Columbia MATH UN1102 Calculus II: Topics, Study Guide and Practice

Columbia MATH UN1102 is Calculus II at Columbia University. It finishes single-variable calculus and appears in every one of the department's calculus sequences, before multivariable calculus in UN1201, UN1205 or the honors track. This guide matches each UN1102 topic with a lesson you can practice with instant feedback.

What Columbia MATH UN1102 covers

The Columbia bulletin lists methods of integration, applications of the integral, Taylor's theorem and infinite series. The prerequisite is MATH UN1101 or the equivalent.

That is the core of a standard Calculus 2: integration techniques, applications such as volume and arc length, convergence of series, and Taylor polynomials and series. Polar and parametric curves are not named in the bulletin description. If your section covers them, the power series, parametric and polar unit on PracticeCalculus has lessons for both.

The two halves of the course feel quite different. The integration half rewards pattern recognition: seeing which method fits an integrand. The series half rewards careful reasoning about limits: deciding whether an infinite sum settles down, and how well a Taylor polynomial approximates a function. Practicing both kinds of problem every week, rather than one unit at a time, makes the final exam much less of a shock.

UN1102 topic map

Where UN1102 leads

Calculus II is also required before UN1202 Calculus IV. The course before it is UN1101 Calculus I.

Example problems at the UN1102 level

1. Trig substitution. With x = 2\sin\theta, dx = 2\cos\theta\,d\theta and \sqrt{4 - x^2} = 2\cos\theta:

2. Estimating with a Taylor polynomial. The degree-3 Maclaurin polynomial of \ln(1 + x) is x - \frac{x^2}{2} + \frac{x^3}{3}, so

which matches the true value 0.09531 to four decimal places.

3. 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. It converges conditionally, since \sum \frac{1}{n} diverges.

How to do well in MATH UN1102

• Practice picking the method. The choosing a strategy lesson mixes techniques the way exams do.
• Learn series tests in a fixed order: nth-term, geometric or p-series, comparison, ratio, alternating. The choosing a test lesson drills that order.
• Memorize the core Maclaurin series for e^x, \sin x, \cos x, \ln(1 + x) and \frac{1}{1-x}. Most Taylor problems are built from them.
• Use the formula sheet for trig identities, scan common calculus mistakes, and take timed practice exams.

Tip: the error in an alternating series estimate is at most the first term you left out. It is the fastest error bound in the course.