Question

Sketch graphs of f (x)= 1/x and its first five derivatives. Then use the slope-height behavior of the graphs to verify that each is the associated slope function of the one before.

Short answer

The first five derivateive of the function is the associated slope function of the one before.

Step-by-step solution

Step 1. Given information is:

f(x) = 1/x

Step 2. Drawing graphs

$$\begin{gathered} \mathrm{The}\mathrm{first}\mathrm{five}\mathrm{derivative}\mathrm{are}: \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=-\frac{1}{{\mathrm{x}}^2},\mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right)=\frac{2}{{\mathrm{x}}^3},\mathrm{f}\text{'}\text{'}\text{'}\left(\mathrm{x}\right)=-\frac{6}{{\mathrm{x}}^4}, \\ \mathrm{f}\text{'}\text{'}\text{'}\text{'}\left(\mathrm{x}\right)=\frac{24}{{\mathrm{x}}^5},\mathrm{f}\text{'}\text{'}\text{'}\text{'}\text{'}\left(\mathrm{x}\right)=-\frac{120}{{\mathrm{x}}^6} \end{gathered}$$

The graph of the function can be drawn as below:

Step 3. Observing slope function:

$$\begin{gathered} \mathrm{For}\mathrm{the}\mathrm{graph}\mathrm{of}\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\mathrm{x}},\mathrm{f}\left(\mathrm{x}\right)\mathrm{decreases}\mathrm{from}0\mathrm{to}\mathrm{infinity}\mathrm{as}\mathrm{x}\mathrm{increases}\mathrm{to}0. \\ \mathrm{For}\mathrm{x}>0,\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{f}\left(\mathrm{x}\right)\mathrm{decreases}\mathrm{from}\mathrm{infinity}\mathrm{to}0. \\ \mathrm{Therefore},\mathrm{slope}\mathrm{function}\mathrm{can}\mathrm{be}\mathrm{drawn}\mathrm{as}\mathrm{below}: \end{gathered}$$

$$\begin{gathered} \mathrm{For}\mathrm{the}\mathrm{graph}\mathrm{of}\mathrm{f}\text{'}\left(\mathrm{x}\right)=-\frac{1}{{\mathrm{x}}^2},\mathrm{f}\text{'}\left(\mathrm{x}\right)\mathrm{decreases}\mathrm{from}0\mathrm{to}\mathrm{infinity}\mathrm{as}\mathrm{x}\mathrm{increases}\mathrm{to}0. \\ \mathrm{For}\mathrm{x}>0,\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{f}\left(\mathrm{x}\right)\mathrm{increases}\mathrm{from}\mathrm{negative}\mathrm{infinity}\mathrm{to}0. \\ \mathrm{Therefore},\mathrm{slope}\mathrm{function}\mathrm{can}\mathrm{be}\mathrm{drawn}\mathrm{as}\mathrm{below}: \end{gathered}$$

Step 4. Drawing other Slope functions:

$\mathrm{Similarly},\mathrm{slope}\mathrm{functions}\mathrm{for}\mathrm{other}\mathrm{derivatives}\mathrm{are}:$