Lesson 4-2

Mean Value Theorem and Rolle’s Theorem

Quiz• Homework Problem: Related Rates 3-10

Gravel is being dumped from a conveyor belt at a rate of 30 ft³/min, and forms a pile in shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10ft high?

• Reading questions: – What were the names of the two theorems in section

4.2?– What pre-conditions (hypotheses) do the Theorems

have in common?

Objectives

• Understand Rolle’s Theorem

• Understand Mean Value Theorem

Vocabulary

• Existence Theorem – a theorem that guarantees that there exists a number with a certain property, but it doesn’t tell us how to find it.

Theorems

Mean Value Theorem: Let f be a function that is a) continuous on the closed interval [a,b]b) differentiable on the open interval (a,b)

then there is a number c in (a,b) such that

f(b) – f(a)f’(c) = --------------- or equivalently: f(b) – f(a) = f’(c)(b – a) b – a

 Rolle’s Theorem: Let f be a function that is

a) continuous on the closed interval [a,b]b) differentiable on the open interval (a,b)c) f(a) = f(b)

then there is a number c in (a,b) such that f’(c) = 0

Mean Value Theorem (MVT)

For a differentiable function f(x), the slope of the secant line through (a, f(a)) and (b, f(b)) equals the slope of the tangent line at some point c between a and b. In other words, the average rate of change of f(x) over [a,b] equals the instantaneous rate of change at some point c in (a,b).

P(c,f(c))

a

A(a,f(a))

b

P1

P2

a

y

b

B(b,f(b))

c2c1x

y

xc

Let f be a function that is a) continuous on the closed interval [a,b]b) differentiable on the open interval (a,b)

then there is a number c in (a,b) such that f(b) – f(a)f’(c) = --------------- (instantaneous rate of change, mtangent = average rate of change, msecant) b – a

or equivalently: f(b) – f(a) = f’(c)(b – a)

Example 1Verify that the mean value theorem (MVT) holds for f(x) = -x² + 6x – 6 on [1,3].

a) continuous on the closed interval [a,b] polynomialb) differentiable on the open interval (a,b) polynomial

f(b) – f(a) = f(3) – f(1) = 3 – (-1) = 4

f(b) – f(a) / (b – a) = 4/2 = 2

f’(x) = 2 = 6 – 2x so -4 = -2x 2 = x

f’(x) = -2x + 6

Example 2Find the number that satisfies the MVT on the given interval or state why the theorem does not apply.

f(x) = x2/5 on [0,32]

a) continuous on the closed interval [a,b] okb) differentiable on the open interval (a,b) ok on open

f(b) – f(a) = f(32) – f(0) = 4 – 0 = 4

f(b) – f(a) / (b – a) = 4/32 = 0.125

f’(x) = 0.125 = (2/5)x-3/5 so x = 6.94891

f’(x) = (2/5)x-3/5

Example 3Find the number that satisfies the MVT on the given interval or state why the theorem does not apply.

f(x) = x + (1/x) on [1,3]

f’(x) = 1 – x-2

f(b) – f(a) = f(3) – f(1) = 10/3 – 2 = 4/3

f(b) – f(a) / (b – a) = (7/3)/2 = 2/3

f’(x) = 2/3 = 1 – x-2 so x-2 = 1/3 x² = 3 x=3 = 1.732

a) continuous on the closed interval [a,b] okb) differentiable on the open interval (a,b) ok on open

Example 4Find the number that satisfies the MVT on the given interval or state why the theorem does not apply.

f(x) = x1/2 + 2(x – 3)1/3 on [1,9]

f’(x) = 1/2x-1/2 + 2/3(x – 3)-2/3

f’(x) undefined at x = 3 (vertical tangent) MVT does not apply

a) continuous on the closed interval [a,b] okb) differentiable on the open interval (a,b) vertical tan

Rolle’s TheoremLet f be a function that is

a) continuous on the closed interval [a,b]b) differentiable on the open interval (a,b)c) f(a) = f(b)

then there is a number c in (a,b) such that f’(c) = 0

a b

y

xc a b

y

xc a b

y

xc

A. B. C. D.

Case 1: f(x) = k (constant) [picture A]

Case 2: f(x) > f(a) for some x in (a,b) [picture B or C] Extreme value theorem guarantees a max value somewhere in [a,b]. Since f(a) = f(b), then at some c in (a,b) this max must occur. Fermat’s Theorem says that f’(c) =0.

Case 3: f(x) < f(a) for some x in (a,b) [picture C or D] Extreme value theorem guarantees a min value somewhere in [a,b]. Since f(a) = f(b), then at some c in (a,b) this min must occur. Fermat’s Theorem says that f’(c) =0.

a b

y

xc1 c2

Example 5Determine whether Rolle’s Theorem’s hypotheses are satisfied &, if so, find a number c for which f’(c) = 0. f(x) = x² + 9 on [-3,3]

f’(x) = 2x

0 in the interval [-3,3]

f’(x) = 0 when x = 0

a) continuous on the closed interval [a,b] polynomialb) differentiable on the open interval (a,b) polynomialc) f(a) = f(b) f(-3) = 18 f(3) = 18

Example 6Determine whether Rolle’s Theorem’s hypotheses are satisfied &, if so, find a number c for which f’(c) = 0. f(x) = x³ - 2x² - x + 2 on [-1,2]

f’(x) = 3x² - 4x – 1

-0.2153 and 1.5486 in the interval [-1,2]

f’(x) = 0 when x = (7 + 2)/3 = 1.5486

a) continuous on the closed interval [a,b] polynomialb) differentiable on the open interval (a,b) polynomialc) f(a) = f(b) f(-1) = 0 f(2) = 0

f’(x) = 0 when x = -(7 - 2)/3 = -0.2153

Example 7Determine whether Rolle’s Theorem’s hypotheses are satisfied &, if so, find a number c for which f’(c) = 0. f(x) = (x² - 1) / x on [-1,1]

a) continuous on the closed interval [a,b] no at x = 0b) differentiable on the open interval (a,b) no at x = 0c) f(a) = f(b) f(-1) = 0 f(1) = 0

Example 8Determine whether Rolle’s Theorem’s hypotheses are satisfied &, if so, find a number c for which f’(c) = 0. f(x) = sin x on [0,π]

f’(x) = cos x

π/2 in the interval [0,π]

f’(x) = 0 (or undefined) when x = π/2

a) continuous on the closed interval [a,b] okb) differentiable on the open interval (a,b) okc) f(a) = f(b) f(0) = 0 f(π) = 0

Summary & Homework

• Summary:– Mean Value and Rolle’s theorems are

existance theorems– Each has some preconditions that must be

met to be used

• Homework: – pg 295-296: 2, 7, 11, 12, 24