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Chapter 4: Problem 94

Use a graphing utility to graph the function. $$f(x)=\frac{\pi}{2}+\cos ^{-1}\left(\frac{1}{\pi}\right)$$

Short Answer

Expert verified
The function \(f(x) = \frac{\pi}{2} + \cos^{-1}\left(\frac{1}{\pi}\right)\) graphs as a straight line in a graphing utility at \(y = 2.8239\), indicating it's a constant function

Step by step solution

01

Find the value of the constant term

The constant term is \(\frac{\pi}{2}\). This value is approximately 1.5708. This can be done by using a calculator or known approximations of pi.
02

Find the value of \(\cos^{-1}\left(\frac{1}{\pi}\right)\)

First find the value of \(\frac{1}{\pi}\), which is approximately 0.3183. Then find the inverse cosine (or arc cosine) of this value. This can be done using a calculator. The value of \(\cos^{-1}\left(\frac{1}{\pi}\right)\) is approximately 1.2531.
03

Add the two values together

Now add together the two values to get the output of the function \(f(x)\), \(f(x) = 1.5708 + 1.2531 = 2.8239\). This means the function \(f(x)\) is constant and equals to 2.8239 for every \(x\).
04

Graph the function

Using a graphing utility, graph the function. Since the function is constant, the graph is a straight line at \(y = 2.8239\), which means that regardless of the \(x\) value, the output is always the same, 2.8239.

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Most popular questions from this chapter

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) As \(x \rightarrow 0^{+},\) the value of \(f(x) \rightarrow\) (b) As \(x \rightarrow 0^{-},\) the value of \(f(x) \rightarrow\) (c) As \(x \rightarrow \pi^{+},\) the value of \(f(x) \rightarrow\) (d) As \(x \rightarrow \pi^{-},\) the value of \(f(x) \rightarrow\) $$f(x)=\cot x$$

Converting to \(\mathrm{D}^{\circ} \mathrm{M}^{\prime} \mathrm{S}^{\prime \prime}\) Form \(\quad\) Convert each angle measure to degrees, minutes, and seconds without using a calculator. Then check your answers using a calculator. (a) \(240.6^{\circ}\) (b) \(-145.8^{\circ}\)

Graph the functions \(f\) and \(g .\) Use the graphs to make a conjecture about the relationship between the functions. $$f(x)=\sin ^{2} x, \quad g(x)=\frac{1}{2}(1-\cos 2 x)$$

Fill in the blank. If not possible, state the reason. $$\text { As } x \rightarrow-1^{+}, \text {the value of } \arccos x \rightarrow\text { _____ } .$$

Fill in the blank. If not possible, state the reason. $$\text { As } x \rightarrow-1^{+}, \text {the value of } \arcsin x \rightarrow\text { _____ } .$$
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