Explanation Given: The function is f ( x ) = e − x 2 and the approximate value of I is 0.461281 . Concept used: The n th Maclaurin polynomial can be expressed as, T n ( x ) = f ( 0 ) + f ′ ( 0 ) 1 ! x + f ″ ( 0 ) 2 ! x 2 + ⋯ + f n ( x ) n ! x n ...... (1) Calculation: The given function is, f ( x ) = e − x 2 Substitute 0 for x in the function f ( x ) to obtain the value of f ( 0 ) as, f ( 0 ) = e − 0 2 = 1 Differentiate the function f ( x ) with respect to x as, d d x f ( x ) = d d x ( e − x 2 ) f ′ ( x ) = − 2 x e − x 2 Substitute 0 for x in the above function to obtain the value of f ′ ( 0 ) as, f ′ ( 0 ) = − 2 ⋅ 0 ⋅ e − 0 2 = 0 Differentiate the function f ′ ( x ) with respect to x as, d d x f ′ ( x ) = d d x ( − 2 x e − x 2 ) f ″ ( x ) = − 2 [ e − x 2 − 2 x 2 e − x 2 ] = − 2 e − x 2 [ 1 − 2 x 2 ] Substitute 0 for x in the above function to obtain the value of f ″ ( 0 ) as, f ″ ( 0 ) = − 2 e − 0 2 [ 1 − 2 ⋅ 0 2 ] = − 2 Differentiate the function f ″ ( x ) with respect to x as, d d x f ″ ( x ) = d d x [ − 2 e − x 2 ( 1 − 2 x 2 ) ] f ‴ ( x ) = − 2 [ e − x 2 ( − 4 x ) + ( 1 − 2 x 2 ) ( − 2 x e − x 2 ) ] = − 2 x e − x 2 [ − 4 − 2 + 4 x 2 ] = 2 x e − x 2 [ 6 − 4 x 2 ] Substitute 0 for x in the above function to obtain the value of f ‴ ( 0 ) as, f ‴ ( 0 ) = 2 ⋅ 0 ⋅ e − 0 2 [ 6 − 4 ⋅ 0 2 ] = 0 Differentiate the function f ‴ ( x ) with respect to x as, d d x f ‴ ( x ) = d d x [ 2 e − x 2 ( 6 x − 4 x 3 ) ] f ( 4 ) ( x ) = 2 [ e − x 2 ( 6 − 12 x 2 ) + ( 6 x − 4 x 3 ) (<

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Chapter 11.7, Problem 71E

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To find: The fourth Maclaurin polynomial T4(x) for the function f(x)=e−x2 and the value of I=∫012T4(x)dx . Find the error in the approximation.

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Chapter 11.7, Problem 70E

Chapter 11.7, Problem 72E

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Chapter 11 Solutions

CALCULUS (CLOTH)

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E

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The fourth Maclaurin polynomial T 4 ( x ) for the function f ( x ) = e − x 2 and the value of I = ∫ 0 1 2 T 4 ( x ) d x . Find the error in the approximation.

Solution Summary: The author calculates the fourth Maclaurin polynomial using T_4(x) and I.

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To find: The fourth Maclaurin polynomial T4(x) for the function f(x)=e−x2 and the value of I=∫012T4(x)dx . Find the error in the approximation.

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Chapter 11 Solutions