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), higherorder 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.
Prerequisites
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).
Text
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.)
Lectures
Unlike some of the other entrylevel math classes at Stanford,
such as the Math 40 and 50 series, Math 19 has no TAled 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 writeup 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 writeups 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.
Calculators
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.
Exams
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 courserelated 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.
Grading
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%

Resources
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.
Classmates
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.
Tutoring
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.
Assignments
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, writeup 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
Solutions
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 anglesum 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:
\[ \lim_{x \to \infty} x^{3/2} (\sqrt{x+1} + \sqrt{x1}  2\sqrt{x}) \,. \]
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 doubleangle 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):
\[ \lim_{x \to 1} \frac{x^{3/5}  1}{x^6  1} \,. \]