---
title: Calculus Theorems
description: Calculus theorems provide the theoretical foundation for limits, derivatives, and integrals. You will understand key principles like the Mean Value Theorem and the Fundamental Theorem of Calculus, enabling you to rigorously prove mathematical relationships and solve complex analytical problems.
category: mathematics
subcategory: calculus
difficulty: beginner, intermediate, advanced
url: /subject/calculus-theorems
---

# Calculus Theorems

Calculus theorems provide the theoretical foundation for limits, derivatives, and integrals. You will understand key principles like the Mean Value Theorem and the Fundamental Theorem of Calculus, enabling you to rigorously prove mathematical relationships and solve complex analytical problems.

## Available Resources

2 Courses • 4 Websites

## Websites

### 1. ProofWiki - Calculus

Community-edited index of formal proofs for calculus results, from limits and the Mean Value Theorem to the Fundamental Theorem of Calculus. Each page states a theorem precisely and proves it step by step, linking every definition and lemma used.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://proofwiki.org/wiki/Category:Calculus

**Tags:** calculus, proofs, theorems, real-analysis, reference

### 2. Paul's Online Math Notes

Paul Dawkins' Calculus I lecture notes from Lamar University, covering limits, derivatives, applications and integrals. Worked examples and practice problems accompany each section, and an extras chapter proves the main limit, derivative and integral theorems used in a first course.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://tutorial.math.lamar.edu/classes/calci/calci.aspx

**Tags:** calculus, limits, derivatives, integrals, lecture-notes

### 3. Khan Academy - Calculus Applications

Khan Academy's Calculus 1 course covers derivatives, integrals, and their applications to optimization, related rates, and area/volume problems, with practice exercises and video explanations for each topic.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://www.khanacademy.org/math/calculus-1

**Tags:** calculus, derivatives, integrals, optimization

### 4. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Courses

### 1. Multivariable Calculus (MIT 18.02SC)

**Author:** Denis Auroux

Differential, integral and vector calculus in several variables: vectors and matrices, partial derivatives, optimization, double and triple integrals, line integrals, and the theorems of Green, Stokes and Gauss. Includes 106 lecture and recitation videos, notes, and problems and exams with solutions, built for independent study.

**Difficulty:** Intermediate | **Price:** Free

**Link:** https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/

**Tags:** multivariable-calculus, partial-derivatives, multiple-integrals, vector-calculus, line-integrals

### 2. Calculus for Engineers

**Author:** Jeffrey R. Chasnov

Hong Kong University of Science and Technology course presenting university calculus for engineering and science students in 59 short lectures. Reviews precalculus, then covers derivatives, integrals, Taylor series and applications, leaving learners able to set up and solve standard single-variable calculus problems.

**Difficulty:** Beginner | **Language:** English | **Price:** Free

**Link:** https://www.coursera.org/learn/calculus-for-engineers

**Tags:** calculus, derivatives, integrals, taylor-series, engineering-mathematics

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