Explanation Given Information: The given definite integral is ∫ 0 1 x 3 d x . Formula used: Simpson’s Rule: If f is continuous on [ a , b ] and n is an even integer, then ∫ a b f ( x ) d x , n = ( b − a 3 n ) [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + … + 4 f ( x n − 1 ) + f ( x n ) ] . The errors E in approximating ∫ a b f ( x ) d x by the Simpson’s Rule: | E | ≤ ( b − a ) 5 180 n 2 [ max | f 4 ( x ) | ] , a ≤ x ≤ b Proof: Consider the definite integral ∫ 0 1 x 3 d x . To prove Simpson’s Rule is exact when approximating the integral of a cubic polynomial function. Let P 3 ( x ) = a x 3 + b x 2 + c x + d is a cubic polynomial function. The first derivative of P 3 ( x ) = a x 3 + b x 2 + c x + d is, P 3 1 ( x ) = d d x ( a x 3 + b x 2 + c x + d ) = 3 a x 2 + 2 b x + c The second derivative of the function is, P 3 2 ( x ) = d d x ( 3 a x 2 + 2 b x + c ) = 6 a x + 2 b The third derivative of the function is, P 3 3 <

Calculus: An Applied Approach (MindTap Course List)

10th Edition

ISBN: 9781305860919

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 6.3, Problem 48E

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To prove: The Simpson’s Rule is exact when approximating the integral of a cubic polynomial function, and also demonstrates the result with n=2 for the function ∫01x3dx.

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Chapter 6.3, Problem 47E

Chapter 6.4, Problem 1CP

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Find the accumulated amount A, if the principal P is invested at an interest rate of r per year for t years. (Round your answer to the nearest cent.) P = $13,000, r = 6%, t = 10, compounded quarterly A = $ 31902 Need Help? Read It Watch It Viewing Saved Work Revert to Last Response SUBMIT ANSWER O/6.66 Points] DETAILS MY NOTES TANAPCALC10 5.3.003. EVIOUS ANSWERS ASK YOUR TEACHER PRACTICE ANOTHER Find the accumulated amount A, if the principal P is invested at an interest rate of r per year for t years. (Round your answer to the nearest cent.) P = $140,000, r = 8%, t = 8, compounded monthly A = $259130.20 X Need Help? Read It Watch It

Find the present value of $20,000 due in 3 years at the given rate of interest. (Round your answers to the nearest cent.) (a) 2%/year compounded monthly (b) 5%/year compounded daily $ Need Help? Read It Watch It SUBMIT ANSWER [-/6.66 Points] DETAILS MY NOTES TANAPCALC10 5.3.009. ASK YOUR TEACHER PRACTICE ANC Find the accumulated amount after 3 years if $4000 is invested at 3%/year compounded continuously. (Round your answer to the nearest cent.) Need Help? Read It Watch It

Chapter 6 Solutions

Calculus: An Applied Approach (MindTap Course List)

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Ch. 6.1 - Prob. 4SWU Ch. 6.1 - Prob. 5SWU Ch. 6.1 - Prob. 6SWU Ch. 6.1 - Prob. 7SWU Ch. 6.1 - Prob. 8SWU Ch. 6.1 - Prob. 9SWU Ch. 6.1 - Prob. 10SWU Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Integration by Parts In Exercises 5-16, use...Ch. 6.1 - Prob. 6E Ch. 6.1 - Integration by Parts In Exercises 5-16, use...Ch. 6.1 - Prob. 8E Ch. 6.1 - Integration by Parts In Exercises 5-16, use...Ch. 6.1 - Prob. 10E Ch. 6.1 - Integration by Parts In Exercises 5-16, use...Ch. 6.1 - Prob. 12E Ch. 6.1 - Integration by Parts In Exercises 5-16, use...Ch. 6.1 - Prob. 14E Ch. 6.1 - Integration by Parts In Exercises 5-16, use...Ch. 6.1 - Prob. 16E Ch. 6.1 - Prob. 17E Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Finding an Indefinite Integral In Exercises 17-38,...Ch. 6.1 - Finding an Indefinite Integral In Exercises 17-38,...Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Finding an Indefinite Integral In Exercises 17-38,...Ch. 6.1 - Prob. 26E Ch. 6.1 - Prob. 27E Ch. 6.1 - Prob. 28E Ch. 6.1 - Prob. 29E Ch. 6.1 - Prob. 30E Ch. 6.1 - Prob. 31E Ch. 6.1 - Prob. 32E Ch. 6.1 - Prob. 33E Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Finding an Indefinite Integral In Exercises 17-38,...Ch. 6.1 - Finding an Indefinite Integral In Exercises 17-38,...Ch. 6.1 - Prob. 39E Ch. 6.1 - Prob. 40E Ch. 6.1 - Prob. 41E Ch. 6.1 - Evaluating a Definite Integral In Exercises 39-46,...Ch. 6.1 - Prob. 43E Ch. 6.1 - Prob. 44E Ch. 6.1 - Prob. 45E Ch. 6.1 - Prob. 46E Ch. 6.1 - Prob. 47E Ch. 6.1 - Prob. 48E Ch. 6.1 - Prob. 49E Ch. 6.1 - Prob. 50E Ch. 6.1 - Prob. 51E Ch. 6.1 - Prob. 52E Ch. 6.1 - Verifying Formulas In Exercises 53 and 54, use...Ch. 6.1 - Verifying Formulas In Exercises 53 and 54, use...Ch. 6.1 - Prob. 55E Ch. 6.1 - Prob. 56E Ch. 6.1 - Prob. 57E Ch. 6.1 - Prob. 58E Ch. 6.1 - Prob. 59E Ch. 6.1 - Prob. 60E Ch. 6.1 - Prob. 61E Ch. 6.1 - Prob. 62E Ch. 6.1 - Prob. 63E Ch. 6.1 - Prob. 64E Ch. 6.1 - Prob. 65E Ch. 6.1 - Revenue A company sells a seasonal product. 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To prove: The Simpson’s Rule is exact when approximating the integral of a cubic polynomial function, and also demonstrates the result with n = 2 for the function ∫ 0 1 x 3 d x .

Solution Summary: The author proves Simpson's Rule is exact when approximating the integral of a cubic polynomial function.

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To prove: The Simpson’s Rule is exact when approximating the integral of a cubic polynomial function, and also demonstrates the result with n=2 for the function ∫01x3dx.

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Chapter 6 Solutions