To determine To sketch: The graph of the given function and check by displaying the graph on a calculator. 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: In example 2, the given function, change cos to sin. That is y = e − x sin x . Obtain the x -intercept as follows. Substitute 0 for y in y = e − x sin x . 0 = e − x sin x sin x = 0 ( ∵ e − x ≠ 0 ) x = n π where n is integer Thus, the x -intercept is ( 0 , 0 ) , ( π , 0 ) and ( 2 π , 0 ) . Obtain the y -intercept as follows. Substitute 0 for x in y = e − x sin x . y = e 0 sin ( 0 ) = 0 Thus, the y -intercept is (0, 0). 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 sin x All the x ’s are replaced with – x in the given function. y = e − ( − x ) sin ( − x ) = − e x sin 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 ) sin ( − x ) − y = − e x sin x y = e x sin x By the above result, there is no symmetry. Obtain the horizontal asymptote as follows. lim x → + ∞ y = lim x → + ∞ e − x sin x = lim x → + ∞ sin x e x = 0 Here, the range of the sine function is − 1 ≤ y ≤ 1 . Thus, the horizontal asymptote is y = 0. It is observed that the given function is no denominators. So, no vertical asymptotes. Obtain the first derivative of y as follows. d y d x = d d x ( e − x sin x ) = e − x d d x ( sin x ) + ( sin x ) d d x ( e − x ) = e − x ( cos x ) + ( sin x ) ( − e − x ) = e − x ( cos x − sin x ) Obtain the second derivative of y as follows

11th Edition

ISBN: 9780134437705

Author: Washington

Publisher: PEARSON

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Chapter 27.8, Problem 1E

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Chapter 27 Solutions

Basic Technical Mathematics

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