📚

All Subjects

>

♾️

AP Calc

>

Unit 5

# 5.1 Using the Mean Value Theorem Sumi Vora

### AP Calculus AB/BC♾️

Bookmarked 7.6k • 259 resources
See Units ## Okay, let’s break this down: 🔍

The first condition states that f must be continuous on the closed interval [a, b]. From our first unit, we know that that simply means that there are no holes, asymptotes, or jump discontinuities in the graph between points a and b. Because they are closed brackets, the graph must be continuous at the points a and b.
The second condition states that  f must be differentiable on (a, b). Note that this time, it’s an open interval. An equation is differentiable at a if it is continuous at a and if lim x->a [f(x) - f(a)]/(x - a) exists. Most continuous equations are differentiable unless they have a corner (as in an absolute value function)
If we meet these two conditions, then we can conclude that there exists a point c on (a, b) such that f'(c) = [f(b)-f(a)]/b-a. In other words, there is a point where the slope of the tangent line is equivalent to the slope of the secant line between a and b. 😁   Note: [f(b)-f(a)]/(b-a) is equivalent to the slope of the line that connects points a and b (recognize the Point-Slope formula from Algebra I?) which is known as the secant line between a and b. The slope of the secant line is also the average rate of change between two points, while the derivative is the instantaneous rate of change or the slope of the tangent line at one point. I’ll use them interchangeably so that you can start to recognize them in context since the College Board likes to switch up the vocabulary to confuse you. ⛰

## Rolle's Theorem

From the Mean Value Theorem, we can derive Rolle’s Theorem, which simply states that if f(a) and f(b) are equal to each other, then there will be some point on the graph where the slope of the tangent line is equal to 0. ✔ Thousands of students are studying with us for the AP Calculus AB/BC exam.
#####   Studying with Hours = the ultimate focus mode
Start a free study session
##### 🔍 Are you ready for college apps?
Take this quiz and find out!
Start Quiz
Browse Study Guides By Unit
📆Big Reviews: Finals & Exam Prep
✍️Free Response Questions (FRQ)
🧐Multiple Choice Questions (MCQ)
Unit 10: Infinite Sequences and Series (BC Only)
👑Unit 1: Limits & Continuity
🤓Unit 2: Differentiation: Definition & Fundamental Properties
🤙🏽Unit 3: Differentiation: Composite, Implicit & Inverse Functions
👀Unit 4: Contextual Applications of the Differentiation
Unit 5: Analytical Applications of Differentiation
🔥Unit 6: Integration and Accumulation of Change
💎Unit 7: Differential Equations
🐶Unit 8: Applications of Integration
🦖Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)
#####  FREE AP calc Survival Pack + Cram Chart PDF
Thousands of students are studying with us for the AP Calculus AB/BC exam.
##### 💪🏽 Are you ready for the Calc AB exam?
Take this quiz for a progress check on what you’ve learned this year and get a personalized study plan to grab that 5!
START QUIZ
##### 💪🏽 Are you ready for the Calc BC exam?
Take this quiz for a progress check on what you’ve learned this year and get a personalized study plan to grab that 5!
START QUIZ
##### Play this on HyperTyper  