Section 5.5 The Intermediate Value Theorem Rolle’s Theorem The Mean Value Theorem 3.6.
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Transcript of Section 5.5 The Intermediate Value Theorem Rolle’s Theorem The Mean Value Theorem 3.6.
Section 5.5The Intermediate Value Theorem
Rolle’s TheoremThe Mean Value Theorem
3.6
Intermediate Value Theorem (IVT)
If f is continuous on [a, b] and N is a value between f(a) and f(b), then there is at least one point c between a and b where f takes on the value N.
a
f(a)
b
f(b)N
c
Rolle’s Theorem
If f is continuous on [a, b], if f(a) = 0, f(b) = 0, then there is at least one number c on (a, b) where f ‘ (c ) = 0
a b
slope = 0
c
f ‘ (c ) = 0
Given the curve:
x
f(x)
x+h
f(x+h)
sec
f x h f xm
h
tan
h 0
f x h f xm lim
h
The Mean Value Theorem (MVT)aka the ‘crooked’ Rolle’s Theorem
If f is continuous on [a, b] and differentiable on (a, b)There is at least one number c on (a, b) at which
f b f af ' c
b a
ab
f(a)
f(b)
c
Conclusion:Slope of Secant Line
EqualsSlope of Tangent Line
2 f b f aIf f x x 2x 1, a 0, b 1, and f ' c , find c.
b a
f(0) = -1 f(1) = 2
f b f a 2 13
b a 1 0
f ' x 2x 2
3 2x 2 1
x2
Find the value(s) of c which satisfy Rolle’s Theorem for on the interval [0, 1]. 4f x x x
Verify…..f(0) = 0 – 0 = 0 f(1) = 1 – 1 = 0
3f ' x 4x 1 30 4x 1
31
c4
which is on [0, 1]
Find the value(s) of c that satisfy the Mean Value Theorem for
1f x x on 4, 4
x
1 17 1 17f 4 4 f 4 4
4 4 4 4
17 17f b f a 174 4
b a 4 4 16
2
17 1f ' c 1
16 x
Find the value(s) of c that satisfy the Mean Value Theorem for
1f x x on 4, 4
x
Note: The Mean Value Theorem requires the function to be continuous on [-4, 4] and differentiable on (-4, 4). Therefore, sincef(x) is discontinuous at x = 0 which is on [-4, 4], there may be no value of c which satisfies the Mean Value Theorem
Since has no real solution, there is no value of c on
[-4, 4] which satisfies the Mean Value Theorem
2
1 1
16 x
Given the graph of f(x) below, use the graph of f to estimate thenumbers on [0, 3.5] which satisfy the conclusion of the Mean ValueTheorem.
2Determine whether f x x 2x 2 satisfies the hypothesis of
the Mean Value Theorem on -2, 2 . If it does, find all numbers
f b f ac in (a, b) such that f ' c
b a
f(x) is continuous and differentiable on [-2, 2]
f 2 f 2 6 2
2 2 42
f ' x 2x 2
2x 2 2 c 0
On the interval [-2, 2], c = 0 satisfies the conclusion of MVT
2x 1Determine whether f x satisfies the hypothesis of
x 2the Mean Value Theorem on -2, 1 . If it does, find all numbers
f b f ac in (a, b) such that f ' c
b a
f(x) is continuous and differentiable on [-2, 1]
30
f 1 f 2 41 2 3
1
4
2
2
2x x 2 1 x 1
xf ' x
2
2
2
x 4x 1 1
x 4x 4 4
2 24x 16x 4 x 4x 4 23x 12x 0
3x x 4 0 On the interval [-2, 1], c = 0 satisfies the conclusion of MVT
2x 1Determine whether f x satisfies the hypothesis of
x 2the Mean Value Theorem on 0, 4 . If it does, find all numbers
f b f ac in (a, b) such that f ' c
b a
Since f(x) is discontinuous at x = 2, which is part of the interval[0, 4], the Mean Value Theorem does not apply
3Determine whether f x x 3x 1 satisfies the hypothesis of
the Mean Value Theorem on -1, 2 . If it does, find all numbers
f b f ac in (a, b) such that f ' c
b a
f(x) is continuous and differentiable on [-1, 2]
f0
2 f 1 3 3
2 1 3
23x 3f x' 23x 3 0
c = 1 satisfies the conclusion of MVT
3 x 1 x 1 0
f(3) = 39 f(-2) = 64
f b f a 64 395
b a 2 3
For how many value(s) of c is f ‘ (c ) = -5?
If , how many numbers on [-2, 3] satisfythe conclusion of the Mean Value Theorem.
2 2f x x 12 x 4
A. 0 B. 1 C. 2 D. 3 E. 4
CALCULATOR REQUIRED
X X X