AP CALCULUS AB
OBJECTIVE: UNDERSTAND AND USE
ROLLE’S THEOREM AND THE MEAN
VALUE THEOREM
3.2 Rolle’s Theorem and Mean Value Theorem
Rolle’s Theorem



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
French Mathematician Michel Rolle (1652-1719)
Gives conditions that guarantee the existence of an
extreme value in the interior of a closed interval.
Let f(x) be continuous on the closed interval [a, b]
and differentiable on the open interval (a, b).
If 𝑓 𝑎 = 𝑓(𝑏), then there is at least one number, c,
in (a, b) such that 𝑓 ′ 𝑐 = 0.
Graphs on Pg. 172
Example of Rolle’s Theorem

Find (2) x-intercepts of 𝑓 𝑥 = 𝑥 2 − 3𝑥 + 2 and
show 𝑓 ′ 𝑥 = 0 at some point between the 2intercepts
𝑥 2 − 3𝑥 + 2 = 0
𝑥 = 1, 2
𝑓 ′ 𝑥 = 2𝑥 − 3 = 0
3
𝑥=
2
MEAN VALUE THEOREM

If f(x)is continuous on the closed interval [a, b] and
differentiable on the open interval (a, b), then there
exists a number, c, in (a, b) such that:
𝑓 𝑏 − 𝑓(𝑎)
′
𝑓 𝑐 =
𝑏−𝑎

Slope of the tangent = Slope of the secant

Instantaneous Rate of Change = Average Rate of Change
Example of the MVT

Find the x-value that satisfies the mean value
theorem
4
𝑓 𝑥 =5−
𝑜𝑛 [1, 4]
𝑥
𝑓 𝑏 − 𝑓(𝑎)
𝑓 𝑥 =
𝑏−𝑎
′
4𝑥 −2
4
4
5− − 5−
4
1
=
4−1
Example of the MVT
4
5−1 − 5−4
=
2
𝑥
4−1
4
4−1 3
=
= =1
2
𝑥
3
3
4 = 𝑥2
𝑥 = ±2
𝑥 = 2 𝑖𝑛 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 [1, 4]
Example of MVT


Find the x-value that satisfies the mean value
theorem.
Sketch a graph of the secant and tangent lines
along with f(x).
𝑓 𝑥 = 𝑥2 + 1
𝑜𝑛 [1, 5]
𝑓 𝑏 − 𝑓(𝑎)
𝑓 𝑥 =
𝑏−𝑎
′
Example of MVT
52 + 1 − 12 + 1
2𝑥 =
5−1
26 − 2 24
2𝑥 =
=
=6
4
4
𝑥=3
#46 Hints
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What to do?????????????

𝑓 𝑥 = 2 sin 𝑥 + sin 2𝑥
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



′
𝑓 𝑐 =
𝑓 𝑏 −𝑓(𝑎)
𝑏−𝑎
𝑜𝑛 [0, 𝜋]
=0
𝑓 ′ 𝑥 = 2 cos 𝑥 + 2 cos 2𝑥 = 0
Divide by 2: 𝑓 ′ 𝑥 = cos 𝑥 + cos 2𝑥 = 0
Use Substitution (Dbl Angle Formula):
cos 2𝑥 = 2𝑐𝑜𝑠 2 𝑥 − 1
Factor: 2𝑐𝑜𝑠 2 𝑥 + cos 𝑥 − 1 = 0
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Formative Assessment

Pg. 177 (39-46, 48, 49)