Berkeley Math 52 (formerly Math 1B) Calculus II: Study Guide

UC Berkeley Math 52, long known as Math 1B, is the second semester of single-variable calculus at Berkeley. It picks up where Math 51 (formerly 1A) ends and goes further than many Calculus 2 courses by including both first and second-order differential equations. This guide maps every Math 52 topic to free practice with instant feedback.

What Berkeley Math 52 covers

The Berkeley math department describes Math 52 as a continuation of Math 51 covering techniques of integration and applications of integration, infinite sequences and series, first-order ordinary differential equations, and second-order linear ordinary differential equations. In the official outline, integration techniques take about nine class hours, a short unit covers further applications of integration, the sequences and series unit takes about twelve hours, and the two differential equations units take about eight hours each.

So Math 52 is a Calculus 2 course without the parametric and polar unit, which Berkeley teaches at the start of Math 53, plus a solid introduction to differential equations.

Math 52 (1B) topic map

Second-order linear differential equations do not have a dedicated lesson yet. They rely on the exponential and trig derivatives from Calculus 1.

Where Math 52 fits at Berkeley

Continue with the Math 53 guide for multivariable calculus.

Example problems at the Math 52 level

1. Trig substitution. With x = \sin\theta, dx = \cos\theta\,d\theta and \sqrt{1 - x^2} = \cos\theta:

That is the area of a quarter of the unit circle, which checks out.

2. An alternating series. \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} converges by the alternating series test because \frac{1}{n} decreases to 0, even though \sum \frac{1}{n} diverges. So it converges conditionally.

3. A second-order linear equation. Solve y'' - 3y' + 2y = 0. The characteristic equation r^2 - 3r + 2 = 0 has roots r = 1 and r = 2, so

How to do well in Berkeley Math 52

• Decide on a technique before you write. Exams mix integrals, so practice recognizing substitution, parts, trig substitution and partial fractions on sight with the techniques practice set.
• Give series the most time. It has the largest block in the outline. Learn the geometric series formula and nth-term test first, then build up to the ratio and comparison tests.
• Check differential equation solutions by substituting them back. It takes seconds and catches most errors.
• Keep the formula sheet close, read the common calculus mistakes, and test yourself with timed practice exams.