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3.1b Notes on MVT and Rolle’s
Mean Value Theorem:
The Mean Value Theorem (MVT) states that if f ( x ) is _____________________ at every point of the
closed interval [a, b] and ____________________ at every point of its interior (a, b), then there is at
least one number c between a and b at which
f (b)  f (a)
 f '(c)
ba
Think about it graphically.
Another way to think about it:
The difference quotient
f (b)  f (a)
is the ________________________________ in f ( x ) over [a, b].
ba
The derivative f '(c ) is the _______________________________________________.
Examples:
1. f ( x)  x 2 on [0, 2]
Step 1: Check if it is continuous on [0, 2]
Step 5: Set last two steps equal and solve for c.
Step 2: Check if it is differentiable on (0, 2)
Step 3: Find f '(c)
Step 4: Find
f (2)  f (0)
20
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2. f ( x) 
3. f ( x) 
x 2  1 on [-1, 1]
3
on [-1, 1]
x2
 x3  3, x  1
4. f ( x)   2
on [-1, 1]
 x  1, x  1
5. A car drives 408 miles in 6 hours. The maximum speed limit on the trip was 65 mph. Show that the car
must have been speeding at some point on the trip.
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Rolle’s Theorem:
This is a special type of MVT.
If f ( x ) is ____________________________ at every point on the closed interval [a, b] and
______________________ at every point of its interior (a, b), and if f (a)  f (b)  0 , then there is at
least one point c in (a, b) such that f '(c)  0 .
Examples:
1. f ( x)   x 2  4 [-2, 2]
Step 1: Check if it is continuous on [-2, 2]
Step 2: Check if it is differentiable on [-2, 2]
Step 3: Find f '(c)
Step 4: Set f '(c)  0 and solve for c.
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2. f ( x) 
1
[2, 3]
x 1
3. f ( x)  x3  4 x 2  3x [1, 3]
Homework: p. 233: 35-41odd, 47-55odd
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