Activity 5.1.4.
Point your browser to the Desmos worksheet at http://gvsu.edu/s/0zv. In what follows, we explore the behavior of power functions \(y = x^n\) where \(n \le -1\text{.}\)
(a)
Press the “play” button next to the slider labeled “\(n\text{.}\)” Watch two loops of the animation and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.
(b)
Click the icons next to each of the following 8 functions so that you can see all of \(y = x^{-1}\text{,}\) \(y = x^{-2}\text{,}\) \(\ldots\text{,}\) \(y = x^{-8}\) graphed at once. On the interval \(1 \lt x\text{,}\) how do the functions \(x^a\) and \(x^b\) compare if \(a \lt b\text{?}\) (Be careful with negative numbers here: e.g., \(-3 \lt -2\text{.}\))
(c)
How do your answers change on the interval \(0 \lt x \lt 1\text{?}\)
(d)
Uncheck the icons on each of the 8 functions to hide their graphs. Click the settings icon to change the domain settings for the axes, and change them to \(-10 \le x \le 10\) and \(-10,000 \le y \le 10,000\text{.}\) Play the animation through twice and then discuss the trends that you observe. Write a careful sentence each for at least two different trends.
(e)
Explain why \(\lim_{x \to \infty} \frac{1}{x^n} = 0\) for any choice of \(n = 1, 2, \ldots\text{.}\)
