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

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 and the UT Austin calculus options

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

If you are on the accelerated track, the M 408D guide covers what comes next. If you are deciding between tracks, compare the M 408K guide and the M 408L guide, 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}.

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

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

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 follows the course order.
• Make derivative rules automatic. You will use the power rule and 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.
• Keep a formula sheet handy. The calculus formula sheet puts every derivative and integral you need on one page.
• Rehearse exams. Use timed practice and the guide on 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.