. Do not let this dissuade you from doing your best on these assignments — you should treat your own homework write-ups as your personal study notes for the midterm and final examinations.

Collaboration policy

Students who can solve homework problems on their own and in a timely manner will do well in this class. We permit, and even encourage, you to discuss the homework problems with whomever you like, but you must write up solutions independently, on your own. In particular, this means that homework solutions may not be written during office hours or group discussions; you may take notes when discussing problems with other people, but should put these notes away before writing up the homework solution.

Homework due dates and grading policy

There are 7 homework assignments. No late work will be accepted; however, we will drop your lowest homework score and compute your homework average from the remaining 6 homework assignments. Homework should be handed in to Xin Zhou every Friday by 5pm in his office at 381B or in his mailbox near the lobby of the math building. Again, late homework will not be accepted.


We will design the homework and exam problems such that there will be no need for calculators. As a result, calculators or other electronic aids will be forbidden during the midterm and final exam.

Weekly Reading

As the pace of this class may be faster than you are accustomed to, the goal of the weekly reading assignments is to help you learn and retain the material at this pace. It will also serve to develop your skills at reading a technical subject. You are expected read the sections we cover during the week for review purposes.


This offering of Math 19 has one midterm exam and a final exam. We will update this section with information on exact locations and times, as well as sample exams.

Midterm conflicts

If you have a course-related schedule conflict with the midterm exam, you must contact one the instructors at least a week in advance of the exam to make arrangements for an alternate (early) sitting. No other schedule conflicts are accommodated.

Final exam conflicts

In particular, the final exam date cannot be changed — if you have a conflict (even if it's the final exam for a different class); do not sign up for this class.


Your grade will be computed by combining scores from your written homework, midterm exam, and final exam according to the following percentages:
Quiz: 5%
Homework: 15%
Midterm 1: 30%
Final Exam: 50%


Office hours

Please bring any questions you have to office hours, or ask them in lecture, or email the questions to one of us (Xiannan, Jon or Xin).

Course assistant

We are lucky to have Xin Zhou as the course assistant for this class. His office is 381B, and this website will be updated with his office hours in a few days.


You are encouraged to discuss questions you have about the class with other students in Math 19. Discussions with your peers are a great way to hear new ideas and to help clarify your understanding of the material.

Emailing instructors

The instructors will generally respond to emails within 24 hours.


Private tutoring is also available from graduate students in the mathematics department. See the Math Department Tutoring website for more information. Note that private tutoring can be expensive.

Stanford Policies

You are expected to follow the Stanford Honor Code, accessible online at the Stanford University Honor Code website. We do not anticipate any honor code issues this summer; however, in order to be fair to all students, we will be very strict in dealing with any honor code violations.

Americans with Disabilities Act:

Please let your instructor (Xiannan or Jon) know if you need any special accommodations for this class, as specified by the Office of Accessible Education at Stanford.

More math from us to you

Solutions to final exam

The solutions to the final exam are here.

Final preparation

Background material


Midterm solution

The solutions to our midterm are here.

Midterm preparation

Below are actual midterms used in previous offerings of this class, intended to give you practice, help gauge your understanding on the concepts and acquaint you with the standards of university examinations. To get the most of these midterms, work through them under simulated examination conditions, and do not simply memorize problems and their solutions — do not expect that you'll be asked questions of an identical nature.

Background material


Homework assignments will be posted here — make sure you have the right edition of the textbook! (Or just grab the problems from a friend.)

Be sure to consult the syllabus for homework policies, especially with regards to collaboration, write-up standards and submission. Turn in homework by 5:00pm on Friday to Xin, or at the end of Thursday's lecture.

Independent reading

You should read the following sections in the book, if you haven't done so already:
  • 4.2, 4.3, 4.6
  • 3.1, 3.2, 3.3, 3.4, 3.5, 3.6
  • 2.2, 2.3, 2.4, 2.5, 2.6, 2.7
  • if you need the review: 1.2, 1.5, 1.6
  • if you need the review: sections A, B, C in the appendix
  • if you need the review: the background handouts


You can learn a lot by reading through the solutions; here they are!

Assignment 7 (due August 12)

  • plot the function \( f(x) = 2x^3 + 3x^2 - 36x - 32 \) — pay attention to intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. (Hint: \( f(4) = 0 \).)
  • from section 4.6: 40, 46

Assignment 6 (due August 5)

  • from section 3.6: 20, 28
  • from section 4.3: 6, 8, 16, 22, 30, 62a, 64, 66
  • from section 4.6: 12, 14, 34, 38

Assignment 5 (due July 29)

  • from section 3.5: 1, 5, 11, 19, 21, 23
  • from section 4.2: 23, 31, 35, 41, 51, 53, 55, 62

Assignment 4 (due July 22)

  • from section 2.6: 48
  • from section 3.3: 4, 6, 10, 15, 16, 17
  • from section 3.4: 12, 26, 28, 34

Assignment 3 (due July 15)

Note: you will not get points for the wrong usage of the Squeeze Theorem: after you show that \[ f(x) \leq g(x) \leq h(x) \,, \] it is incorrect to claim that \[ f(0) \leq g(x) \leq h(0) \quad\text{for all $x$} \] and then to take the limit of \( g(x) \).

