Explanation Given: The function f 1 ( x ) = x and f 2 ( x ) = x 2 are orthogonal on the interval [ − 2 , 2 ] . Definition used: Orthogonal Functions: The functions f ( x ) and g ( x ) are said to be orthogonal functions over the interval [ a , b ] with weighting function w ( x ) if ∫ a b f ( x ) g ( x ) w ( x ) d x = 0 . Calculation: A definite integral is defined as the difference between the value of integral at the specified limit and upper limits of the independent variable. Two functions f 1 ( x ) and f 2 ( x ) are said to be orthogonal on [ a , b ] if ( f 1 ( x ) , f 2 ( x ) ) is ∫ a b f 1 ( x ) ⋅ f 2 ( x ) = 0 . It is given that the function f 1 ( x ) = x and f 2 ( x ) = x 2 are orthogonal on the interval [ − 2 , 2 ] . Since the given function f 1 ( x ) = x and f 2 ( x ) = x 2 are orthogonal function, the value of integral is zero. ∫ − 2 2 f 1 ( x ) ⋅ f 2 ( x ) d x = ∫ − 2 2 x 3 d x = 0 Now, the expression ∫ − 2 2 f 1 ( x ) ⋅ f 3 ( x ) d x is evaluated as follows: ∫ − 2 2 f 1 ( x ) ⋅ f 3 ( x ) d x = ∫ − 2 2 x ( x + c 1 x 2 + c 2 x 3 ) d x = ∫ − 2 2 x 2 + c 1 x 3 + c 2 x 4 d x = ∫ − 2 2 x 2 d x + c 1 ∫ − 2 2 x 3 d x + c 2 ∫ − 2 2 x 4 d x = [ x 3 3 ] − 2 2 + c 1 [ x 4 4 ] − 2 2 + c 2 [ x 5 5 ] − 2 2 Further simplify the above equation

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ISBN: 9781284127003

Author: ZILL

Publisher: JONES+BARTLETT LEARNING,LLC (CC)

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Chapter 12.1, Problem 18E

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The constants c1 and c2 if the function f3(x)=x+c1x2+c2x3 is orthogonal to both f1 and f2 on the same interval.

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Chapter 12.1, Problem 17E

Chapter 12.1, Problem 19E

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Chapter 12 Solutions

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