Given:
f(x) = 2Ln(x)
x1 = 9
Let's find the slope given different values of x2.
Apply the formula:
[tex]m=\frac{f(x_2)-f(x_1)}{x_2-x_1}[/tex]• When x2 = 14.
First find the value of f(x) when x = 9.
Substitute x for 9 in f(x) and solve.
[tex]f(x_1)=f(9)=2\ln (9)=4.3944[/tex]Now, solve for f(x) when x = 14:
[tex]f(x_2)=f(14)=2\ln (14)=5.2781[/tex]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:
[tex]\begin{gathered} m=\frac{5.2871-4.3944}{14-9} \\ \\ m=\frac{0.8927}{5} \\ \\ m=0.1785 \end{gathered}[/tex]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:
[tex]f(x_2)=f(11)=2\ln (11)=4.7958[/tex]Now, to find the slope, we have:
[tex]\begin{gathered} m=\frac{4.7958-4.3944}{11-9} \\ \\ m=0.2007 \end{gathered}[/tex]When x2 = 11, m = 0.2007
• When x2 = 10:
[tex]f(x_2)=f(10)=2\ln (10)=4.6052[/tex]To find the slope, we have:
[tex]\begin{gathered} m=\frac{4.6052-4.3944}{10-9} \\ \\ m=0.2108 \end{gathered}[/tex]When x2 = 10, m = 0.2108
• When x2 = 9.1
[tex]f(x_2)=f(9.1)=2\ln (9.1)=4.4165[/tex]To find the slope, we have:
[tex]\begin{gathered} m=\frac{4.4165-4.3944}{9.1-9} \\ \\ m=0.2210 \end{gathered}[/tex]When x2 = 9.1, m = 0.2210
• When x2 = 9.01:
[tex]f(x_2)=f(9.01)=2\ln (9.01)=4.3967[/tex]To find the slope, we have:
[tex]\begin{gathered} m=\frac{4.3967-4.3944}{9.01-9} \\ \\ m=0.2270 \end{gathered}[/tex]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