Stanford Math 19 Calculus: Topics, Study Guide and Practice
A study guide for Stanford Math 19, the first quarter of calculus: limits, rates of change, derivatives and their applications, mapped to free practice.
Stanford University Updated 3 min read
At a glance
- Course code
- MATH 19
- Units
- 3
- Offered
- Autumn and winter quarters (2026-27)
- Prerequisites
- Trigonometric functions, advanced algebra, and exponential and logarithmic functions
- Covers
- A review of elementary functions, limits, rates of change, the derivative and its properties, and applications such as maxima, minima and curve sketching
- Next course
- MATH 20 (Calculus)
- Practice here
- Calculus 1, the limits and derivative units
On this page
Stanford Math 19 is the first course in Stanford's single-variable calculus series. Because Stanford runs on quarters, one year of calculus is split into three shorter courses: Math 19, Math 20 and Math 21. Math 19 handles the differential half of a typical Calculus 1 course. This guide maps every Math 19 topic to free practice with instant feedback.
What Stanford Math 19 covers
Stanford's course catalog describes Math 19 as an introduction to differential calculus of functions of one variable. It begins by reviewing elementary functions, including exponentials and logarithms, then develops limits, rates of change, and the derivative with its properties, finishing with applications of the derivative. The math department's course overview adds trigonometric, exponential and logarithmic functions, maximum and minimum problems, and curve sketching to that picture.
Compared with a semester Calculus 1 course, Math 19 covers everything up to integration. Antiderivatives and the definite integral wait for Math 20.
Math 19 topic map
| Math 19 topic | Practice | Read |
|---|---|---|
| Review of elementary functions | Functions and notation | Polynomial functions |
| Exponentials and logarithms | Exponentials and logarithms | Natural logarithm |
| Trigonometric functions | Key trig values | Sine function |
| Limits | What is a limit? | How to evaluate limits, limit laws |
| Continuity | Continuity | Continuity guide |
| Rates of change | Instantaneous rate of change | Limit definition of the derivative |
| Derivative rules | The power rule | Power rule, product rule |
| Chain rule | Chain rule: any function | Chain rule guide |
| Derivatives of trig, exp and log | Exponentials and logarithms | Derivatives of trig functions |
| Maxima and minima | Absolute extrema | Optimization problems |
| Curve sketching | Increasing and decreasing | Curve sketching with derivatives |
The Stanford calculus sequence
| Course | Units | Main content |
|---|---|---|
| Math 19 | 3 | Limits and differential calculus |
| Math 20 | 3 | Integrals, the Fundamental Theorem, volumes, intro to differential equations |
| Math 21 | 4 | Improper integrals, sequences, series and Taylor series |
| Math 51 | 5 | Linear algebra and multivariable differential calculus |
Together, Math 19, 20 and 21 make up a full year of single-variable calculus. The department also offers Math 18, Foundations for Calculus, for students who want to strengthen precalculus skills first.
Example problems at the Math 19 level
1. A limit by factoring.
2. A product rule derivative. Differentiate f(x) = x^2 e^x:
3. Absolute extrema. Find the extreme values of f(x) = x^3 - 3x on [0, 2]. Since f'(x) = 3x^2 - 3 = 0 at x = 1, check f(0) = 0, f(1) = -2 and f(2) = 2. The minimum is -2 and the maximum is 2.
How to do well in Stanford Math 19
- Fix prerequisite gaps in week one. The catalog lists trig, algebra, and exponentials and logs as prerequisites. Our functions warm-up drills each in a few minutes a day.
- Keep up with the quarter pace. Ten weeks goes quickly, so practice a little after every class rather than catching up before a midterm.
- Learn the rules as patterns. Use the mixed rules practice to recognize product, quotient and chain rule problems on sight.
- Review with the formula sheet, read the common calculus mistakes, and finish with timed practice exams.
Practice it now
What is a limit?
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