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UT Austin M 408K Differential Calculus: Study Guide and Practice

A study guide for UT Austin M 408K, Differential Calculus: limits, continuity, derivatives and the mean value theorem, with practice for every topic.

UT Austin Updated 3 min read

At a glance

Course code
M 408K
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, and the mean value theorem and its applications
Credit overlap
Only one of M 403K, 408C, 408K, 408N, 408Q and 408R may be counted
Next course
M 408L (Integral Calculus)
Practice here
Calculus 1, units 1 to 5
On this page
  1. What UT Austin M 408K covers
  2. M 408K topic map
  3. M 408K, M 408L and M 408M vs the accelerated track
  4. Example problems at the M 408K level
  5. How to do well in UT Austin M 408K

M 408K is UT Austin's one-semester course in differential calculus. It covers the first half of single-variable calculus, from limits to the mean value theorem, and leaves integration to its follow-up, M 408L. This guide maps each M 408K topic to free practice so you can learn by solving problems with instant feedback.

What UT Austin M 408K covers

The UT Austin catalog describes M 408K as an introduction to the theory and applications of differential calculus of one variable. It names limits, continuity, differentiation, and the mean value theorem and its applications.

In standard terms, M 408K is the derivative half of Calculus 1. It matches the first five units of our Calculus 1 path: the function warm-up, limits, the derivative, derivative rules and applications of derivatives. The integral half of Calculus 1 comes in M 408L.

M 408K topic map

M 408K topic Practice Read
Functions review Functions and notation Exponential functions, logarithmic functions
Limits What is a limit? What is a limit?
One-sided and infinite limits One-sided and infinite limits Limits at infinity, squeeze theorem
Continuity Continuity Continuity, Intermediate Value Theorem
Definition of the derivative The limit definition Limit definition of the derivative
Differentiation rules The power rule Power rule, quotient rule
Chain rule Chain rule: any function Chain rule
Implicit differentiation Implicit differentiation Implicit differentiation
The mean value theorem Tangent lines at work Mean value theorem, Rolle's theorem
Extrema and curve sketching Critical points Curve sketching, Extreme Value Theorem
Optimization and related rates Related rates Related rates, optimization problems
Linear approximation Tangent lines at work Linear approximation

M 408K, M 408L and M 408M vs the accelerated track

Course Title Prerequisite
M 408K Differential Calculus Placement exam or M 305G with a B-
M 408L Integral Calculus M 408C, 408K or 408N with a C-
M 408M Multivariable Calculus M 408L or 408S with a C-
M 408C Differential and Integral Calculus Placement exam or M 305G with a B-
M 408D Sequences, Series, and Multivariable Calculus M 408C, 408L or 408S with a C-

M 408K covers the first part of what M 408C covers. After it, the M 408L guide picks up with integration. If you want the faster route, see the M 408C guide.

Example problems at the M 408K level

1. A limit at infinity. Evaluate \lim_{x \to \infty} \frac{4x^2 + 1}{2x^2 - x} by dividing through by x^2:

\lim_{x \to \infty} \frac{4 + \frac{1}{x^2}}{2 - \frac{1}{x}} = \frac{4}{2} = 2

2. The quotient rule. Differentiate f(x) = \frac{x}{x^2 + 1}.

f'(x) = \frac{(1)(x^2 + 1) - x(2x)}{(x^2 + 1)^2} = \frac{1 - x^2}{(x^2 + 1)^2}

3. The mean value theorem. For f(x) = x^2 on [1, 3], the average slope is \frac{9 - 1}{3 - 1} = 4. Setting f'(c) = 2c = 4 gives c = 2, which lies inside (1, 3) as the theorem promises.

How to do well in UT Austin M 408K

  • Do not rush limits. Factoring, rationalizing and dividing by the highest power come back in every later topic. Practice them in the limits unit.
  • Drill the rules until they are automatic. The chain rule causes the most lost points, so practice it in short, frequent sessions.
  • Draw the picture for the mean value theorem. Sketch the secant line and find where a tangent is parallel to it before you compute.
  • Review the usual traps. Our list of common calculus mistakes covers sign errors and misused rules.
  • Keep the formulas in one place. Use the formula sheet and test yourself with timed practice before exams.

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What is a limit?

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Frequently asked questions

What is M 408K at UT Austin?
M 408K, Differential Calculus, introduces the theory and applications of differential calculus of one variable. Its catalog topics are limits, continuity, differentiation, and the mean value theorem with its applications.

What comes after M 408K?
M 408K is one of the listed prerequisites for M 408L, Integral Calculus, which covers integration, the fundamental theorem, transcendental functions, sequences and series. After M 408L, M 408M covers multivariable calculus.

Should I take M 408K or M 408C?
M 408C covers differential and integral calculus in one semester, while M 408K covers only the differential half and leaves integration to M 408L. Only one of M 408K and M 408C counts for credit, so the choice depends on how fast you want to move; talk to your adviser about which fits your degree plan.

Is M 408K hard?
It covers less ground than M 408C, which gives you more time for each topic. The parts students usually find tricky are limits that need algebra and word problems such as optimization, and steady practice on both helps most.

How do I prepare for M 408K?
Make sure function notation, exponents, logarithms and the unit circle feel comfortable. The free functions warm-up and limits units on PracticeCalculus are a good starting point.

Written by the PracticeCalculus team

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Related university courses

UT Austin M 408L Calculus 1 and 2 Integral Calculus A study guide for UT Austin M 408L, Integral Calculus: integration, the fundamental theorem, transcendental functions, sequences and series, with practice. UT Austin M 408C Calculus 1 Differential and Integral Calculus 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. UT Austin M 408M Calculus 3 Multivariable Calculus A study guide for UT Austin M 408M, Multivariable Calculus: vectors, parametric and polar curves, partial derivatives, gradients and multiple integrals.