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Chapter 4: Problem 86

Sketch a graph of the function. $$f(x)=\frac{\pi}{2}+\arctan x$$

Short Answer

Expert verified
The graph of the function will be a curve, shifted upwards by 蟺/2 units. The range of the y-values will now be from 0 to 蟺. As x approaches infinity, the function will tend towards 蟺, and as x approaches negative infinity, the function will tend towards 0. At x=0, the y-value will be 蟺/2.

Step by step solution

01

Understanding arctan graph

The arctan function, also known as the inverse tangent function, has a typical curved shape. As the values of x approach negative infinity, the function tends to -蟺/2 and as x tends to positive infinity, the function tends towards 蟺/2. In other words, we can say that the range of an arctan function is from -蟺/2 to 蟺/2.
02

Adding a constant to the function

Adding a constant to a function, \(f(x) = \arctan x + \frac{\pi}{2}\), will shift the function up by that constant value. This means the function will move up by 蟺/2 units in the y-direction.
03

Sketching the final graph

Now that we have this information, we can sketch the function. Like the arctan graph, this graph will have a curved shape. But because of the constant, the range will now be from 0 to 蟺. The function will flatten out to y=蟺 as x approaches infinity and will flatten out to y=0 as x approaches negative infinity. As x=0, f(x) will be 蟺/2. Plot a few more points as needed, applying the arctan to x-values, and then sketch the shifted curve connecting them.

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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) \(x \rightarrow\left(\frac{\pi}{2}\right)^{+}\) (b) \(x \rightarrow\left(\frac{\pi}{2}\right)^{-}\) (c) \(x \rightarrow\left(-\frac{\pi}{2}\right)^{+}\) (d) \(x \rightarrow\left(-\frac{\pi}{2}\right)^{-}\) $$f(x)=\sec x$$

Find the length of the sides of a regular hexagon inscribed in a circle of radius 25 inches.

Use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as \(x\) increases without bound. $$f(x)=\frac{1-\cos x}{x}$$

Determine whether the statement is true or false. Justify your answer. The Leaning Tower of Pisa is not vertical, but when you know the angle of elevation \(\theta\) to the top of the tower as you stand \(d\) feet away from it, you can find its height \(h\) using the formula \(h=d \tan \theta\)

Determine whether the statement is true or false. Justify your answer. You can obtain the graph of \(y=\csc x\) on a calculator by graphing the reciprocal of \(y=\sin x\)
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