Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Calculus I
Topic: Analysis of Curves
Subtopic: Mean Value Theorem for Derivatives

Overview

This lesson covers two theorems, the Mean Value Theorem for Derivatives (MVT) and Rolle's Theorem. Rolle's Theorem is a special case of the MVT. Both theorems require continuous differentiable functions and show that there is a point at which the tangent line has a specific given slope. These theorems are primarily used to prove other (more practical) calculus theorems.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 3.2.1 Understand the hypothesis and conclusion of Rolle's Theorem
• 3.2.2. Apply Rolle's Theorem to show that there exists a c-value where the tangent line is horizontal
• 3.2.3 Understand the hypothesis and conclusion of the Mean Value Theorem for Derivatives
• 3.2.4. Apply the Mean Value Theorem to show that there exists a c-value where the tangent line is of the required slope
• 3.2.5 Understand Rolle's Theorem and the Mean Value Theorem from both an algebraic and graphical perspective

Terminology

Terms you should be able to define: hypothesis, conclusion, imply (i.e. mathematical implication), theorem, proof, existence theorem

Mini-Lectures and Examples

Supplementary Resources (optional)

rev. 2020-10-31