# Berkeley Math 51 (formerly Math 1A) Calculus I: Study Guide

> A study guide for UC Berkeley Math 51, formerly Math 1A: limits, derivatives, applications, integrals and their applications, with free practice for each.

Source: https://practicecalculus.com/courses/uc-berkeley-math-51
Updated: 2026-11-03

## At a glance

- **Course code:** Math 51 (formerly Math 1A)
- **Units:** 4 (three hours of lecture and three of discussion per week)
- **Offered:** Fall and spring
- **Prerequisites:** Three and one-half years of high school math, including trigonometry and analytic geometry
- **Covers:** Functions and models, limits and derivatives, differentiation rules, applications of differentiation, integrals and applications of integrals
- **Next course:** Math 52 (formerly Math 1B), Calculus II
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- Math 51 is the new number for Berkeley's Math 1A, the first semester of calculus for engineering and physical science majors.
- It introduces differential and integral calculus of one variable, with applications and transcendental functions.
- It is a 4-unit course with three hours of lecture and three hours of discussion each week.
- It is followed by Math 52 (formerly 1B), then Math 53 for multivariable calculus.

**UC Berkeley Math 51**, which most students still know as **Math 1A**, is Berkeley's first semester of calculus for engineering and physical science majors. The math department has renumbered the course from 1A to 51, so you will see both numbers on syllabi, advising pages and prerequisite lists. This guide maps every Math 51 topic to free practice, so you can learn each idea by solving problems with instant feedback.

## What Berkeley Math 51 covers

The Berkeley math department describes Math 51 as an introduction to differential and integral calculus of functions of one variable, with applications and an introduction to transcendental functions. Its official outline has six parts: functions and models, limits and derivatives, differentiation rules, applications of differentiation, integrals, and applications of integrals. About two thirds of the class hours go to derivatives and their applications, with the last few weeks on integration.

That matches a standard **Calculus 1** course from limits through the Fundamental Theorem and basic applications of the integral.

## Math 51 (1A) topic map

| Math 51 topic | Practice | Read |
| --- | --- | --- |
| Functions and models | [Functions and notation](/learn/calculus-1/functions-warmup/function-notation) | [Exponential functions](/functions/exponential-functions), [logarithmic functions](/functions/logarithmic-functions) |
| Limits | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [What is a limit?](/blog/what-is-a-limit), [how to evaluate limits](/blog/how-to-evaluate-limits) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Intermediate Value Theorem](/theorems/intermediate-value-theorem) |
| The derivative | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Differentiation rules | [The product rule](/learn/calculus-1/derivative-rules/product-rule) | [Power rule](/blog/power-rule), [product and quotient rules](/blog/product-and-quotient-rules) |
| Chain rule and implicit differentiation | [Implicit differentiation](/learn/calculus-1/derivative-rules/implicit-differentiation) | [Chain rule guide](/blog/chain-rule), [implicit differentiation](/blog/implicit-differentiation) |
| Transcendental functions | [Exponentials and logarithms](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Derivatives of exponential and log functions](/blog/derivatives-of-exponential-and-logarithmic-functions) |
| Applications of differentiation | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems), [Mean Value Theorem](/theorems/mean-value-theorem) |
| Applications of differentiation: shape of a graph | [Concavity and the second derivative](/learn/calculus-1/applications-of-derivatives/second-derivative) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Integrals | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Substitution | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [U-substitution guide](/blog/u-substitution) |
| Applications of integrals | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves) | [Volumes of revolution](/blog/volumes-of-revolution) |

## Math 51 vs Math 52 vs Math 16A

| Course | Former number | Units | For whom |
| --- | --- | --- | --- |
| Math 51 Calculus I | 1A | 4 | Engineering and physical science majors |
| Math 52 Calculus II | 1B | 4 | Continues Math 51 |
| Math 16A Analytic Geometry and Calculus | 16A | 3 | Students outside the physical sciences and engineering |

Students who take Math 51 continue to the [Math 52 guide](/courses/uc-berkeley-math-52) and then [Math 53](/courses/uc-berkeley-math-53). If your major only needs a short calculus sequence, compare with the [Math 16A guide](/courses/uc-berkeley-math-16a).

## Example problems at the Math 51 level

**1. A limit with a square root.** Rationalize the numerator:

$$\lim_{x\to 0}\frac{\sqrt{x + 4} - 2}{x} = \lim_{x\to 0}\frac{1}{\sqrt{x + 4} + 2} = \frac{1}{4}$$

**2. Implicit differentiation.** For $x^2 + y^2 = 25$, differentiate both sides: $2x + 2y\,y' = 0$, so $y' = -\frac{x}{y}$. At $(3, 4)$ the slope of the tangent line is $-\frac{3}{4}$.

**3. Area between curves.** The curves $y = x$ and $y = x^2$ meet at $x = 0$ and $x = 1$:

$$\int_0^1 (x - x^2)\,dx = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$$

## How to do well in Berkeley Math 51

- **Use discussion sections well.** With three hours of discussion a week, come with problems you attempted and could not finish. Our [limits practice](/learn/calculus-1/limits/limits-practice) and [mixed derivative practice](/learn/calculus-1/derivative-rules/derivative-rules-practice) are good warm-ups.
- **Fix algebra and trig early.** The prerequisites include trigonometry and analytic geometry, and our [functions warm-up](/learn/calculus-1/functions-warmup/key-trig-values) reviews both.
- **Keep the [formula sheet](/formulas) beside you** while practicing, then work without it before exams.
- **Simulate exams** with timed [practice exams](/practice), and read the [common calculus mistakes](/blog/common-calculus-mistakes) list to avoid easy point losses.

## Frequently asked questions

### Is Berkeley Math 51 the same as Math 1A?

Yes. The Berkeley math department lists the course as Math 51, formerly 1A. Its successor, formerly Math 1B, is now Math 52, and older pages and prerequisites may still use the old numbers.

### What does Berkeley Math 51 cover?

It introduces differential and integral calculus of functions of one variable, with applications and transcendental functions. The outline runs through functions and models, limits and derivatives, differentiation rules, applications of differentiation, integrals and applications of integrals.

### What comes after Math 51?

Math 52 (formerly 1B), which covers integration techniques, infinite series and differential equations. After that, Math 53 covers multivariable calculus.

### Is Math 51 (1A) hard?

It is a full first-semester calculus course at a fast pace, aimed at engineering and science majors. Students with solid algebra and trig who practice a little every day are in good shape.

### Math 51 vs Math 16A: which should I take?

Math 51 is intended for majors in engineering and the physical sciences. Math 16A is meant for students outside those fields whose majors only need 16A or the 16A and 16B sequence. There is no credit for 16A after taking 1A.