section 5.5 the intermediate value theorem rolle’s theorem the mean value theorem 3.6
TRANSCRIPT
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