Explanation Given information: A definite integral of the form ∫ a b f ( x ) d x is approximated by trapezoidal rule as shown in the figure below, Explanation: The input for calculating definite integral ∫ a b f ( x ) d x using trapezoidal rule, the format on ti- 84 calculator is given below: ( Calculate first term saperately + sum ( seq ( 2 f ( x ) ) , X , a + Δ x , b − Δ x , Δ x ) + Calculate second term saperately ) ⋅ Δ x 2 After implementing trapezoidal rule, store the result in variable T . By comparing the format of trapezoidal rule with the input given in above figure, First term = sin ( 2 2 ) , f ( x ) = sin ( x 2 ) , a + Δ x = 2.5 , b − Δ x = 5.5 Δ x = 0.5 , Last term = sin ( 2 2 ) Substitute the value of Δ x = 0.5 in a + Δ x = 2.5 , ⇒ a + 0.5 = 2

Calculus & Its Applications (14th Edition)

14th Edition

ISBN: 9780134437774

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar

Publisher: PEARSON

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Chapter 9.4, Problem 36E

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f(x), a, b and n using following figure whichgives a definite integral of the form ∫abf(x) dx is approximated by trapezoidal rule.

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Chapter 9.4, Problem 35E

Chapter 9.4, Problem 37E

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(b) Find the (instantaneous) rate of change of y at x = 5. In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the following limit. lim h→0 - f(x + h) − f(x) h The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule defining f. f(x + h) = (x + h)² - 5(x+ h) = 2xh+h2_ x² + 2xh + h² 5✔ - 5 )x - 5h Step 4 - The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x). - f(x + h) f(x) = = (x² x² + 2xh + h² - ])- = 2x + h² - 5h ])x-5h) - (x² - 5x) = ]) (2x + h - 5) Macbook Pro

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f ( x ) , a , b and n using following figure whichgives a definite integral of the form ∫ a b f ( x ) d x is approximated by trapezoidal rule.

Solution Summary: The author illustrates how a definite integral is approximated by trapezoidal rule. The format of the ti-84 calculator is given below.

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f(x), a, b and n using following figure whichgives a definite integral of the form ∫abf(x) dx is approximated by trapezoidal rule.

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