To determine To calculate: The equation of the curve. Explanation Given: A tangent line and its slope are: f ′ ( x ) = x 2 3 . And, curve passes through a point which is ( 1 , 3 5 ) Formula used: Power rule, ∫ x n d x = x n + 1 n + 1 + c Calculation: Consider the slope, f ′ ( x ) = x 2 3 To get the equation of the curve then takes antiderivative of the slope which is the integration of the above expression. Use power rule formula in the above expression, ∫ x n d x = x n + 1 n + 1 + c Where c is arbitrary constant Integrate both sides as: f ′ ( x ) = x 2 3 ∫ f ′ ( x ) = ∫ x 2 3 As the integration of a derivative cancel out each other and left with the function which is: ∫ f ′ ( x ) = f ( x ) Apply the formula to the above expression which is: ∫ f ′ ( x )

Mathematics with Applications In the Management, Natural and Social Sciences (11th Edition)

11th Edition

ISBN: 9780321931078

Author: Margaret L. Lial, Thomas W. Hungerford, John P. Holcomb, Bernadette Mullins

Publisher: PEARSON

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Chapter 13.1, Problem 43E

To determine

To calculate: The equation of the curve.

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Chapter 13.1, Problem 42E

Chapter 13.1, Problem 44E

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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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Chapter 13 Solutions

Mathematics with Applications In the Management, Natural and Social Sciences (11th Edition)

Ch. 13.1 - Checkpoint 1 Find an antiderivative for each of...Ch. 13.1 - Checkpoint 2 Find each of the...Ch. 13.1 - Prob. 3CP Ch. 13.1 - Prob. 4CP Ch. 13.1 - Prob. 5CP Ch. 13.1 - Prob. 6CP Ch. 13.1 - Prob. 7CP Ch. 13.1 - Checkpoint 8 The marginal cost at a level of...Ch. 13.1 - Prob. 9CP Ch. 13.1 - 1. What must be true of F(x) and G(x) if both are...

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