AVERAGE VALUE AND MEAN VALUE THEOREM
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1 AVERAGE VALUE AND MEAN VALUE THEOREM Section 4.4A Calculus AP/Dual, Revised 017 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 1
2 MATERIALS NEEDED A. Grid Paper B. Compass C. Graphing Calculator 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem
3 EXPERIMENT A. Identify the x and y-axis B. Graph y = 9 x where the semicircle is at the origin and r = 3 C. Use each grid point as 0. 5 to remain consistent 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 3
4 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. 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 4
5 EXPERIMENT 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 5
6 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. 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 6
7 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! 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 7
8 SHOULD LOOK LIKE y x 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 8
9 EXPERIMENT 3 A. Use the graphing calculator to integrate the equation, 3 π r and find out the area of half circle using, A = B. Take the answer and divide the length of the interval of 6. C. Then, compare the answers to the height. Is it close? 9 x dx 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 9
10 SHOULD LOOK LIKE y x 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 10
11 COMPARISON A A = A Geometry A = r = ( 3) 9 units units Calculus 9 x ( x ) 1/ 9 3 A dx dx 9 x x arcsin + 9 x ( ) units 3 3 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 11
12 SPLIT IT UP TO SIX OF YOUR FRIENDS y 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? A units A units x A.356 units 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 1
13 AVERAGE VALUE EQUATION A. If f is integrable on the closed interval a, b, then the average value 1 b of f on the interval is f x dx b a a 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 13
14 EXAMPLES OF AVERAGE VALUE 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 14
15 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. 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 15
16 EXAMPLE 1 Determine the average value of h x = 4x x on 0, 4 11 b f ( x ) b a 4 ( dx 4 x x ) dx a x x ( 0) /30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem
17 EXAMPLE Determine the average value of f x = cos x + x + 1 on 0, 4 1 b ( ) b a f x dx a 1 4 cos x + x + 1 dx ( cos 1) 4 u + x + du x sin ( x) + + x sin ( 8) sin ( 0) 4 u du 1 = x = dx du = dx 1 1 sin ( 8 ) units 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 17
18 EXAMPLE 3 Let f be a function such that f x = 6x + 1. (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. 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 18
19 EXAMPLE 3A x = 0, f '( x) = 4 f ' = ( 0, 4) Let f be a function such that f x = 6x + 1. y = 4x 5 (a) Find f(x) if the graph of f is tangent to the line 4x y = 5 at the point 0, 5. f '' x = 6x + 1 f ''( x) dx = 6x + 1 dx 6x f '( x) dx = + 1x + C1 f ' x dx = 3x + 1x + C 4 = C ; C = 4 f ' x dx = 3x + 1x + 4 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem
20 EXAMPLE 3A Let f be a function such that f x = 6x + 1. (a) Find f(x) if the graph of f is tangent to the line 4x y = 5 at the point 0, 5. f ' x dx = 3x + 1x + 4 f ' x dx = 3x + 1x + 4 dx f x dx = x + 6x + 4x + C = C ; C = 5 f x = x + 6x + 4x 5 3 f = ' ( 0, 0, 4 5) y = 4x 5 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 0
21 EXAMPLE 3B Let f be a function such that f x = 6x + 1. (b) Find the average value of f x on the closed interval 1, 1. 1 b f ( c) = f ( x) dx b a a f c = x + 6x + 4x 5 dx x f c = + x + x 5x f c = 3 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 1 1 1
22 YOUR TURN Find the average value of f x = 3x x on 1, /30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem
23 REVIEW: MEAN VALUE THEOREM FOR DERIVATIVES f ( a) f b f ' ( c) = b a Slope of Tangent Line = Slope of Secant Line Instantaneous ROC = Average ROC 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 3
24 REVIEW: MEAN VALUE THEOREM FOR DERIVATIVES f ' ( c) f b b a f a = f ' c f ( c) f x f ( b) f ( a) [ a c ] b 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 4
25 MEAN VALUE THEOREM FOR INTEGRATION f x Area Under the Curve = Area of Rectangle b = ( ) a f x dx f c b a [ a c ] b f ( c) 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 5
26 MEAN VALUE THEOREM IN INTEGRALS A. Equation: a b f x dx = f c b a 1. f c = Average Height. b a = Width 3. b a f x dx is the calculus area under the curve 4. f c b a is the geometry area of the rectangle 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 6
27 EXAMPLE 4 Identify c guaranteed by the Mean Value Theorem, f x = 9 x, [ 3, 3] ( 3,0) ( 3,0) 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 7
28 EXAMPLE 4 Identify c guaranteed by the Mean Value Theorem, f x = 9 x, [ 3, 3] 3 ( 9 x ) dx 3 9x x ( 7 9) 7 ( 9) ( 18) ( 18) 36 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 8
![EXAMPLE 4 Identify c guaranteed by the Mean Value Theorem, f x = 9 x, [ 3, 3] b f ( x ) dx = f ( c )( b a ) a ( c ) 36 = 9 3 3](/docs-images/95/123023933/images/29-1.jpg "( ) 36 = 9 c 6 ( ) 6= 9 c c = 3 c = 3 ( 3,0) ( 3,0) ( 3,0) ( 3,0) 7/30/018 3:00 AM 4.")
29 EXAMPLE 4 Identify c guaranteed by the Mean Value Theorem, f x = 9 x, [ 3, 3] b f ( x ) dx = f ( c )( b a ) a ( c ) 36 = ( ) 36 = 9 c 6 ( ) 6= 9 c c = 3 c = 3 ( 3,0) ( 3,0) ( 3,0) ( 3,0) 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 9
![EXAMPLE 4 Identify c guaranteed by the Mean Value Theorem, f x = 9 x, [ 3, 3] c = 3 (](/docs-images/95/123023933/images/30-1.jpg "3,0) ( 3,0) ( 3,0) ( 3,0) 7/30/018 3:00 AM 4.")
30 EXAMPLE 4 Identify c guaranteed by the Mean Value Theorem, f x = 9 x, [ 3, 3] c = 3 ( 3,0) ( 3,0) ( 3,0) ( 3,0) 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 30
31 EXAMPLE 5 3 Given 1 f x dx = 8, solve for f c using the mean value theorem in between 1, 3. b f ( x ) dx = f ( c )( b a ) a 8 = f c 3 1 8= ( f ( c) ) 4 = f ( c) f ( c ) = 4 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 31
32 YOUR TURN Identify c guaranteed by the Mean Value Theorem, f x = x 1 x, 0, c = 6 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 3
33 AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON CALCULATOR) What is the average value of y for the part of the curve of y = 3x x which is in the first quadrant? (A) 6 (B) (C) 3 (D) 9 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 33
34 AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON CALCULATOR) What is the average value of y for the part of the curve of y = 3x x which is in the first quadrant? Vocabulary Connections and Process Answer and Justifications Average Value First Quadrant x ( x) 3 = 0 1 b 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 34 f x dx a x= 0, x= 3 b a x x 3x x dx = ( 3) ( 3) = = = = = = C
35 ASSIGNMENT Worksheet 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 35
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