A VALUE AND MEAN VALUE HEOREM · 2018. 12. 9. · EXPERIMENT A. Use the graphing calculator to integrate the equation, and find out the area of half circle using, 𝑨=𝝅𝒓 B

AVERAGE VALUE AND MEAN VALUE THEOREM

Section 4.4A

Calculus AP/Dual, Revised ©2017

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7/30/2018 3:00 AM §4.4A: Average Value and Mean Value Theorem 1

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MATERIALS NEEDED

A. Grid Paper

B. Compass

C. Graphing Calculator


EXPERIMENT

A. Identify the 𝒙 and 𝒚-axis

B. Graph 𝒚 = 𝟗 − 𝒙𝟐 where the semicircle is at the origin and 𝒓 = 𝟑

C. Use each grid point as 𝟎. 𝟐𝟓 to remain consistent


EXPERIMENT

A. Fill in the area under the curve with the M&M’s AS COMPACTLY AS POSSIBLE.

B. DO NOT EAT THE M&M’s… YET.


EXPERIMENT


EXPERIMENT

A. Using exactly the same number of M&M’s, form a rectangle whose length runs along the 𝒙-axis from −𝟑 to 𝟑.

B. When you think you have it, with your pencil, mark the upper left and right hand corners of the rectangle.


EXPERIMENT

A. As you clear out the rectangle, return the candies to the cup OR you can eat them as you please.

B. Draw the rectangle with the ruler and ESTIMATE the height of the rectangle. BE PRECISE as possible!


SHOULD LOOK LIKE…


− − − −

−

−

−

−

x

y

EXPERIMENT

A. Use the graphing calculator to integrate the equation, 𝟑−𝟑

𝟗 − 𝒙𝟐𝒅𝒙

and find out the area of half circle using, 𝑨 =𝝅 𝒓𝟐

𝟐

B. Take the answer and divide the length of the interval of 𝟔.

C. Then, compare the answers to the height. Is it close?


SHOULD LOOK LIKE…


− − − −

−

−

−

−

x

y

COMPARISON

Geometry Calculus


2

2

rA

=

( )

( )

23

2A

=

2

2

9

2

14.1372

A units

A units

=

32

39 x dx

−−

( )1/23

2

39 x dx

−−

214.1372 A units

3

2

3

9arcsin 9

2 3 2

x xx

−

+ −

( )7.0686 7.0686− −

SPLIT IT UP TO SIX OF YOUR FRIENDS…

If you have a party and want to split this up to six of your closest friends… how much of the area of the M&M’s would you share?


− − − −

−

−

−

−

x

y

214.1372 A units

214.1372 6

A units

22.3562 A units

AVERAGE VALUE EQUATION

A. If 𝒇 is integrable on the closed interval 𝒂, 𝒃 , then the average value

of 𝒇 on the interval is 𝟏

𝒃−𝒂𝒂𝒃𝒇 𝒙 𝒅𝒙


EXAMPLES OF AVERAGE VALUE


AVERAGE VALUE VS AVERAGE RATE OF CHANGE

A. For Average Value, the equation is for the TOTAL amount associated with the equation.

B. For Average Value, the integration is used.

C. For Average Rate of Change, the equation is for the CHANGE of the amount associated.

D. For Average Rate of Change, typically the slope equation is used.


EXAMPLE 1

Determine the average value of 𝒉 𝒙 = 𝟒𝒙 − 𝒙𝟐on 𝟎, 𝟒


( )1

b

af x dx

b a− 43

2

0

12

4 3

xx

−

( )4

2

0

14

4 0x x dx−

−

( )( )

( )3 4

2

0

412 4 0

4 3

− −

8

3

EXAMPLE 2

Determine the average value of 𝒇 𝒙 = −𝐜𝐨𝐬 𝟐𝒙 + 𝒙 + 𝟏 on 𝟎, 𝟒


( )1 b

af x dx

b a−

( )42

0

1 1sin 2

4 2 2

xx x

− + +

( )( )4

0

1cos 2 1

4 0x x dx− + +

−

( ) ( )1 1 16 1

sin 8 4 sin 04 2 2 2

− + + − −

2.876units

2

2

1

2

u x

du dx

du dx

=

=

=

( )( )4

0

1cos 1

4u x du

− + +

( )1 1

sin 8 124 2

− +

EXAMPLE 3

Let 𝒇 be a function such that 𝒇′′ 𝒙 = 𝟔𝒙 + 𝟏𝟐.

(a) Find 𝒇(𝒙) if the graph of 𝒇 is tangent to the line 𝟒𝒙 − 𝒚 = 𝟓 at the point 𝟎,−𝟓 .

(b) Find the average value of 𝒇 𝒙 on the closed interval −𝟏, 𝟏 .


