# Harvard Math 1a Introduction to Calculus: Study Guide, Practice

> A study guide for Harvard Math 1a (Introduction to Calculus): derivatives, optimization, related rates and the definite integral, mapped to free practice.

Source: https://practicecalculus.com/courses/harvard-math-1a
Updated: 2026-11-03

## At a glance

- **Course code:** Mathematics 1A (Math 1a)
- **Credits:** 4
- **Recommended prep:** Strong precalculus skills
- **Covers:** Derivatives read from graphs and tables, the product, quotient, chain, implicit and logarithmic rules, optimization, related rates, the definite integral and the Fundamental Theorem of Calculus
- **Next course:** Mathematics 1B (Integration, Series and Differential Equations)
- **Placement:** Students enroll through Math PL and take the Math Placement Verification Exam
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- Math 1a is the first half of single-variable calculus at Harvard: differentiation and the basics of integration.
- Its three linked ideas are differential calculus, integral calculus and the Fundamental Theorem of Calculus that connects them.
- Students are expected to bring strong precalculus skills, and Math Ma is the route for those who need more preparation.
- The next course is Math 1b, then multivariable calculus in Math 21a or Math 19a.

**Harvard Math 1a**, officially Mathematics 1A: Introduction to Calculus, is where most Harvard students without a year of high school calculus begin. It is a one-semester course on the first half of single-variable calculus, taught in small sections, and it feeds directly into Math 1b. This guide maps each Math 1a topic to free practice so you can learn by solving problems with instant feedback.

## What Harvard Math 1a covers

The Harvard Mathematics Department describes Math 1a as three linked ideas: differential calculus, which measures rates of change, integral calculus, which measures accumulation, and the Fundamental Theorem of Calculus, which ties the two together. On the differential side, students interpret derivatives graphically and numerically, connect them to the shape of a graph, and use the product, quotient, chain, implicit and logarithmic differentiation rules to solve optimization and related rates problems. The course then introduces the definite integral as a way of adding up small pieces of change. Examples come from the sciences, economics and everyday life, and students are asked to reason with algebraic, graphical, numerical and symbolic descriptions of functions.

That lines up closely with a standard **Calculus 1** course.

## Math 1a topic map

| Math 1a topic | Practice | Read |
| --- | --- | --- |
| Functions and rates of change | [Slope and rate of change](/learn/calculus-1/functions-warmup/slope-and-rate-of-change) | [Exponential functions](/functions/exponential-functions) |
| Limits and the derivative | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [What is a limit?](/blog/what-is-a-limit), [limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Derivatives from graphs | [Notation and meaning](/learn/calculus-1/the-derivative/derivative-notation-and-meaning) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Product and quotient rules | [The product rule](/learn/calculus-1/derivative-rules/product-rule) | [Product and quotient rules](/blog/product-and-quotient-rules) |
| Chain rule | [Chain rule: any function](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule guide](/blog/chain-rule) |
| Implicit and logarithmic differentiation | [Implicit differentiation](/learn/calculus-1/derivative-rules/implicit-differentiation) | [Logarithmic differentiation](/rules/logarithmic-differentiation) |
| Shape of a graph | [Concavity and the second derivative](/learn/calculus-1/applications-of-derivatives/second-derivative) | [Mean Value Theorem](/blog/mean-value-theorem) |
| Optimization | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems) |
| Related rates | [Related rates](/learn/calculus-1/applications-of-derivatives/related-rates) | [Related rates guide](/blog/related-rates) |
| The definite integral as accumulation | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums) |
| Fundamental Theorem of Calculus | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1), [FTC part 2](/theorems/fundamental-theorem-of-calculus-part-2) |

## Math Ma vs Math 1a vs Math 1b

| Course | Starting point | What it covers |
| --- | --- | --- |
| Math Ma and Mb | Precalculus needs strengthening | Precalculus combined with single-variable calculus over two semesters, then on to Math 1b |
| Math 1a | Strong precalculus | First half of single-variable calculus |
| Math 1b | Math 1a or equivalent | Integration, series and differential equations |

Your Harvard Math Placement recommendation points you to one of these starting points. After Math 1a, continue with the [Math 1b guide](/courses/harvard-math-1b), then multivariable calculus in [Math 21a](/courses/harvard-math-21a) or [Math 19a](/courses/harvard-math-19a).

## Example problems at the Math 1a level

**1. Logarithmic differentiation.** Differentiate $y = x^x$ for $x > 0$. Take logs: $\ln y = x \ln x$. Differentiate both sides:

$$\frac{y'}{y} = \ln x + 1 \quad\Rightarrow\quad y' = x^x(\ln x + 1)$$

**2. Optimization.** A rectangle has perimeter 20. Its area is $A = x(10 - x)$, so $A'(x) = 10 - 2x = 0$ gives $x = 5$. The maximum area is $25$, from a $5$ by $5$ square.

**3. Related rates.** A circle's radius grows at 2 cm/s. How fast is the area growing when $r = 3$?

$$\frac{dA}{dt} = 2\pi r\frac{dr}{dt} = 2\pi(3)(2) = 12\pi \text{ cm}^2/\text{s}$$

## How to do well in Harvard Math 1a

- **Shore up precalculus first.** The department calls strong precalculus essential. Our [functions warm-up](/learn/calculus-1/functions-warmup/function-notation) covers exponents, logs and trig values in short drills.
- **Read every derivative three ways.** Math 1a asks you to reason from graphs, tables and formulas, so practice explaining what a derivative means, not only computing it.
- **Draw a picture for every word problem.** Optimization and related rates go wrong at the setup, and our [common calculus mistakes](/blog/common-calculus-mistakes) list shows the usual traps.
- **Prepare for evening midterms** with timed [practice exams](/practice) and the [calculus formula sheet](/formulas) at hand.

## Frequently asked questions

### What is Harvard Math 1a?

Math 1a, Introduction to Calculus, is a one-semester course covering the first half of single-variable calculus. It teaches derivatives and their uses, such as optimization and related rates, then introduces the definite integral and the Fundamental Theorem of Calculus.

### What comes after Math 1a?

Math 1b, which covers integration techniques and applications, infinite series and an introduction to differential equations. After 1b, students usually continue to Math 21a or, for the life sciences, Math 19a.

### Is Math 1a hard?

The department stresses that strong precalculus skills are essential, and most difficulty in a first calculus course comes from algebra and function skills rather than the new ideas. Students who keep those sharp and practice regularly usually find it manageable.

### How do I get into Math 1a?

Harvard places students using the Harvard Math Placement recommendation. Students enroll in the placeholder course Math PL, fill in a sectioning form, and take a Math Placement Verification Exam before sections are assigned.

### Math 1a vs Math Ma: which should I take?

Math Ma and Mb combine precalculus with single-variable calculus over two semesters and then lead to Math 1b. Math 1a assumes your precalculus is already strong. Your placement recommendation is the best guide.