📚

All Subjects

 > 

♾️ 

AP Calc

 > 

✨

Unit 5

5.1 Using the Mean Value Theorem

2 min read•june 8, 2020

sumi-vora

Sumi Vora


AP Calculus AB/BC ♾️

Bookmarked 7.6k • 259 resources
See Units

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(505).png?alt=media&token=024c0138-6105-4a3c-80a1-72fdb31ef8ac

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. 😁
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(509).png?alt=media&token=84ac6dbb-5ca9-4dca-a10c-85fdce40c455
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(510).png?alt=media&token=a85cf0db-9a8c-4d94-92c9-19d04692d3cc
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(511).png?alt=media&token=c1828ddd-f0d0-4083-a983-9c6a0dc440f4
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. ✔
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(513).png?alt=media&token=f581e206-ec59-4fef-adce-0108aadd865d

Was this guide helpful?

Join us on Discord
Thousands of students are studying with us for the AP Calculus AB/BC exam.
join now
Hours Logo
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
Sign up now for instant access to 2 amazing downloads to help you get a 5
Join us on Discord
Thousands of students are studying with us for the AP Calculus AB/BC exam.
join now
💪🏽 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
Practice your typing skills while reading Using the Mean Value Theorem
Start Game
📱 Stressed or struggling and need to talk to someone?
Talk to a trained counselor for free. It's 100% anonymous.
Text FIVEABLE to 741741 to get started.
© 2021 Fiveable, Inc.