5 Steps to a 5 AP Calculus BC 2019

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

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 7.1-1.)

(c, f (c))

f

f′ (c)=0

y

ac b

x

0

Figure 7.1-1

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 7.1-2.)

(c,f(c))

(a,f(a))

(b,f(b))

c x

y

f

0

f′(c) = f(b) – f(a)

b – a

Figure 7.1-2