EXAMPLE 3A




( )'' 6 12f x x= +

4 5y x= −

( )' 0, 4f =

( )'' 6 12 f x dx x dx= +

( )2

1

6' 12

2

xf x dx x C= + +

( ) 2 1' 3 12f x dx x x C= + +

( ) ( )2

14 3 0 12 0 ; 4C C= + + =

( ) 2' 3 12 4f x dx x x= + +

( )0, ' 4x f x= =

EXAMPLE 3A




( ) 2' 3 12 4f x dx x x= + +

4 5y x= −

( )' 0, 4f =

( ) 2' 3 12 4 f x dx x x dx= + + ( ) 3 2 26 4f x dx x x x C= + + +

( ) ( ) ( )3 2

25 0 6 0 4 0 ; 5C C− = + + + = −

( ) 3 26 4 5f x x x x= + + −

( )0, 5f = −

EXAMPLE 3B


(b) Find the average value of 𝒇 𝒙 on the closed interval −𝟏, 𝟏 .


( ) ( )1

b

af c f x dx

b a=

−

( )( )

13 2

1

16 4 5

1 1f c x x x dx

−= + + −

− −

( )4

3 2

1

1

12 2 5

2 4

xf c x x x

−

= + + −

( ) 3f c = −

YOUR TURN

Find the average value of 𝒇 𝒙 = 𝟑𝒙𝟐 − 𝟐𝒙 on 𝟏, 𝟒


16

REVIEW: MEAN VALUE THEOREM FOR DERIVATIVES


( )( ) ( )

'f b f a

f cb a

−=

−

Slope of Tangent Line Slope of Secant Line=

Instantaneous ROC Average ROC=

REVIEW: MEAN VALUE THEOREM FOR DERIVATIVES


[

a

]

b

|

c

( )f x

( )f a −

( )f b −

( )f c −

( )'f c

( ) ( )f b f a

b a

−=

−( )'f c

MEAN VALUE THEOREM FOR INTEGRATION


[

a

Area Under the Curve = Area of Rectangle

( ) ( ) ( )b

af x dx f c b a= −

( )f c

( )f x

]

bc

MEAN VALUE THEOREM IN INTEGRALS

A. Equation: 𝒂𝒃𝒇 𝒙 𝒅𝒙 = 𝒇 𝒄 𝒃 − 𝒂

1. 𝒇 𝒄 = Average Height

2. 𝒃 – 𝒂 = Width

3. 𝒂𝒃𝒇 𝒙 𝒅𝒙 is the calculus area under the curve

4. 𝒇 𝒄 𝒃 – 𝒂 is the geometry area of the rectangle


EXAMPLE 4

Identify 𝒄 guaranteed by the Mean Value Theorem, 𝒇 𝒙 = 𝟗 − 𝒙𝟐,[−𝟑, 𝟑]

( )

3,0

•

− ( )

3,0

•


EXAMPLE 4



( )3

2

39 x dx

−−

33

3

93

xx

−

−

( ) ( )( )27 9 27 9− − − − −

36( ) ( )18 18− −

EXAMPLE 4


( ) ( ) ( )b


( ) ( )( )236 9 3 3c= − − −

( ) ( )236 9 6c= −

( )26 9 c= −2 3c =

3c =

( )

3,0

•

− ( )

3,0

•

( )3,0

−

•

( )3,0 •


EXAMPLE 4


3c =

( )

3,0

•

− ( )

3,0

•

( )3,0

−

•

( )3,0 •


EXAMPLE 5

Given 𝟏𝟑𝒇 𝒙 𝒅𝒙 = 𝟖, solve for 𝒇 𝒄 using the mean value theorem in

between 𝟏, 𝟑 .


( ) ( )( ) b


( )( )8 3 1f c= −

( )( )8 2 f c=

( )4 f c=

( ) 4f c =

YOUR TURN

Identify 𝒄 guaranteed by the Mean Value Theorem, 𝒇 𝒙 = 𝒙 𝟏 − 𝒙 ,𝟎, 𝟏


1 3

2 6c =

What is the average value of 𝒚 for the part of the curve of 𝒚 = 𝟑𝒙 − 𝒙𝟐

which is in the first quadrant?

(A) −𝟔

(B) −𝟐

(C) 𝟑

𝟐

(D) 𝟗

𝟐


AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON–CALCULATOR)

What is the average value of 𝒚 for the part of the curve of 𝒚 = 𝟑𝒙 − 𝒙𝟐

which is in the first quadrant?


AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON–CALCULATOR)

Vocabulary Connections and Process Answer and Justifications

Average Value

First Quadrant( )3 0

0, 3

x x

x x

− =

= =( )

1 b

af x dx

b a−

32 33

2

00

1 1 33

3 0 3 2 3

x xx x dx

− = −

−

( ) ( )2 3

3 3 310

3 2 3

1 27 27 1 27 18 1 9 9 3

3 2 3 3 2 2 3 2 6 2

= − −

= − = − = = =

C

ASSIGNMENT

Worksheet


Documents

A VALUE AND MEAN VALUE HEOREM · 2018. 12. 9. · EXPERIMENT A. Use the graphing calculator to integrate the equation, and find out the area of half circle using, 𝑨=𝝅𝒓 B