Explanation Trapezoid Rule: The Trapezoid Rule approximation to ∫ a b f ( x ) d x that uses n equally spaced subintervals on [ a , b ] is T ( n ) = ( 1 2 f ( x 0 ) + ∑ k = 1 n − 1 f ( x k ) + 1 2 f ( x n ) ) Δ x , where Δ x = ( b − a ) n and x k = a + k Δ x , for k = 0 , 1 , ... , n . Simpson’s Rule The Simpson’s Rule approximation to ∫ a b f ( x ) d x that uses n equally spaced subintervals on [ a , b ] is S ( n ) = ( f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + ⋯ 4 f ( x n − 1 ) + f ( x n ) ) Δ x 3 , where Δ x = ( b − a ) n , n is an even integer and x k = a + k Δ x , for k = 0 , 1 , ... , n . Calculation: By the Trapezoid rule, T ( 2 n ) = ( 1 2 f ( x 0 ) + ∑ k = 1 2 n − 1 f ( x k ) + 1 2 f ( x 2 n ) ) ( b − a 2 n ) , x 0 = a , x n = b , 1 ≤ i ≤ n and 1 ≤ j ≤ 2 n . Simply 4 T ( 2 n ) − T ( n ) by using the Trapezoid rule as follows. 4 T ( 2 n ) − T ( n ) = 4 ( 1 2 f ( a ) + ∑ j = 1 2 n − 1 f ( x j ) + 1 2 f ( b ) ) Δ x − ( 1 2 f ( a ) + ∑ i = 1 n − 1 f ( x i ) + 1 2 f ( b ) ) Δ x = ( 2 f ( a ) + 4 ∑ j = 1 2 n − 1 f ( x j )

Calculus: Single Variable, Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)

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ISBN: 9780134996714

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 8.8, Problem 78E

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To prove: For n≥1, S(2n)=4T(2n)−T(n)3 that uses the notation of the text.

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Chapter 8.8, Problem 77E

Chapter 8.8, Problem 79E

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Calculus: Single Variable, Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)

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To prove: For n ≥ 1 , S ( 2 n ) = 4 T ( 2 n ) − T ( n ) 3 that uses the notation of the text.

Solution Summary: The author explains the Trapezoid Rule approximation to displaystyle 'int', which uses n equally spaced subintervals.

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To prove: For n≥1, S(2n)=4T(2n)−T(n)3 that uses the notation of the text.

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Chapter 8 Solutions