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Chapter 5: Problem 99

Use a graphing utility to graph \(y_{1}\) and \(y_{2}\) in the same viewing window. Use the graphs to determine whether \(y_{1}=y_{2}\) Explain your reasoning. $$y_{1}=\cos (x+2), \quad y_{2}=\cos x+\cos 2$$

Short Answer

Expert verified
The comparison of the graphs will either affirm that \(y_{1} = y_{2}\) if they coincide for all x, or deny it \(y_{1} \ne y_{2}\) if they do not.

Step by step solution

01

Graph the first function

To start, graph the first function \(y_{1}=\cos (x+2)\). This is the cosine function shifted to the left by two units. Using a graphing utility, input the function and generate the graph.
02

Graph the second function

Next, graph the second function \(y_{2}=\cos x+\cos 2\). This function seems to be a sum of the functions \(\cos x\) (cosine of x) and \(\cos 2\) (a constant). Generate the graph of \(y_{2}\) using the same graphing utility.
03

Compare the two graphs

With both graphs displayed on the same viewing window, visually inspect to see if \(y_{1}\) and \(y_{2}\) produce the same outputs for the same inputs across the domain.
04

Analyze and explain your reasoning

After comparing both graphs, if they coincide for all values of x, that will mean \(y_{1} = y_{2}\). Otherwise, if graphs don't coincide, it means \(y_{1} \ne y_{2}\), explain your observation based on how the graphs look.

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Most popular questions from this chapter

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\sin \left(x+\frac{\pi}{6}\right)-\sin \left(x-\frac{7 \pi}{6}\right)=\frac{\sqrt{3}}{2}$$

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Let \(x=\pi / 3\) in the identity in Example 8 and define the functions \(f\) and \(g\) as follows. \begin{array}{l}f(h)=\frac{\sin [(\pi / 3)+h]-\sin (\pi / 3)}{h} \\\g(h)=\cos \frac{\pi}{3}\left(\frac{\sin h}{h}\right)-\sin \frac{\pi}{3}\left(\frac{1-\cos h}{h}\right)\end{array} (a) What are the domains of the functions \(f\) and \(g ?\) (b) Use a graphing utility to complete the table.$$\begin{array}{|l|l|l|l|l|l|l|}\hline h & 0.5 & 0.2 & 0.1 & 0.05 & 0.02 & 0.01 \\\\\hline f(h) & & & & & & \\\\\hline g(h) & & & & & & \\\\\hline\end{array}$$. (c) Use the graphing utility to graph the functions \(f\) and \(g\). (d) Use the table and the graphs to make a conjecture about the values of the functions \(f\) and \(g\) as \(h \rightarrow 0^{+}\).

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