Theoretical problems

Justify your solutions carefully, quoting the appropriate theorems when appropriate.
  • from section 2.3: 29, 31, 32
  • from section 2.4: 40, 44, 45(a), 46(a) (do by hand), 53, 55
  • from section 2.6: 34, 36

Computational problems

Just do these with whichever methods you wish; there are a lot, but they're meant to be quick. (If they're not fast, do more until they are.)
  • from section 2.6: 8, 14, 20
  • from section 3.1: 4, 6, 8, 16, 20
  • from section 3.2: 4, 14, 18
  • from section 3.4: 2, 4, 10

Assignment 2 (due July 8)

  • from section 2.3: 34, 38, 42, 50
  • from section 2.4: 4, 12, 16, 20, 24, 34, 36
  • additional problem — compute the following limit: \[ \lim_{x \to 0} \sin (x + e^{1/x^2}) - \sin (e^{1/x^2}) \,. \] (Hint: use the squeeze theorem twice, and an angle-sum formula from trigonometry.)
  • bonus problem — without using differentiation (which we haven't taught yet), show that for all values of \( \theta \): \[ | 3 \sin \theta + 4 \cos \theta | \leq 5 \,. \] Solutions using differentiation will earn zero credit.
  • bonus problem (typo fixed) — without using l'Hôpital's rule, compute the following limit: \[ \lim_{x \to \infty} x^{3/2} (\sqrt{x+1} + \sqrt{x-1} - 2\sqrt{x}) \,. \] Solutions using l'Hôpital's rule will earn zero credit.

Assignment 1 (due July 1)

  • from section 2.2: 4, 8, 15
  • from section 2.3: 2, 4, 8, 14, 20, 24, 30, 48
  • additional problem — solve for \( \theta \) in the equation: \[ \cos \theta + \sin \theta = \sqrt{2} \,. \] (Hint: Use the double-angle identity \( \sin 2\theta = 2 \cos\theta \sin\theta \).)
  • bonus problem — for \( a, b, c > 1 \), prove that: \[ \log_a(bc) \cdot \log_b(ac) \cdot \log_c(ab) = \log_a(bc) + \log_b(ac) + \log_c(ab) + 2 \,. \]
  • bonus problem — compute the following limit (without using l'Hôpital's rule): \[ \lim_{x \to 1} \frac{x^{3/5} - 1}{x^6 - 1} \,. \]

Breaking news!

Important announcements will be posted here as they come up in class. To ensure you don't miss anything, it is your responsibility to check this page regularly!

  • July 8 — the midterm is set for July 19, 7:00–9:00pm in the Mudd Chemistry Building's Braun Auditorium; sample midterms for practice have been posted
  • July 6 — homework solutions are now being posted
  • June 29 — the quiz has been graded --- see results here
  • June 23 — background review problem handouts posted
  • June 20 — syllabus updated with links to course advice

Midterm information

Location and time

The midterm will be held Tuesday, July 19, 7:00–9:00pm, in the Mudd Chemistry Building's Braun Auditorium. (Note that this is not the auditorium in the Braun Music Center, or any of the similarly named Braun buildings on campus.)

Things you should know

Note that the following is a guideline, not an exhaustive list.
  • by now, you should be adept with the high-school algebra that is a pre-requisite for this class; if not, seek help soon
    • be comfortable with absolute value manipulations
    • be able to bound a function from above and below
  • limits and one-sided limits, both at a point and at infinity:
    • understand what they mean
    • be able to recognize them, and compute them via algebra
    • be able to transform them to simpler limits via variable substitutions
  • continuity
    • be able to state the formal mathematical definition
    • be able to determine (with full justification) whether a piece-wise defined function is continuous
  • derivatives
    • be able to state their formal mathematical definition
    • be able to compute derivatives using the formal definition
    • be able to compute derivatives using the shortcut methods (power rule, quotient rule, chain rule, etc.)
    • be able to take derivatives of polynomials, rational functions, basic trigonometric functions, logarithms, exponential functions and any combination of them
    • be able to use the derivative to write the equation of a tangent line to a curve at a specified point
  • theorems (squeeze theorem, intermediate value theorem)
    • be able to state these theorems precisely (hypotheses and conclusion)
    • be able to recognize when these theorems apply (and do not apply)
    • be able to properly quote these theorems, using a properly written sentence after explicitly verifying hypotheses
  • given information about a function, be able to sketch it if necessary!

    Grading information

    This exam will not be curved. Rather, your overall course average (quiz, homeworks, midterm and final) will be computed as-is after the final examination; it is these raw averages that will be curved. Midterm statistics will be posted here after the midterms have been graded.

    Resources for preparation

    See the handouts tab for old midterms to work through.

    Scheduling conflicts

    Pending room confirmation by the Registrar's Office, the exam will be Tuesday, July 19 from 7:00pm to 9:00pm. You must notify one of the instructors by Wednesday, July 13 at noon if this time doesn't work for you — we will then make alternate arrangements.

    Jonathan Lee

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