To determine [tex]\( g'(9) \)[/tex], given that the tangent line to the graph of the function [tex]\( g \)[/tex] at the point [tex]\((9, 2)\)[/tex] passes through the point [tex]\((5, 7)\)[/tex], we'll follow these steps:
1. Identify the coordinates:
- Point 1: [tex]\((9, 2)\)[/tex]
- Point 2: [tex]\((5, 7)\)[/tex]
2. Calculate the difference in the y-coordinates ([tex]\(\Delta y\)[/tex]):
[tex]\[
\Delta y = 7 - 2 = 5
\][/tex]
3. Calculate the difference in the x-coordinates ([tex]\(\Delta x\)[/tex]):
[tex]\[
\Delta x = 5 - 9 = -4
\][/tex]
4. Calculate the slope of the line passing through the two points:
The slope ([tex]\(m\)[/tex]) is given by the formula:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{5}{-4} = -1.25
\][/tex]
The slope of the tangent line at [tex]\( x = 9 \)[/tex] is the same as the derivative of [tex]\( g \)[/tex] at [tex]\( x = 9 \)[/tex]. Hence,
[tex]\[
g'(9) = -1.25
\][/tex]
