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Chapter 3: Problem 32

Determine all significant features (approximately if necessary) and sketch a graph. $$f(x)=\tan ^{-1}\left(\frac{1}{x^{2}-1}\right)$$

Short Answer

Expert verified
This is a continuous process that is meant to provide an approximate sketch. The exact values will be dependent on methods used in steps 1 through 4. You will get the roots, local extrema, y-intercept, and where the function increases or decreases.

Step by step solution

01

Finding the Roots of the Function

Set the function equal to zero and solve for \(x\). This can be done by factoring or using the rational root theorem, synthetic division, or long division.
02

Finding the Local Extrema

Take the derivative of the function to obtain \(f'(x)\) and find its roots by setting it equal to zero. These points are where the slope of the function is zero, making them potential locations of local maxima and minima. Instance, if \(f'(a) > 0\) and \(f'(a) < 0\), then \(a\) is a local maximum.
03

Determine Where the Function is Increasing or Decreasing

Further analyze the intervals determined by the roots of \(f'(x)\) by selecting a test value in each interval and substituting it into \(f'(x)\). If \(f'(test value) > 0\), the function is increasing on that interval. If \(f'(test value) < 0\), the function is decreasing on that interval.
04

Find the Y-Intercept

The y-intercept can be found by setting \(x\) equal to zero in the original function to you get \(y\). This gives the point (0,\(y\)) where the function intersects the y-axis.
05

Sketching the Graph

Plot the roots (x-intercepts), y-intercept, and local extrema on a graph, indicating where the function is increasing and decreasing. Connect these points with a smooth curve to draw the graph of the function. Make sure the curve matches the behavior described by the derivative on each interval.

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Most popular questions from this chapter

In this exercise, we will explore the family of functions \(f(x)=x^{3}+c x+1,\) where \(c\) is constant. How many and what types of local extrema are there? (Your answer will depend on the value of \(c .\) ) Assuming that this family is indicative of all cubic functions, list all types of cubic functions.

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