CMU 21-120 Differential and Integral Calculus: Topics and Practice

CMU 21-120, Differential and Integral Calculus, is the first calculus course at Carnegie Mellon. The Department of Mathematical Sciences publishes a detailed topic list for it, written for the 21-120 placement exam, and that list is the clearest public picture of what the course expects you to master. This guide walks through it and links every topic to free practice with instant feedback.

What 21-120 covers

The department's 21-120 topic list starts with limits: one-sided and two-sided limits from graphs, algebraic techniques for limits at a point and at infinity, the Squeeze Theorem, classifying discontinuities, the Intermediate Value Theorem, and vertical and horizontal asymptotes.

It then covers the derivative from its limit definition, tangent lines, and what the derivative says about a function. Differentiation rules follow: power, product and quotient rules, the special trig limits, derivatives of trig, inverse trig, exponential and log functions, the chain rule, implicit differentiation, derivatives of inverse functions, and logarithmic differentiation for functions like f(x)^{g(x)}.

Applications come next: related rates, linear approximation and differentials for error estimates, global and local extrema, Rolle's Theorem and the Mean Value Theorem, first and second derivative tests, L'Hospital's rule including the forms 0 \cdot \infty, \infty - \infty, 0^0, \infty^0 and 1^\infty, and applied optimization.

The list ends with integration: antiderivatives and initial value problems, sigma notation and Riemann sums, properties of definite integrals, both parts of the Fundamental Theorem, u-substitution, and integration by parts.

21-120 topic map

Placement into 21-120 and beyond

The department places incoming students using standardized test scores and placement exams given during orientation week. AP, IB or Cambridge scores can earn credit for 21-120 under CMU's credit rules. Passing the 21-120 placement exam, on the other hand, gives a prerequisite waiver rather than course credit. The topic list notes that calculators and all other technology are prohibited on the exam.

Example problems at the 21-120 level

1. A 1^\infty form. Since \ln\left(1 + \frac{1}{x}\right)^x = x \ln\left(1 + \frac{1}{x}\right) \to 1, we get \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e.

2. Logarithmic differentiation. For y = x^x, \ln y = x \ln x, so y' = x^x(\ln x + 1).

3. FTC part 1.

How to do well in 21-120

• Use the topic list as a checklist. Tick off each line once you can do it without notes.
• Master the indeterminate forms. The list names five forms beyond 0/0, so practice rewriting each one.
• Practice without a calculator. The placement exam bans technology, and hand fluency helps in the course too.
• Do not skip integration by parts. It is the last item on the list and leads straight into 21-122.