# Berkeley Math 16A Analytic Geometry and Calculus: Study Guide

> A study guide for UC Berkeley Math 16A: derivatives, optimization, exponential and log functions, and integrals with business uses, mapped to free practice.

Source: https://practicecalculus.com/courses/uc-berkeley-math-16a
Updated: 2026-11-03

## At a glance

- **Course code:** Math 16A
- **Units:** 3 (three hours of lecture and one and a half of discussion per week)
- **Offered:** Fall and spring
- **Prerequisites:** Three years of high school math including trigonometry, plus a satisfactory placement result (an AP test, the UC/CSU diagnostic exam or similar)
- **Covers:** Linear, quadratic, exponential and log functions, limits and continuity, derivative rules, curve sketching, optimization, related rates, antiderivatives, substitution, the definite integral and consumer surplus
- **Next course:** Math 16B (Analytic Geometry and Calculus)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- Math 16A is Berkeley's one-variable calculus course for students outside the physical sciences and engineering.
- It covers derivatives, maxima and minima, exponential and log functions, and definite integrals with applications.
- Applications lean toward economics: supply and demand, revenue, profit, cost and consumer surplus.
- Math 16B completes the sequence, and there is no credit for 16A after taking Math 1A (now 51).

**UC Berkeley Math 16A** is the first half of Berkeley's applied calculus sequence, designed for students whose majors are outside the physical sciences and engineering. It covers the core ideas of single-variable calculus with an emphasis on applications, especially in economics and business. Students often search for it as Berkeley's business calculus. This guide maps every Math 16A topic to free practice with instant feedback.

## What Berkeley Math 16A covers

The Berkeley math department describes Math 16A as calculus of one variable: derivatives, definite integrals and their applications, maxima and minima, and applications of exponential and logarithmic functions. The official outline starts with linear functions, including supply and demand, revenue, profit and cost, then nonlinear functions such as quadratics, exponentials, logarithms and compound interest, plus a brief definition of the trig functions. It then develops the derivative through limits, continuity and rates of change, the product, quotient and chain rules, and derivatives of exponential, log and trig functions. Graphs and derivatives, optimization, implicit differentiation and related rates follow, and the course ends with integration: antiderivatives, substitution, the definite integral, the Fundamental Theorem, area between curves and consumer surplus.

That is a standard applied **Calculus 1** course, sometimes called business calculus.

## Math 16A topic map

| Math 16A topic | Practice | Read |
| --- | --- | --- |
| Linear functions and rates | [Slope and rate of change](/learn/calculus-1/functions-warmup/slope-and-rate-of-change) | [Linear function](/functions/linear-function) |
| Quadratic, exponential and log functions | [Exponentials and logarithms](/learn/calculus-1/functions-warmup/exponentials-and-logs) | [Quadratic function](/functions/quadratic-function), [exponential functions](/functions/exponential-functions) |
| Limits and continuity | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [What is a limit?](/blog/what-is-a-limit), [continuity](/blog/continuity) |
| Rate of change and the derivative | [Instantaneous rate of change](/learn/calculus-1/the-derivative/instantaneous-rate-of-change) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Product, quotient and chain rules | [The quotient rule](/learn/calculus-1/derivative-rules/quotient-rule) | [Product and quotient rules](/blog/product-and-quotient-rules), [chain rule](/blog/chain-rule) |
| Exponential, log and trig derivatives | [Chain rule: e and ln](/learn/calculus-1/derivative-rules/chain-rule-exponentials) | [Derivatives of exponential and log functions](/blog/derivatives-of-exponential-and-logarithmic-functions) |
| Graphs, extrema and concavity | [Increasing and decreasing](/learn/calculus-1/applications-of-derivatives/increasing-and-decreasing) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Optimization and economics | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems) |
| Implicit differentiation and related rates | [Related rates](/learn/calculus-1/applications-of-derivatives/related-rates) | [Related rates guide](/blog/related-rates) |
| Antiderivatives and substitution | [The reverse power rule](/learn/calculus-1/integrals/reverse-power-rule) | [Antiderivatives](/blog/antiderivatives), [u-substitution](/blog/u-substitution) |
| Definite integral and Fundamental Theorem | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Area between curves and consumer surplus | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves) | [Area between curves guide](/blog/area-between-curves) |

## Math 16A vs Math 16B vs Math 51

| Course | Units | For whom | Content |
| --- | --- | --- | --- |
| Math 16A | 3 | Majors outside physical sciences and engineering | One-variable calculus with applications |
| Math 16B | 3 | Continues 16A | Differential equations, probability and calculus, series, multivariable calculus |
| Math 51 (formerly 1A) | 4 | Engineering and physical science majors | One-variable calculus with transcendental functions |

There is no credit for 16A after Math 1A. If you might later need multivariable calculus for your major, check whether you should take [Math 51](/courses/uc-berkeley-math-51) instead.

## Example problems at the Math 16A level

**1. Marginal profit.** If profit is $P(x) = -2x^2 + 80x - 300$ dollars for $x$ units, then $P'(x) = -4x + 80$, which is zero at $x = 20$. Profit is greatest at 20 units, where $P(20) = 500$ dollars.

**2. Continuous compounding.** The value of 1000 dollars at 5 percent compounded continuously is $A(t) = 1000e^{0.05t}$, so $A'(t) = 50e^{0.05t}$, and the account grows at 50 dollars per year at $t = 0$.

**3. Consumer surplus.** With demand $p = 100 - x$ and market price 60, the quantity sold is $x = 40$, and

$$CS = \int_0^{40}\big((100 - x) - 60\big)\,dx = 40x - \frac{x^2}{2}\Big|_0^{40} = 800$$

## How to do well in Berkeley Math 16A

- **Translate every word problem into a function first.** Revenue, cost and profit questions are mostly about setting up the right function, which our [function notation lesson](/learn/calculus-1/functions-warmup/function-notation) practices.
- **Know exponentials and logs cold.** Compound interest and growth models appear throughout, so review the [natural logarithm](/functions/natural-logarithm) and [exponent rules](/learn/calculus-1/functions-warmup/exponent-rules).
- **Check units in every answer.** A derivative of profit is in dollars per unit, which helps catch setup errors.
- **Use the [formula sheet](/formulas)**, timed [practice exams](/practice) and the [common calculus mistakes](/blog/common-calculus-mistakes) list as you prepare.

## Frequently asked questions

### What is Berkeley Math 16A?

Math 16A, Analytic Geometry and Calculus, is a 3-unit course in one-variable calculus. It covers derivatives, definite integrals and their applications, maxima and minima, and applications of exponential and logarithmic functions.

### Who should take Math 16A instead of Math 51 (1A)?

Math 16A is intended for students outside the physical sciences and engineering whose major is satisfied by 16A, or by 16A and 16B, and who do not plan further math. Engineering and physical science majors take Math 51, formerly 1A.

### What comes after Math 16A?

Math 16B, which completes the sequence. Its topics include differential equations, probability and calculus, sequences and series, and multivariable calculus.

### Is Math 16A easier than Math 1A?

They are aimed at different students. Math 16A is a 3-unit course built around applied examples, many from business and economics, while Math 1A (now Math 51) is a 4-unit course for engineering and physical science majors that includes an introduction to transcendental functions. Either one needs regular practice.