# Princeton MAT 103 Calculus I: Syllabus, Topics and Practice

> A study guide for Princeton MAT 103 Calculus I: limits, derivatives, integrals and their applications, how placement works, and where to practice each topic.

Source: https://practicecalculus.com/courses/princeton-mat-103
Updated: 2026-11-08

## At a glance

- **Course code:** MAT 103
- **Credits:** Not stated on the department course page
- **Offered:** Not stated on the department course page
- **Prerequisites:** MAT 100 or equivalent, with a solid pre-calculus background from high school
- **Covers:** Limits and continuity, derivatives and integrals of one variable, optimization, curve sketching, approximation, area and the rudiments of differential equations
- **Next course:** MAT 104 (Calculus II)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MAT 103 is the first semester of Princeton's standard three-semester calculus sequence: MAT 103, MAT 104 and MAT 201.
- It covers limits, continuity, derivatives and integrals in one variable, with applications such as optimization, curve sketching and area.
- No calculators or internet resources are allowed on homework or exams, so hand computation matters.
- Princeton accepts a 5 on AP Calculus BC as equivalent to passing MAT 103, but a 5 on AP Calculus AB is not enough on its own.

**Princeton MAT 103** is Calculus I, the first course in the university's standard three-semester calculus sequence. The math department calls it an introduction to the mathematics of change and a gateway to further work in the natural sciences, engineering, economics, finance and the social sciences. This guide lists what the course covers and links every topic to free practice with instant feedback.

## What MAT 103 covers

According to the Princeton math department's course page, MAT 103 develops the technical concepts of limits and continuity, derivatives and integrals in the single variable setting. The applications named on the page are optimization, the geometry of curves, approximation and error analysis, counting and estimating roots, average behavior and estimating change, and computing the area of planar regions.

The course also includes the rudiments of differential equations and modeling, through curve sketching and initial value problems. The department says it emphasizes critical reasoning and communication skills in a quantitative setting.

One rule stands out: no computational technology or internet resources are allowed on homework or exams. You need to differentiate, integrate and sketch graphs by hand.

## MAT 103 topic map

| MAT 103 topic | Practice | Read |
| --- | --- | --- |
| Limits | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [How to evaluate limits](/blog/how-to-evaluate-limits) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity guide](/blog/continuity) |
| The derivative | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Differentiation rules | [Product rule](/learn/calculus-1/derivative-rules/product-rule) | [Chain rule](/rules/chain-rule), [quotient rule](/rules/quotient-rule) |
| Geometry of curves and curve sketching | [Second derivative](/learn/calculus-1/applications-of-derivatives/second-derivative) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Optimization | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems) |
| Approximation and error | [Linear approximation and the MVT](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Linear approximation](/blog/linear-approximation) |
| Estimating roots | [Intermediate Value Theorem](/learn/calculus-1/limits/intermediate-value-theorem) | [Intermediate Value Theorem](/theorems/intermediate-value-theorem) |
| Average behavior and estimating change | [Average value](/learn/calculus-1/applications-of-integrals/average-value) | [Mean Value Theorem](/theorems/mean-value-theorem) |
| Integrals | [Fundamental Theorem, part 2](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Area of planar regions | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves) | [Area between curves](/blog/area-between-curves) |
| Initial value problems | [Undoing derivatives](/learn/calculus-1/integrals/undoing-derivatives) | [Antiderivatives](/blog/antiderivatives) |

## Placement: is MAT 103 the right course?

The department is direct about placement. MAT 103 covers similar topics to AP Calculus AB, but a 5 on the AB exam is not treated as equivalent to passing the course. A 5 on AP Calculus BC, a 7 on IB Math HL or an A at A-level is accepted as equivalent. The department runs placement workshops at fall registration and gives a placement quiz early in the semester, and it advises students to take the course that matches their current knowledge rather than jumping ahead.

Most students in MAT 103 are first-year students, and the page says the course is a must for anyone considering economics or a natural science.

## Example problems at the MAT 103 level

**1. A limit that needs factoring.**

$$\lim_{x \to 3} \frac{x^2 - 9}{x - 3} = \lim_{x \to 3} (x + 3) = 6$$

**2. An optimization problem.** A rectangle has perimeter 20. Its area is $A = x(10 - x)$, so $A'(x) = 10 - 2x = 0$ gives $x = 5$, a $5 \times 5$ square with area 25.

**3. Area of a planar region.** The area between $y = x$ and $y = x^2$ on $[0, 1]$ is

$$\int_0^1 (x - x^2)\,dx = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$$

## How to do well in MAT 103

- **Practice without a calculator from day one.** Since none is allowed, fluency with [exponent rules](/learn/calculus-1/functions-warmup/exponent-rules) and [key trig values](/learn/calculus-1/functions-warmup/key-trig-values) pays off quickly.
- **Do a little every day.** Short daily sessions on [derivative rules](/learn/calculus-1/derivative-rules/power-rule) build the speed you need on exams.
- **Write out your reasoning.** The course stresses communication, so practice explaining why each step works, not just the final number.
- **Watch for common slips.** Our list of [common calculus mistakes](/blog/common-calculus-mistakes) covers the errors that cost the most points.

When you finish, the [MAT 104 guide](/courses/princeton-mat-104) shows what Calculus II adds.

## Frequently asked questions

### What is Princeton MAT 103?

MAT 103 is Calculus I at Princeton. The department describes it as an introduction to the mathematics of change, covering limits, continuity, derivatives and integrals of one variable and their applications.

### What are the prerequisites for MAT 103?

The department lists MAT 100 or equivalent, plus a solid pre-calculus background from high school.

### Does AP Calculus AB place me out of MAT 103?

No. The department says MAT 103 covers similar topics to AB calculus, but a 5 on the AB exam is not equivalent to a passing grade. A 5 on BC, a 7 on IB Math HL or an A at A-level is accepted as equivalent.

### Can I use a calculator in MAT 103?

No. The course page says no computational technology or internet resources are allowed on homework or exams.

### What comes after MAT 103?

The next course in the standard sequence is MAT 104, Calculus II, followed by MAT 201, Multivariable Calculus.