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

> A study guide for Carnegie Mellon 21-120 Differential and Integral Calculus: every topic on the official list, from limits to integration by parts.

Source: https://practicecalculus.com/courses/carnegie-mellon-21-120
Updated: 2026-11-08

## At a glance

- **Course code:** 21-120
- **Units:** 10 (per the department's sample schedule)
- **Offered:** Fall of Year 1 in the department's sample schedule; other terms are not stated on the department pages
- **Prerequisites:** Not stated on the department topic list; placement uses test scores and orientation-week placement exams
- **Covers:** Limits and continuity, the derivative and its rules, implicit and logarithmic differentiation, related rates, linear approximation, the Mean Value Theorem, curve analysis, L'Hospital's rule, optimization, Riemann sums, the Fundamental Theorem, u-substitution and integration by parts
- **Next course:** 21-122 (Integration and Approximation)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- 21-120 is Carnegie Mellon's first calculus course and is followed by 21-122.
- The department's topic list runs from one-sided limits to u-substitution and integration by parts.
- Passing the 21-120 placement exam gives a prerequisite waiver, not course credit.
- No calculators or other technology are allowed on the placement exam.

**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

| 21-120 topic | Practice | Read |
| --- | --- | --- |
| One-sided limits and asymptotes | [One-sided and infinite limits](/learn/calculus-1/limits/one-sided-and-infinite-limits) | [Limits at infinity](/blog/limits-at-infinity) |
| Squeeze Theorem | [Squeeze theorem](/learn/calculus-1/limits/squeeze-theorem) | [Squeeze theorem](/theorems/squeeze-theorem) |
| Continuity and the IVT | [Intermediate Value Theorem](/learn/calculus-1/limits/intermediate-value-theorem) | [Continuity guide](/blog/continuity) |
| Limit definition and tangent lines | [Tangent lines](/learn/calculus-1/the-derivative/tangent-lines) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Power, product and quotient rules | [Quotient rule](/learn/calculus-1/derivative-rules/quotient-rule) | [Product and quotient rules](/blog/product-and-quotient-rules) |
| Trig, exponential and log derivatives | [Trig derivatives](/learn/calculus-1/derivative-rules/trig-derivatives) | [Derivatives of trig functions](/blog/derivatives-of-trig-functions) |
| Chain rule | [Chain rule in general](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule](/rules/chain-rule) |
| Implicit and inverse function derivatives | [Implicit differentiation](/learn/calculus-1/derivative-rules/implicit-differentiation) | [Inverse function rule](/rules/inverse-function-rule) |
| Logarithmic differentiation | [Exponential and log derivatives](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Logarithmic differentiation](/rules/logarithmic-differentiation), [derivative of x^x](/derivatives/x-to-the-x) |
| Related rates | [Related rates](/learn/calculus-1/applications-of-derivatives/related-rates) | [Related rates guide](/blog/related-rates) |
| Linear approximation and the MVT | [Linear approximation and the MVT](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean Value Theorem](/theorems/mean-value-theorem), [Rolle's theorem](/theorems/rolles-theorem) |
| Extrema and derivative tests | [Absolute extrema](/learn/calculus-1/applications-of-derivatives/absolute-extrema) | [Extreme Value Theorem](/theorems/extreme-value-theorem) |
| L'Hospital's rule | [L'Hôpital's rule](/learn/calculus-1/applications-of-derivatives/lhopitals-rule) | [L'Hôpital's rule](/rules/lhopitals-rule) |
| Optimization | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems) |
| Riemann sums | [Riemann sums](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums guide](/blog/riemann-sums) |
| Fundamental Theorem | [Fundamental Theorem, part 1](/learn/calculus-1/integrals/fundamental-theorem-part-1) | [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1) |
| u-substitution | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [U-substitution guide](/blog/u-substitution) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts rule](/rules/integration-by-parts-rule) |

## 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.**

$$\frac{d}{dx}\int_0^{x^2} \cos t\,dt = 2x\cos(x^2)$$

## 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](/courses/carnegie-mellon-21-122).

## Frequently asked questions

### What is CMU 21-120?

21-120 Differential and Integral Calculus is Carnegie Mellon's first-semester calculus course. The department's topic list covers limits, derivatives and their applications, and integrals through u-substitution and integration by parts.

### How many units is 21-120?

The Department of Mathematical Sciences sample schedule lists 21-120 at 10 units.

### Can I place out of 21-120?

Students can earn credit from AP, IB or Cambridge exams under CMU's credit rules, or take a placement exam during orientation. The department says passing a placement exam gives a prerequisite waiver, not course credit.

### Is integration by parts in 21-120?

Yes. The official topic list ends with using integration by parts on definite and indefinite integrals, so the course goes a little further than many Calculus 1 courses.

### What comes after 21-120?

21-122 Integration and Approximation, which covers integration techniques, improper integrals, series, Taylor series, differential equations, parametric curves and polar coordinates.