Question

For each slope below, enter the value correct to four decimal places.

Answer 1

Given:

f(x) = 2Ln(x)

x1 = 9

Let's find the slope given different values of x2.

Apply the formula:

`m=(f(x_2)-f(x_1))/(x_2-x_1)`

• When x2 = 14.

First find the value of f(x) when x = 9.

Substitute x for 9 in f(x) and solve.

`f(x_1)=f(9)=2\ln (9)=4.3944`

Now, solve for f(x) when x = 14:

`f(x_2)=f(14)=2\ln (14)=5.2781`

To find the slope, subtitute 4.3944 for f(x1), 5.2781 for f(x2), 9 for x1, and 14 for x2 in the slope formula and solve for m.

Thus, we have:

`\begin{gathered} m=(5.2871-4.3944)/(14-9) \\ \\ m=(0.8927)/(5) \\ \\ m=0.1785 \end{gathered}`

When x2 = 14, m = 0.1785

• When x2 = 11

Using the same method used in the first part above, we have:

f(x1) = 4.3944

To solve for f(x2), substitute 11 for x:

`f(x_2)=f(11)=2\ln (11)=4.7958`

Now, to find the slope, we have:

`\begin{gathered} m=(4.7958-4.3944)/(11-9) \\ \\ m=0.2007 \end{gathered}`

When x2 = 11, m = 0.2007

• When x2 = 10:

`f(x_2)=f(10)=2\ln (10)=4.6052`

To find the slope, we have:

`\begin{gathered} m=(4.6052-4.3944)/(10-9) \\ \\ m=0.2108 \end{gathered}`

When x2 = 10, m = 0.2108

• When x2 = 9.1

`f(x_2)=f(9.1)=2\ln (9.1)=4.4165`

To find the slope, we have:

`\begin{gathered} m=(4.4165-4.3944)/(9.1-9) \\ \\ m=0.2210 \end{gathered}`

When x2 = 9.1, m = 0.2210

• When x2 = 9.01:

`f(x_2)=f(9.01)=2\ln (9.01)=4.3967`

To find the slope, we have:

`\begin{gathered} m=(4.3967-4.3944)/(9.01-9) \\ \\ m=0.2270 \end{gathered}`

When x2 = 9.01, m = 0.2270

ANSWER:

• When x2 = 14, m = , 0.1785

,

• When x2 = 11, m = , 0.2007

,

• When x2 = 10, m = , 0.2108

,

• When x2 = 9.1, m = , 0.2210

,

• When x2 = 9.01, m = , 0.2270