CMU 21-122 Integration and Approximation: Topics and Practice

CMU 21-122, Integration and Approximation, is the second calculus course at Carnegie Mellon, following 21-120. The Department of Mathematical Sciences publishes a full topic list for its placement exam, and it shows a course that combines integration techniques with a strong theme of approximation: numerical integration, series, Taylor polynomials and Newton's method. This guide maps every topic on that list to free practice.

What 21-122 covers

The department's 21-122 topic list begins with integration: standard formulas, u-substitution, integration by parts, trig identities for products of powers of trig functions, trig substitution for \sqrt{a^2 - x^2}, \sqrt{x^2 + a^2} and \sqrt{x^2 - a^2}, and partial fractions. Improper integrals follow, including the p-integral \int_1^\infty \frac{1}{x^p}\,dx and the Comparison Theorem. The Midpoint and Trapezoid Rules come with their error bounds.

The longest part of the list is sequences and series: convergence of sequences with the Squeeze and Monotone Convergence Theorems, geometric series, the Test for Divergence, the Integral Test with remainder estimates, direct and limit comparison, the Alternating Series Test with its error bound, the Ratio and Root Tests, absolute and conditional convergence, power series with radius and interval of convergence, building series from \frac{1}{1-x}, differentiating and integrating series, Taylor and Maclaurin series, Taylor's Theorem with Remainder, and the binomial series.

The list ends with Newton's method, arc length and surfaces of revolution, basic and separable differential equations, parametric curves with their derivatives and arc length, and polar coordinates with area and arc length.

21-122 topic map

Newton's method also appears on the list. It uses tangent lines, so the tangent lines lesson is the best warm-up.

Example problems at the 21-122 level

1. Trig substitution. With x = \sin\theta,

2. An alternating series bound. For \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}, the error after 9 terms is at most \frac{1}{100}.

3. Newton's method. To approximate \sqrt{2}, use f(x) = x^2 - 2 and x_{n+1} = x_n - \frac{x_n^2 - 2}{2x_n}. From x_0 = 1: x_1 = 1.5, x_2 \approx 1.4167.

How to do well in 21-122

• Practice choosing a method. Use choosing a strategy until picking a technique is automatic.
• Learn every error bound. Trapezoid, alternating series, integral test and Taylor remainders all appear on the list.
• Build series from known ones. Starting from \frac{1}{1-x} saves a lot of work on power series questions.
• Keep 21-120 skills sharp. L'Hospital's rule reappears in improper integrals, so revisit 21-120 topics as needed.