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 = e x x . Obtain the x -intercept as follows. Substitute 0 for y in y = e x x . 0 = e x x e x ≠ 0 Thus, there is no x -intercept. Obtain the y -intercept as follows. Substitute 0 for x in y = e x x . y = e 0 0 = 1 0 = undefined Thus, there is no y -intercept. 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 = e x x All the x ’s are replaced with – x in the given function. y = e − x − x = − e − x x All the y ’s are replaced with –y and all the x ’s are replaced with – x in the given function. − y = e − x − x − y = − e − x x y = e − x x By the above result, there is no symmetry. Obtain the horizontal asymptote as follows. lim x → + ∞ y = lim x → + ∞ e x x = ∞ lim x → − ∞ y = lim x → − ∞ e x x = 0 Thus, the negatively horizontal asymptote is y = 0. Obtain the vertical asymptote as follows. Take the denominator and equal to zero. x = 0 Thus, the vertical asymptote is x = 0. The given function is defined for all x except x = 0. Thus, the domain of the function is ( − ∞ , 0 ) ∪ ( 0 , ∞ ) . Obtain the first derivative of y as follows. d y d x = d d x ( e x x ) = x d d x ( e x ) − ( e x ) d d x ( x ) x 2 = x ( e x ) − ( e x ) ( 1 ) x 2 = e x ( x − 1 ) x 2 Obtain the second derivative of y as follows

Basic Technical Mathematics with Calculus (11th Edition)

11th Edition

ISBN: 9780134437736

Author: Allyn J. Washington, Richard Evans

Publisher: PEARSON

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Chapter 27.8, Problem 6E

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Chapter 27 Solutions

Basic Technical Mathematics with Calculus (11th Edition)

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