# UT Austin M 408C Calculus: Topics, Study Guide and Practice

> A study guide for UT Austin M 408C, Differential and Integral Calculus: every topic from limits to the fundamental theorem, the sequence after it and practice.

Source: https://practicecalculus.com/courses/ut-austin-m-408c
Updated: 2026-11-03

## At a glance

- **Course code:** M 408C
- **Hours:** 4 (three lecture hours and two discussion hours a week)
- **Prerequisites:** An appropriate score on the mathematics placement exam, or M 305G with at least a B-
- **Covers:** Limits, continuity, differentiation, the mean value theorem, integration, the fundamental theorem of calculus and transcendental functions
- **Credit overlap:** Only one of M 403K, 408C, 408K, 408N, 408Q and 408R may be counted
- **Next course:** M 408D (Sequences, Series, and Multivariable Calculus)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- M 408C is the first semester of the accelerated calculus sequence at UT Austin, covering both derivatives and integrals in one term.
- It counts for 4 hours and meets for three lecture hours and two discussion hours a week.
- The second semester of the accelerated sequence is M 408D, which covers series and multivariable calculus.
- M 408K covers only the differential half of the same material, so you can count only one of M 408C and M 408K.

**M 408C** is the first course of the accelerated calculus sequence at the University of Texas at Austin. In one semester it takes students from limits all the way to the fundamental theorem of calculus, which is ground that the slower M 408K and M 408L pair spreads over two terms. This guide maps every M 408C topic to free practice, so you can learn each idea by working problems with instant feedback.

## What UT Austin M 408C covers

The UT Austin catalog describes M 408C as an introduction to the theory and applications of differential and integral calculus of one variable. The listed topics are limits, continuity, differentiation, the mean value theorem, integration, the fundamental theorem of calculus and transcendental functions such as exponentials, logarithms and trig functions.

That lines up with a complete standard **Calculus 1** course: derivatives first, then the definite integral and the fundamental theorem. The course meets for three lecture hours and two discussion hours a week, and the next semester, M 408D, moves straight on to series and several variables.

## M 408C topic map

Each row links to a lesson you can practice and a reference page or guide that explains the idea.

| M 408C topic | Practice | Read |
| --- | --- | --- |
| Limits | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [What is a limit?](/blog/what-is-a-limit), [limit laws](/rules/limit-laws) |
| Evaluating limits | [Factor and cancel](/learn/calculus-1/limits/factoring-zero-over-zero) | [How to evaluate limits](/blog/how-to-evaluate-limits) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity](/blog/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 | [Product rule](/learn/calculus-1/derivative-rules/product-rule) | [Product and quotient rules](/blog/product-and-quotient-rules), [chain rule](/rules/chain-rule) |
| 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), [natural logarithm](/functions/natural-logarithm) |
| Trig derivatives | [Sine, cosine and tangent](/learn/calculus-1/derivative-rules/trig-derivatives) | [Derivatives of trig functions](/blog/derivatives-of-trig-functions) |
| The mean value theorem | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean Value Theorem](/theorems/mean-value-theorem), [Rolle's theorem](/theorems/rolles-theorem) |
| Applications of derivatives | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems), [curve sketching](/blog/curve-sketching-with-derivatives) |
| Integration | [Riemann sums](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums), [antiderivatives](/blog/antiderivatives) |
| The fundamental theorem | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Substitution | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [u-substitution](/blog/u-substitution) |

## M 408C and the UT Austin calculus options

UT Austin offers two main routes through single-variable calculus. M 408C is the fast one.

| Course | Title | What it covers |
| --- | --- | --- |
| M 408C | Differential and Integral Calculus | Derivatives and integrals in one semester |
| M 408D | Sequences, Series, and Multivariable Calculus | Second semester of the accelerated sequence |
| M 408K | Differential Calculus | Limits, derivatives and the mean value theorem only |
| M 408L | Integral Calculus | Integration, transcendental functions, sequences and series |
| M 408M | Multivariable Calculus | Functions of several variables |

If you are on the accelerated track, the [M 408D guide](/courses/ut-austin-m-408d) covers what comes next. If you are deciding between tracks, compare the [M 408K guide](/courses/ut-austin-m-408k) and the [M 408L guide](/courses/ut-austin-m-408l), which together cover the same single-variable material as M 408C over two semesters.

## Example problems at the M 408C level

**1. A limit that needs factoring.** Evaluate $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$.

$$\lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} = \lim_{x \to 3} (x + 3) = 6$$

**2. A chain rule derivative with a transcendental function.** Differentiate $f(x) = e^{3x^2}$.

$$f'(x) = e^{3x^2} \cdot 6x = 6x e^{3x^2}$$

**3. The fundamental theorem.** Evaluate $\int_0^{\pi} \sin x\,dx$.

$$\int_0^{\pi} \sin x\,dx = \left[-\cos x\right]_0^{\pi} = -\cos\pi + \cos 0 = 1 + 1 = 2$$

## How to do well in UT Austin M 408C

- **Stay ahead of the pace.** M 408C covers derivatives and integrals in one term, so a short practice session every day keeps you from falling behind. The [Calculus 1 path](/learn/calculus-1) follows the course order.
- **Make derivative rules automatic.** You will use the [power rule](/rules/power-rule) and [chain rule](/rules/chain-rule) on nearly every problem in the second half, so drill them until they need no thought.
- **Watch for algebra slips.** Most lost points come from signs and simplification, not calculus. Read our list of [common calculus mistakes](/blog/common-calculus-mistakes).
- **Keep a formula sheet handy.** The [calculus formula sheet](/formulas) puts every derivative and integral you need on one page.
- **Rehearse exams.** Use [timed practice](/practice) and the guide on [how to prepare for a calculus exam](/blog/how-to-prepare-for-a-calculus-exam) before each test.

> **Tip:** the mean value theorem appears by name in the M 408C description. Learn the picture behind it (a secant slope equals some tangent slope) and the proofs and problems built on it get much easier.

## Frequently asked questions

### What is M 408C at UT Austin?

M 408C, Differential and Integral Calculus, is a one-semester introduction to the calculus of one variable. It covers limits, continuity, derivatives, the mean value theorem, integration, the fundamental theorem of calculus and transcendental functions.

### What comes after M 408C?

The catalog lists M 408C as one of the prerequisites for M 408D, Sequences, Series, and Multivariable Calculus, which is the second semester of the accelerated sequence. M 408C also satisfies the prerequisite for M 408L, Integral Calculus.

### What is the difference between M 408C and M 408K?

M 408K, Differential Calculus, covers limits, continuity, derivatives and the mean value theorem, and integration waits for M 408L. M 408C fits both differential and integral calculus into one semester. Only one of the two may be counted for credit.

### Is M 408C hard?

It covers a full year of calculus topics in a single semester, so the pace is quick. Students who keep up with short daily practice, especially on derivative rules and algebra, find it far more manageable than those who cram.

### How do I prepare for M 408C?

Review functions, exponent rules, logarithms and key trig values, then start on limits and derivatives. The free Calculus 1 path on PracticeCalculus covers the same topics in the same order.