Q. 16 - Calculus

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.

Step-by-Step Solution

Answer

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

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:

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