Q. 15 - Calculus

Question

In the text of this section we displayed graphs of f (x) = $x^4$ and its first five derivatives. 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:

$f\left(x\right) = x^4$

Step 2. Drawing graphs

$$\begin{gathered} \mathrm{The} \mathrm{first} \mathrm{five} \mathrm{derivative} \mathrm{are}: \\ \mathrm{f}\text{'}\left(\mathrm{x}\right) = 4x^3, \mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right) = 12{\mathrm{x}}^2, \mathrm{f}\text{'}\text{'}\text{'}\left(\mathrm{x}\right) = 24\mathrm{x}, \\ \mathrm{f}\text{'}\text{'}\text{'}\text{'}\left(\mathrm{x}\right) =24, \mathrm{f}\text{'}\text{'}\text{'}\text{'}\text{'}\left(\mathrm{x}\right) = 0 \end{gathered}$$

The graph of the function can be drawn as below:

Step 3. Slope function for f(x)

$$\begin{gathered} \mathrm{For} \mathrm{the} \mathrm{graph} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^4, \mathrm{f}\left(\mathrm{x}\right) \mathrm{decreases} \mathrm{as} \mathrm{x} \mathrm{increases} \mathrm{to} 0. \\ \mathrm{For} \mathrm{x}>0, \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right) \mathrm{increases}. \\ \mathrm{Therefore}, \mathrm{slope} \mathrm{function} \mathrm{can} \mathrm{be} \mathrm{drawn} \mathrm{as} \mathrm{below}: \end{gathered}$$

Step 4. Slope function for f'(x)

$$\begin{gathered} \mathrm{For} \mathrm{the} \mathrm{graph} \mathrm{of} \mathrm{f}\text{'}\left(\mathrm{x}\right)=4{\mathrm{x}}^3, \mathrm{f}\text{'}\left(\mathrm{x}\right) \mathrm{increases} \mathrm{for} \mathrm{all} \mathrm{x}. \\ \mathrm{Therefore}, \mathrm{slope} \mathrm{function} \mathrm{can} \mathrm{be} \mathrm{drawn} \mathrm{as} \mathrm{below}: \end{gathered}$$

Step 5. Slope function for f''(x)

$$\begin{gathered} \mathrm{For} \mathrm{the} \mathrm{graph} \mathrm{of} \mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right)=12{\mathrm{x}}^2, \mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right) \mathrm{decreases} \mathrm{for} \mathrm{x}<0. \\ \mathrm{For} \mathrm{x}>0, \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right) \mathrm{increases}. \\ \mathrm{Therefore}, \mathrm{slope} \mathrm{function} \mathrm{can} \mathrm{be} \mathrm{drawn} \mathrm{as} \mathrm{below}: \end{gathered}$$

Step 6. Slope function for f'''(x)

$$\begin{gathered} \mathrm{For} \mathrm{the} \mathrm{graph} \mathrm{of} \mathrm{f}\text{'}\text{'}\text{'}\left(\mathrm{x}\right)=24x, \mathrm{f}\text{'}\text{'}\text{'}\left(\mathrm{x}\right) \mathrm{increases} \mathrm{for} \mathrm{all} \mathrm{x}. \\ \mathrm{Therefore}, \mathrm{slope} \mathrm{function} \mathrm{can} \mathrm{be} \mathrm{drawn} \mathrm{as} \mathrm{below}: \end{gathered}$$

Step 7. Slope function for f''''(x) & f'''''(x)

$$\begin{gathered} \mathrm{For} \mathrm{the} \mathrm{graph} \mathrm{of} \mathrm{f}\text{'}\text{'}\text{'}\text{'}\left(\mathrm{x}\right)=24 and \mathrm{f}\text{'}\text{'}\text{'}\text{'}\text{'}\left(\mathrm{x}\right)=0. \\ \mathrm{Therefore}, \mathrm{slope} \mathrm{function} \mathrm{can} \mathrm{be} \mathrm{drawn} \mathrm{as} \mathrm{below}: \end{gathered}$$