Explanation Given Information: The provided function is f ( x ) = − x 2 − 6 x − 5 and the interval is [ − 3 , 1 ] . Formula used: The power rule for differentiation states that, d d x ( x k ) = k x k − 1 The derivative of a constant times a function is, d d x [ c ⋅ f ( x ) ] = c ⋅ d d x ( f ( x ) ) The derivative of a sum/difference of terms in a function, d d x ( f ± g ) = d d x f ± d d x g According to average rate change formula, if f ( x ) be the function defined on interval [ a , b ] then, Δ y Δ x = f ( b ) − f ( a ) b − a Calculation: The average rate of change of the function f ( x ) = − x 2 − 6 x − 5 over the interval [ − 3 , 1 ] is given by Δ f Δ x = f ( 1 ) − f ( − 3 ) 1 − ( − 3 ) Now, substitute 1 for x in the function f ( x ) = − x 2 − 6 x − 5 . f ( 1 ) = − ( 1 ) 2 − 6 ( 1 ) − 5 = − 1 − 6 − 5 = − 12 And, substitute − 3 for x in the function f ( x ) = − x 2 − 6 x − 5 . f ( − 3 ) = − ( − 3 ) 2 − 6 ( − 3 ) − 5 = − 9 + 18 − 5 = − 4 Substituting these values in the above equation to get the average rate of change, Δ f Δ x = f ( 1 ) − f ( − 3 ) 1 − ( − 3 ) = − 12 − 4 1 + 3 = − 16 4 = − 4 The average rate of change of the function f ( x ) = − x 2 − 6 x − 5 over the interval [ − 3 , 1 ] is ( − 4 )

Calculus: An Applied Approach (MindTap Course List)

10th Edition

ISBN: 9781305860919

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 2.3, Problem 6E

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To calculate: The average rate of change of the function f(x)=−x2−6x−5 over the interval [−3,1] and compare it with the instantaneous rate of change at the end points of the interval.

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Chapter 2 Solutions

Calculus: An Applied Approach (MindTap Course List)

Ch. 2.1 - Checkpoint 1 Worked-out solution available at...Ch. 2.1 - Checkpoint2 Worked-out solution available at...Ch. 2.1 - Checkpoint 3 Worked-out solution available at...Ch. 2.1 - Checkpoint 4 Worked-out solution available at...Ch. 2.1 - Checkpoint 5 Worked-out solution available at...Ch. 2.1 - Checkpoint 6 Worked-out solution available at...Ch. 2.1 - Checkpoint 7 Worked-out solution available at...Ch. 2.1 - In Exercises 1-4, find an equation of the line...Ch. 2.1 - In Exercises 1-4, find an equation of the line...Ch. 2.1 - In Exercises 1-4, find an equation of the line...

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The average rate of change of the function f ( x ) = − x 2 − 6 x − 5 over the interval [ − 3 , 1 ] and compare it with the instantaneous rate of change at the end points of the interval.

Solution Summary: The author compares the instantaneous rate of change of the function f(x)=-2-6x-5 over the interval and the power rule for differentiation.

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To calculate: The average rate of change of the function f(x)=−x2−6x−5 over the interval [−3,1] and compare it with the instantaneous rate of change at the end points of the interval.

Chapter 2 Solutions