Explanation Features to be used in graphing functions: 1. Intercepts points for which the graph crosses each axis. 2. Tests for symmetry: a. A graph will have symmetry about the x -axis if it is same equation when all the y ’s are replaced with – y . b. A graph will have symmetry about the y -axis if it is same equation when all the x ’s are replaced with – x . c. A graph will have symmetry about the origin if it is same equation when all the y ’s are replaced with – y and all the x ’s are replaced with – x . 3. An asymptote is a line that the curve approaches as one of the variables approaches some particular value. 4. Vertical asymptotes: To find vertical asymptotes by finding values that make factors in the denominator zero. 5. Domain and range: The complete set of possible values of the independent variable is called the domain of the function and the complete set of all possible resulting values of the dependent variable is called the range of the function. 6. Derivatives we will find information from the first two derivatives. Calculation: The given function is y = 4 e − x 2 . Obtain the x -intercept as follows. Substitute 0 for y in y = 4 e − x 2 . 0 = 4 e − x 2 e − x 2 = 0 There is no x -intercept. Obtain the y -intercept as follows. Substitute 0 for x in y = 4 e − x 2 . y = 4 e − ( 0 ) 2 = 4 e 0 = 4 Thus, the y -intercept is (0, 4). Check whether the given function is symmetry about x -axis, y -axis, and origin: All the y ’s are replaced with – y in the given function. − y = 4 e − x 2 All the x ’s are replaced with – x in the given function. y = 4 e − ( − x ) 2 = 4 e − x 2 All the y ’s are replaced with –y and all the x ’s are replaced with – x in the given function. − y = 4 e − ( − x ) 2 − y = 4 e − x 2 By the above result, the given function is symmetry about y -axis. Obtain the horizontal asymptote as follows. lim x → ± ∞ y = lim x → ± ∞ 4 e − x 2 = lim x → ± ∞ 4 e x 2 = 0 Thus, the horizontal asymptote is y = 0. It is observed that the given function is no denominators. So, no vertical asymptotes. By the definition of domain, there is no undefined points. Thus, the domain of the function is set of all real numbers. Obtain the first derivative of y as follows. d y d x = d d x ( 4 e − x 2 ) = 4 d d x ( e − x 2 ) = 4 ( e − x 2 ( − 2 x ) ) = − 8 x e x 2 Obtain the second derivative of y as follows

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

11th Edition

ISBN: 9780137554843

Author: Allyn Washington, Richard Evans

Publisher: PEARSON+

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Chapter 27.8, Problem 9E

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Chapter 27.8, Problem 8E

Chapter 27.8, Problem 10E

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Chapter 27 Solutions

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

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