Explanation Formula used : The formula of differentiation of exponential function is, d d x e x = e x The formula of differentiation of power component function is, d d x ( x n ) = n x n − 1 The formula of the product chain rule of differentiation is, d d x ( u v ) = u d d x ( v ) + v d d x ( u ) Equation of the line passes through a point ( x 1 , y 1 ) with the slope m is, y − y 1 = m ( x − x 1 ) Given information: The equation of the graph is, f ( x ) = e − x 2 (1) Calculation: Differentiate the equation (1) both side with respect to x , f ′ ( x ) = d d x ( e − x 2 ) [ Chain rule ] = d d x e − x 2 = e − x 2 d d x ( − x 2 ) = − 2 x e − x 2 Differentiate the above equation both side with respect to x for the second derivative of the function f ( x ) , f ″ ( x ) = d d x ( − 2 x e − x 2 ) = − 2 x d d x e − x 2 + ( − 2 e − x 2 ) d d x x = − 2 x ( − 2 x e − x 2 ) + ( − 2 e − x 2 ) = 2 e − x 2 ( 2 x 2 − 1 ) If f ″ ( x ) = 0 , 2 e − x 2 ( 2 x 2 − 1 ) = 0 , 2 x 2 − 1 = 0 x 2 = 1 2 Take square root both side of the above equation, x = ± 1 2 Sign change from negative to positive at the point, x = − 1 2 , x = 1 2 Substitute − 1 2 for x in the equation (1) for the value of f ( x ) , f ( − 1 2 ) = e − ( − 1 2 ) 2 = e − 1 / 2 The inflection point of the function is ( − 1

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 5.4, Problem 41E

To determine

The equation of the tangents line to the graph of the function f(x)=e−x2 at its inflection points.

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Chapter 5.4, Problem 40E

Chapter 5.4, Problem 42E

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Chapter 5 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

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The equation of the tangents line to the graph of the function f ( x ) = e − x 2 at its inflection points.

Solution Summary: The author explains the formula of differentiation of exponential function and power component function.

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The equation of the tangents line to the graph of the function f(x)=e−x2 at its inflection points.

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Chapter 5 Solutions