Explanation Given information: The function f ( x ) = ( 1 − x ) e x and domain − 2 ≤ x ≤ 1 . Graph: Consider the provided function f ( x ) = ( 1 − x ) e x . Evaluate the first derivative of the function, apply the product rule of differentiation , d d x ( f ( x ) g ( x ) ) = f ( x ) d d x g ( x ) + g ( x ) d d x f ( x ) . d d x f ( x ) = d d x ( ( 1 − x ) e x ) = ( 1 − x ) d d x ( e x ) + e x d d x ( 1 − x ) = ( 1 − x ) e x + e x d d x ( 1 − x ) Apply the sum rule of differentiation, d d x ( f ( x ) + g ( x ) ) = d d x f ( x ) + d d x g ( x ) . d d x f ( x ) = ( 1 − x ) e x + e x [ d d x ( 1 ) + d d x ( − x ) ] Apply the constant product rule, d d x ( c f ( x ) ) = c d d x f ( x ) . d d x f ( x ) = ( 1 − x ) e x + e x [ 0 − d d x ( x ) ] = ( 1 − x ) e x − e x = e x − x e x − e x = − x e x Recall if first derivative of the function is greater than zero that is g ' ( x ) > 0 for every x in ( c , d ) , then the function g is increasing on interval [ c , d ] . If first derivative of the function is less than zero that is g ' ( x ) < 0 for every x in ( c , d ) , then the function g is decreasing on interval [ c , d ] . The first derivative of the function is f ' ( x ) = − x e x which is positive if, x is less than or equal to zero and negative if x is greater than zero. Therefore, the function is increasing when x ≤ 0 and decreasing when x > 0 . Evaluate the second derivative of the function, differentiate first derivative again with respect to x . Apply the product rule of differentiation, d d x ( f ( x ) g ( x ) ) = f ( x ) d d x g ( x ) + g ( x ) d d x f ( x )

Modeling the Dynamics of Life: Calculus and Probability for Life Scientists

3rd Edition

ISBN: 9780840064189

Author: Frederick R. Adler

Publisher: Cengage Learning

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Chapter 2.8, Problem 19E

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To graph:The function f(x)=(1−x)ex for −2≤x≤1 with help of first and second derivative of the function and identify the regions in which function is increasing and concave up.

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Chapter 2.8, Problem 18E

Chapter 2.8, Problem 20E

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Chapter 2 Solutions

Modeling the Dynamics of Life: Calculus and Probability for Life Scientists

Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E

Ch. 2.1 - Prob. 11E Ch. 2.1 - Prob. 12E Ch. 2.1 - Prob. 13E Ch. 2.1 - Prob. 14E Ch. 2.1 - Prob. 15E Ch. 2.1 - Prob. 16E Ch. 2.1 - Prob. 17E Ch. 2.1 - Prob. 18E Ch. 2.1 - Prob. 19E Ch. 2.1 - Prob. 20E Ch. 2.1 - Prob. 21E Ch. 2.1 - Prob. 22E Ch. 2.1 - Prob. 23E Ch. 2.1 - Prob. 24E Ch. 2.1 - Prob. 25E Ch. 2.1 - Prob. 26E Ch. 2.1 - Prob. 27E Ch. 2.1 - Prob. 28E Ch. 2.1 - Prob. 29E Ch. 2.1 - Prob. 30E Ch. 2.1 - Prob. 31E Ch. 2.1 - Prob. 32E Ch. 2.1 - Prob. 33E Ch. 2.1 - Prob. 34E Ch. 2.1 - Prob. 35E Ch. 2.1 - Prob. 36E Ch. 2.1 - Prob. 37E Ch. 2.1 - Prob. 38E Ch. 2.1 - Prob. 39E Ch. 2.1 - Prob. 40E Ch. 2.1 - Prob. 41E Ch. 2.1 - Prob. 42E Ch. 2.1 - Prob. 43E Ch. 2.1 - Prob. 44E Ch. 2.1 - Prob. 45E Ch. 2.1 - Prob. 46E Ch. 2.2 - Prob. 1E Ch. 2.2 - Prob. 2E Ch. 2.2 - Prob. 3E Ch. 2.2 - Prob. 4E Ch. 2.2 - Prob. 5E Ch. 2.2 - Prob. 6E Ch. 2.2 - Prob. 7E Ch. 2.2 - Prob. 8E Ch. 2.2 - Prob. 9E Ch. 2.2 - Prob. 10E Ch. 2.2 - Prob. 11E Ch. 2.2 - Prob. 12E Ch. 2.2 - Prob. 13E Ch. 2.2 - Prob. 14E Ch. 2.2 - 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To graph: The function f ( x ) = ( 1 − x ) e x for − 2 ≤ x ≤ 1 with help of first and second derivative of the function and identify the regions in which function is increasing and concave up.

Solution Summary: The author analyzes the function f(1-x)ex with help of first and second derivatives and identify the regions in which function is increasing and concave up.

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To graph:The function f(x)=(1−x)ex for −2≤x≤1 with help of first and second derivative of the function and identify the regions in which function is increasing and concave up.

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Chapter 2 Solutions