Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 4 - Section 4.5 - Summary of Curve Sketching - 4.5 Exercises - Page 322: 34

Answer

A. The domain is $(-\infty, \infty)$ B. The y-intercept is $1$ The x-intercept is $-0.74$ C. The function is not an odd function or an even function. D. $\lim\limits_{x \to -\infty}(x+cos~x) = -\infty$ $\lim\limits_{x \to \infty}(x+cos~x) = \infty$ No asymptotes. E. The function is increasing on the interval $(-\infty, \infty)$ F. There is no local maximum or local minimum. G. The graph is concave down on the intervals $(\frac{3\pi}{2}+2\pi~n, \frac{5\pi}{2}+2\pi~n)$, where $n$ is an integer. The graph is concave up on the intervals $(\frac{\pi}{2}+2\pi~n, \frac{3\pi}{2}+2\pi~n)$, where $n$ is an integer. The points of inflection are $(\frac{\pi}{2}+\pi~n, \frac{\pi}{2}+\pi~n)$, where $n$ is an integer. H. We can see a sketch of the curve below.
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Work Step by Step

$y = x+cos~x$ A. The function is defined for all real numbers. The domain is $(-\infty, \infty)$ B. When $x=0$, then $y = (0)+cos~0 = 1$ The y-intercept is $1$ When $y = 0$: $x+cos~x = 0$ $cos~x = -x$ $x \approx -0.74$ The x-intercept is $-0.74$ C. The function is not an odd function or an even function. D. $\lim\limits_{x \to -\infty}(x+cos~x) = -\infty$ $\lim\limits_{x \to \infty}(x+cos~x) = \infty$ There are no asymptotes. E. We can find values of $x$ such that $y' = 0$: $y' = 1-sin~x = 0$ $sin~x = 1$ $x = \frac{\pi}{2}+2\pi~n,$ where $n$ is an integer When $0 \lt x \lt \frac{\pi}{2}$ or $\frac{\pi}{2} \lt x \lt 2\pi$, then $y' \gt 0$ The function is increasing on the interval $(-\infty, \infty)$ F. Since the function is increasing on all intervals, there is no local maximum or local minimum. G. We can find the values of $x$ such that $y'' = 0$: $y'' = -cos~x = 0$ $cox~x = 0$ $x = \frac{\pi}{2}+\pi~n,$ where $n$ is an integer The graph is concave down on the intervals $(\frac{3\pi}{2}+2\pi~n, \frac{5\pi}{2}+2\pi~n)$, where $n$ is an integer. The graph is concave up on the intervals $(\frac{\pi}{2}+2\pi~n, \frac{3\pi}{2}+2\pi~n)$, where $n$ is an integer. When $x= \frac{\pi}{2}+\pi~n$: $y = (\frac{\pi}{2}+\pi~n)+cos~(\frac{\pi}{2}+\pi~n)$ $y = \frac{\pi}{2}+\pi~n+0$ $y = \frac{\pi}{2}+\pi~n$ The points of inflection are $(\frac{\pi}{2}+\pi~n, \frac{\pi}{2}+\pi~n)$, where $n$ is an integer. H. We can see a sketch of the curve below.
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