Princeton MAT 104 Calculus II: Syllabus, Topics and Practice

Princeton MAT 104 is Calculus II, the second semester of Princeton's standard three-semester calculus sequence. It builds on MAT 103 with integration techniques, series and Taylor approximation, topics the department calls fundamental to the natural sciences, engineering and finance. This guide maps each MAT 104 topic to free practice with instant feedback.

What MAT 104 covers

The math department's course page lists these topics: techniques and applications of integration, including area, length and volume; an introduction to differential equations; polar coordinates; analysis of convergence of infinite series and improper integrals; and approximation using Taylor's theorem with remainder.

The page also explains the style of the course. It emphasizes mature problem-solving skills through problems that need sophisticated pattern recognition and the ability to combine several techniques in one problem. As in MAT 103, no computational technology or internet resources are allowed on homework or exams.

MAT 104 topic map

Placement: MAT 104 or something else?

The department compares MAT 104 to AP Calculus BC but is clear that test scores are not a reliable measure: a 5 on BC, a 7 on IB Math HL or an A at A-level is not equivalent to passing MAT 104. It suggests that two years of high school calculus with good grades may be comparable, and it uses a placement quiz early in the semester to help students confirm their choice.

Most students in MAT 104 are first-year students heading toward a natural science, engineering, math-track economics or finance. The prerequisite note is worth reading twice: the course expects you to solve equations, differentiate and sketch graphs with the standard functions quickly and without a calculator.

Example problems at the MAT 104 level

1. Integration by parts.

2. A convergence question. The series \sum_{n=1}^{\infty} \frac{1}{n^2 + 1} converges by comparison with the p-series \sum \frac{1}{n^2}, since \frac{1}{n^2+1} < \frac{1}{n^2}.

3. Taylor approximation. Near 0, e^x \approx 1 + x + \frac{x^2}{2}, so e^{0.1} \approx 1.105. Taylor's theorem bounds the error by roughly \frac{e^{0.1}(0.1)^3}{6} < 0.0002.

How to do well in MAT 104

• Keep a decision checklist for integrals. Most exam integrals are hard because you must pick the method. Practice choosing a strategy until it feels routine.
• Learn the convergence tests as a set. Work through choosing a test so you know which test fits which series.
• Refresh MAT 103 skills early. A quick pass through u-substitution and inverse trig derivatives saves time later.
• Practice under exam conditions with the practice exams, since calculators are not allowed.

After MAT 104, see the MAT 201 guide for multivariable calculus.