Duke MATH 112L Laboratory Calculus II: Topics and Practice

Duke MATH 112L, Laboratory Calculus II, is the second semester of Duke's laboratory calculus sequence. Like MATH 111L, it pairs calculus with lab projects, group work and written reports. Its topic list is wider than most Calculus 2 courses, reaching probability, systems of differential equations and Fourier series. This guide maps each topic to free practice with instant feedback.

What MATH 112L covers

The Duke math department lists the topics as methods of integration, applications of integrals, functions defined by integration, improper integrals, an introduction to probability and distributions, infinite series, Taylor polynomials, series solutions of differential equations, systems of differential equations, and Fourier series.

The prerequisite is MATH 106L or 111L, or consent. The course is not open to students who have had MATH 122 or 122L, and the department notes that AP or placement credit for MATH 21 is not sufficient if MATH 122L is being offered. MATH 112L is typically offered in the fall and/or spring.

MATH 112L topic map

Systems of differential equations and Fourier series are also on the list. They go beyond the lessons on this site, but both build on the integration, series and differential equations skills above.

MATH 112L or MATH 122L?

Duke has two routes into the second semester. MATH 112L continues the laboratory sequence from MATH 106L or 111L. MATH 122L, Introductory Calculus II with Applications, is for students with advanced placement credit for MATH 21, and covers sequences and series, the definition of the integral and its uses, Taylor and Fourier series, and differential equations with weekly labs. You cannot take both, and either one can lead to MATH 212.

Example problems at the MATH 112L level

1. A probability density. For f(x) = 2e^{-2x} on [0, \infty), the total probability is

2. A Taylor polynomial. The degree 3 Taylor polynomial of \cos x at 0 is 1 - \frac{x^2}{2}, since the odd terms vanish.

3. A series solution idea. If y' = y and y = \sum a_n x^n, matching coefficients gives a_{n+1} = \frac{a_n}{n+1}, so with a_0 = 1, y = e^x.

How to do well in MATH 112L

• Master integration methods early. Every later topic, from probability to improper integrals, uses them. The techniques practice mixes them like an exam.
• Connect improper integrals and probability. A density must integrate to 1, which makes improper integrals feel concrete.
• Treat series as a toolkit. Taylor polynomials, series solutions and Fourier series all approximate functions with simpler pieces.
• Write clearly in lab reports. Explain what each result means in the model, not just how you got it.

Next, see the MATH 212 guide for multivariable calculus.