5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 17:21

138 STEP 4. Review the Knowledge You Need to Score High

8.1 Rolle’s Theorem, Mean Value Theorem, and Extreme

Value Theorem

Main Concepts:Rolle’s Theorem, Mean Value Theorem, Extreme Value Theorem

TIP • Set your calculator to Radians and change it to Degrees if/when you need to. Do not

forget to change it back to Radians after you have finished using it in Degrees.

Rolle’s Theorem

Iff is a function that satisfies the following three conditions:

1. fis continuous on a closed interval [a,b]


2. fis differentiable on the open interval (a,b)


3. f(a)= f(b)= 0

then there exists a numbercin (a,b) such thatf′(c)=0. (See Figure 8.1-1.)

(c, f(c))

f

f′(c) = 0

y

ac b

x

0

Figure 8.1-1

Note that if you change condition 3 fromf(a)=f(b)=0tof(a)= f(b), the conclusion

of Rolle’s Theorem is still valid.

Mean Value Theorem

Iff is a function that satisfies the following conditions:

1. fis continuous on a closed interval [a,b]


2. fis differentiable on the open interval (a,b)
   then there exists a numbercin (a,b) such thatf′(c)=
   f(b)−f(a)
   b−a
   . (See Figure 8.1-2.)