(a) Explanation Given Information: The data set is Number of Years (since 1990), x 0 10 13 17 21 Life expectancy of men (years), y 71.8 74.1 74.8 75.4 76.3 Formula used: If the function is differentiable function of x and y , then the relative maximum and minimum values of the function is determined by following the below steps: Step 1: Find f x , f y , f x x , f y y and f x y . Step 2: Solve the system of equations f x = 0 , f y = 0 . Let ( a , b ) represent a solution. Step 3: Evaluate D , where D = f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 . Step 4: Then, (a) The function f has a maximum at ( a , b ) if D > 0 and f x x ( a , b ) < 0 . (b) The function f has a minimum at ( a , b ) if D > 0 and f x x ( a , b ) > 0 . (c) The function f has neither a maximum nor a minimum at ( a , b ) if D < 0 . In this case, f has a saddle point at ( a , b ) . (d) This test is not applicable if D = 0 . Calculation: Consider the provided data set. The data points are ( 0 , 71.8 ) , ( 10 , 74.1 ) , ( 13 , 74.8 ) , ( 17 , 75.4 ) and ( 21 , 76.3 ) . The points on the regression line are ( 0 , y 1 ) , ( 10 , y 2 ) , ( 13 , y 3 ) , ( 17 , y 4 ) and ( 21 , y 5 ) . The y -deviations are y 1 − 71.8 , y 2 − 74.1 , y 3 − 74.8 , y 4 − 75.4 and y 5 − 76.3 . Now use the above data, to minimize the function; S = ( y 1 − 71.8 ) 2 + ( y 2 − 74.1 ) 2 + ( y 3 − 74.8 ) 2 + ( y 4 − 75.4 ) 2 + ( y 5 − 76.3 ) 2 …… (1) Where: y 1 = m ⋅ 0 + b , y 2 = m ⋅ 10 + b , y 3 = m ⋅ 13 + b , y 4 = m ⋅ 17 + b and y 5 = m ⋅ 21 + b Substitute the value of y 1 , y 2 , y 3 , y 4 and y 5 in equation (1), to get S as; S = ( b − 71.8 ) 2 + ( 10 m + b − 74 (b) To determine The life expectancy of men in 2020 and 2025.

Calculus and Its Applications (11th Edition)

11th Edition

ISBN: 9780321979391

Author: Marvin L. Bittinger, David J. Ellenbogen, Scott J. Surgent

Publisher: PEARSON

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Chapter 6.4, Problem 12E

(a)

To determine

To calculate: The regression line for the data set represented by the table,

Number of Years (since 1990), x 0 10 13 17 21

Life expectancy of men (years), y 71.8 74.1 74.8 75.4 76.3

(b)

To determine

The life expectancy of men in 2020 and 2025.

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Chapter 6.4, Problem 11E

Chapter 6.4, Problem 13E

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Number of Years (since 1990), x	0	10	13	17	21
Life expectancy of men (years), y	71.8	74.1	74.8	75.4	76.3

The regression line for the data set represented by the table, Number of Years (since 1990), x 0 10 13 17 21 Life expectancy of men (years), y 71.8 74.1 74.8 75.4 76.3

Solution Summary: The author calculates the regression line for the given data set: Number of Years (since 1990), x 0 10 13 17 21 Life expectancy of men (years).