4.2 Mean Value Theorem Quick Review What you’ll learn about Mean Value Theorem Physical Interpretation Increasing and Decreasing Functions Other Consequences

4.2

Mean Value Theorem

Quick Review

2

2

2

2

Find exact solutions to the inequality.

1. 2 4 0

2. 4 4 0

Let ( ) 6 - 2 .

3. Find the domain of .

4. Where is continuous?

5. Where is differentiable?

6. Given ( ) 3 . Find so that the

x

x

f x x

f

f

f

f x x x C C

graph of the

function passes through the point (1, 1).f

2 ,2

1 , ,1

3 ,3

3 ,3

3 ,3

5c

What you’ll learn about Mean Value Theorem Physical Interpretation Increasing and Decreasing Functions Other Consequences

Essential QuestionsHow can we use the Mean Value Theorem as atool to connect the average and instantaneous rates ofChange?

Mean Value Theorem for Derivatives

If ( ) is continuous at every point of the closed interval , and

differentiable at every point of its interior , , then there is at least

( ) - ( )one point in , at which '( ) .

-

y f x a b

a b

f b f ac a b f c

b a

Example Explore the Mean Value Theorem

2Show that the function ( ) satisfies the hypothesis of the Mean Value

Theorem on the interval 0,2 . Then find a solution to the equation

( ) - ( )'( ) on this interval.

-

f x x

c

f b f af c

b a

1.

The function f (x) = x2 is continuous on [0, 2]and differentiable on (0, 2).

,00 f ,42 f xxf 2 and

ab

afbfcf

c202

04

22 c1c


At some point her instantaneous speed was 79.5 mph.

ab

afbfcf

02

0159

5.79

2. A trucker handed in a ticket at a toll booth showing that in two hours she covered 159 mi on a toll road with a speed limit of 65 mph. The trucker was cited for speeding. Why?

Increasing Function, Decreasing Function

1 2

1 2 1 2

1 2 1 2

Let be a function defined on an interval and let and

be any two points in .

1. on if ( ) ( ).

2. on if ( ) ( ).

f I x x

I

f I x x f x f x

f I x x f x f x

increases

decreases

Corollary: Increasing and Decreasing Functions

Let be continuous on , and differentiable on , .

1. If ' 0 at each point of , , then increases on , .

2. If ' 0 at each point of , , then decreases on , .

f a b a b

f a b f a b

f a b f a b

3. Where is the function f (x) = 2x3 – 12x increasing and where is it decreasing?

The function is increasing when f ‘(x) > 0.

0126 2 x126 2 x22 x

2x 2x

The function is decreasing when f ‘(x) < 0.

0126 2 x126 2 x22 x

22 x

Example Determining Where Graphs Rise or Fall

,22,

2,2

Example Determining Where Graphs Rise or Fall4. Use analytic methods to find (a) the local extrema, (b) the

intervals on which the function is increasing, (c) the intervals on which the function is decreasing.

The function is increasing when f ‘(x) > 0.

2, The function is decreasing when f ‘(x) < 0.

42

x

xxh

4 2 x

xh x x2

22 4x

2x

4

22 4x0

042 x 2 ,2 xx

Maximum at x = – 2. Minimum at x = 2.

l l2 2

,2

2 ,2

Corollary: Functions with f’ = 0 are Constant

If '( ) 0 at each point of an interval , then there is a

constant for which ( ) for all in .

f x I

C f x C x I

Corollary: Functions with the Same Derivative Differ by a Constant

If '( ) '( ) at each point of an interval , then there is a constant

such that ( ) ( ) for all in .

f x g x I

C f x g x C x I

AntiderivativeA function ( ) is an of a function ( ) if '( ) ( )

for all in the domain of . The process of finding an antiderivative

is .

F x f x F x f x

x f

antiderivative

antidifferentiation

Example Finding Velocity and Position

5. Find the velocity and position functions of a freely falling body for the following set of conditions:The acceleration is 9.8 m/sec2 and the body falls from rest.

Assume that the body is released at time t = 0.Velocity: We know that a(t) = 9.8 and v(0) = 0, so

Cttv 8.9 Cv 00 0C ttv 8.9

Position: We know that v(t) = 9.8t and s(0) = 0, so

Ctts 29.4

Cs 00 0C 29.4 tts

Pg. 202, 4.2 #1-37 odd


12. It took 20 sec for the temperature to rise from 0o F to 212o F when a thermometer was taken from a freezer and placed in boiling water. Explain why at some moment in that interval the mercury was rising at exactly 10.6o F/sec.



xexf 5.0



910 24 xxy



83/1 xxxg

Example Determining Where Graphs Rise or Fall34. Find all possible functions f with the given derivative.

1 ,1

1

x

xxf

Example Determining Where Graphs Rise or Fall36. Find the function with the given derivative whose graph

passes through the given point P.

2 ,1 ,4

14/3

Px

xf

Documents

4.2 Mean Value Theorem Quick Review What you’ll learn about Mean Value Theorem Physical Interpretation Increasing and Decreasing Functions Other Consequences