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Activity9.7Making Graphs

Let's put it all together and produce some graphs.

Subsection9.7.1Exercises

Consider the function \(\fe{f}{x}=\frac{8x^2-8}{(2x-4)^2}\).

1

Evaluate each of the following limits: \(\lim\limits_{x\to\infty}\fe{f}{x}\), \(\lim\limits_{x\to-\infty}\fe{f}{x}\), \(\lim\limits_{x\to2^{-}}\fe{f}{x}\), and \(\lim\limits_{x\to2^{+}}\fe{f}{x}\).

2

What are the horizontal and vertical asymptotes for \(f\)'s graph?

3

What are the horizontal and vertical intercepts for \(f\)'s graph?

4

Use the formulas \(\fe{\fd{f}}{x}=\frac{4(1-2x)}{(x-2)^3}\) and \(\fe{\sd{f}}{x}=\frac{4(4x+1)}{(x-2)^4}\) to help you accomplish each of the following.

  • State the critical numbers of \(f\).

  • Create well-documented increasing/decreasing and concavity tables for \(f\).

  • State the local minimum, local maximum, and inflection points on \(f\). Make sure that you explicitly address all three types of points whether they exist or not.

5

Graph \(y=\fe{f}{x}\) onto Figure 9.7.1. Choose a scale that allows you to clearly illustrate each of the features found in Exercises 9.7.1.19.7.1.4. Label all axes and asymptotes well and write the coordinates of each local extreme point and inflection point next to the point.

Figure9.7.1\(y=\fe{f}{x}\)
6

Check your graph using a graphing calculator.

7

Following analysis similar to that implied in Exercises 9.7.1.19.7.1.5 graph the function \begin{equation*}\fe{g}{t}=\frac{1}{(e^t+4)^2}\text{.}\end{equation*} Use the formulas \begin{equation*}\fe{\fd{g}}{t}=\frac{-2e^t}{(e^t+4)^3}\qquad\fe{\sd{g}}{t}=\frac{4e^t(e^t-2)}{(e^t+4)^4}\text{.}\end{equation*} Check your graph using a graphing calculator.

Figure9.7.2\(y=\fe{g}{x}\)