Explanation Definition used: If f ' ( x ) > 0 on an interval ( a , b ) , then the graph of f ( x ) increases on that interval and if f ' ( x ) < 0 , then f ( x ) decreases on that interval. Concavity: For the interval ( a , b ) , if f ′ ′ ( x ) > 0 , then the function f is concave upward and if f ′ ′ ( x ) < 0 , then f is concave downward. Calculation: The given function is f ( x ) = e − 0.5 x 2 . Domain: Since the given function is an exponential function, its domain is all real numbers. The domain of the function is ( − ∞ , ∞ ) . Intercepts: Set x = 0 and find the y −intercepts. f ( 0 ) = e 0 = 1 Thus, the y -intercept is 1. There is no x- intercept. Asymptotes: Obtain the horizontal asymptote as follows, lim x → ∞ ( e − 0.5 x 2 ) = 0 . So, the line y = 0 is the horizontal asymptote. The exponential function does not have vertical asymptotes. Thus, there is no vertical asymptote for this function. First derivative test: The derivative of the function f ( x ) is, f ′ ( x ) = d d x ( e − 0.5 x 2 ) = − x e − 0.5 x 2 Set f ′ ( x ) = 0 and find the critical points. − x e − 0.5 x 2 = 0 x = 0 The critical points of f is x = 0 . Thus, the partitions are ( − ∞ , 0 ) and ( 0 , ∞ ) . The sign chart of f ′ ( x ) is as shown below in Table 1. Interval Test numbers f ′ ( x ) − ∞ < x < 0 −1 0.606 increasing 0 < x < ∞ 1 −0.606 decreasing Table 1 From Table 1, it is observed that the function is decreasing on ( 0 , ∞ ) and increasing on ( − ∞ , 0 ) . The f has maximum value at x = 0 . Therefore, f ( 0 ) = e − 0.5 × 0 = 1 is local maximum. Second derivative test: The second derivative of the function f ( x ) is, f ′ ′ ( x ) = d d x ( − x e − 0

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

15th Edition

ISBN: 9781323047620

Author: Barnett, Ziegler, Byleen

Publisher: Pearson Education

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Chapter 4.4, Problem 53E

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To sketch: The graph of the function f(x)=e−0.5x2 by using graphing strategy.

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Chapter 4.4, Problem 52E

Chapter 4.4, Problem 54E

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Chapter 4 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

Ch. 4.1 - Matched Problem 1 Repeat Example 1 for f(x) = x2 ...Ch. 4.1 - Matches Problem 2 Find the critical numbers of f,...Ch. 4.1 - Matched Problem 3 Find the critical numbers of f,...Ch. 4.1 - Matched Problem 4 Find the critical numbers of f,...Ch. 4.1 - Matched Problem 5 Find the critical numbers of f,...Ch. 4.1 - Prob. 6MP Ch. 4.1 - Matched Problem 7 Given f(x) = x3 9x2 + 24x 10,...Ch. 4.1 - Matched Problem 8 The graph in Figure 13...Ch. 4.1 - Matched Problem 9 The graph of the total revenue...Ch. 4.1 - Skills Warm-up Exercises In Problems 1-8, inspect...

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The graph of the function f ( x ) = e − 0.5 x 2 by using graphing strategy.

Solution Summary: The author illustrates the graph of the function f(x)=e-0.5x

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To sketch: The graph of the function f(x)=e−0.5x2 by using graphing strategy.

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Chapter 4 Solutions