   Chapter 7.7, Problem 2CP ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Find the values of a and b such that the linear model f ( x ) = a x + b has a minimum sum of the squared errors for the points. ( − 2 , 0 ) , ( 0 , 2 ) , ( 2 , 5 ) , ( 4 , 7 ) ,

To determine

To calculate: The values of a and b such that linear model f(x)=ax+b has a minimum sum of the second errors for the points. (2,0), (0,2), (2,5), (4,7).

Explanation

Given Information:

Linear model f(x)=ax+b

The provided points are (2,0),(0,2),(2,5),(4,7).

Formula used:

The sum of squared error for model y=f(x) with respective points (xn,y1)...(xn,yn) is.

S=[f1(y)y1]2+[f2(x)y2]2+...[fn(x)yn]2

Step 1:

Find the sum of square error and get the value of S in terms of a and b.

Step 2 Find derivative of the sum of squared errors with respect to a,

Step 3 Find derivative of the sum of squared errors with respect to a,

Step 4 Substitute, Sa=0 in the equation,

Step 5 Substitute, Sb=0 in the equation,

Step 6 Find value of a and b.

Calculation:

Consider the provided point (2,0), (0,2), (2,5), (4,7). and linear model equation, f(x)=ax+b.

The following table shows the error at different point,

 x −2 0 2 4 y 0 2 5 7 f(x1) −2a+b b 2a+b 4a+b Error=f(x1)−yi −2a+b b−2 2a+b−5 4a+b−7 Error=[f(x1)−yi]2 ( −2a+b)2 (b−2)2 (2a+b−5)2 (4a+b−7)2

The sum of squared errors,

S=(2a+b)2+(b2)2+(2a+b5)2+(4a+b7)2=b2+4a24ab+b2+44b+4a2+(b5)2+2(2a)(b5)+(4a)2+(b7)2+2(4a)(b7)=b2+4a24ab+b2+

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