ap calc study guides

๐Ÿ‘‘ย  Unit 1: Limits & Continuity

๐Ÿถย  Unit 8: Applications of Integration

โ™พย  Unit 10: Infinite Sequences and Series (BC Only)

5.1 Using the Mean Value Theorem

#meanvaluetheorem

#rolle'stheorem

โฑ๏ธย ย 2 min read

written by

Sumi Vora

sumi vora


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, letโ€™s break this down:ย ๐Ÿ”

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.ย 

The second condition states that ย f must be differentiable on (a, b). Note that this time, itโ€™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)ย 

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.ย ๐Ÿ˜

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. Iโ€™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.ย โ›ฐ

Rolle's Theorem

From the Mean Value Theorem, we can derive Rolleโ€™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. โœ”

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(513).png?alt=media&token=f581e206-ec59-4fef-adce-0108aadd865d

continue learning

Slide 1 of 10
Fiveable

Join Our Community

Fiveable Community students are already meeting new friends, starting study groups, and sharing tons of opportunities for other high schoolers. Soon the Fiveable Community will be on a totally new platform where you can share, save, and organize your learning links and lead study groups among other students!๐ŸŽ‰

Fiveable Logo

2550 north lake drive
suite 2
milwaukee, wi 53211

92% of Fiveable students earned a 3 or higher on their 2020 AP Exams.

*apยฎ and advanced placementยฎ are registered trademarks of the college board, which was not involved in the production of, and does not endorse, this product.

ยฉ fiveable 2020 | all rights reserved.