If the line tangent to the graph of the function $f$ at the point $(2,7)$ passes through the point $(-3,-3)$ then $f'(2)$ is...?
A. 5 B.1 C. 2 D.7 E. Undefined
I don't understand how to do this. I know that I can find the slope of the line tangent to the point $(2,7)$ but after I find that equation of the line how do I find $f'(2)$?
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$\begingroup$We can determine $f'(2)$ by knowing that the slope of the line tangent to $f(x)$ at $x=2$ is equal to $f'(2).$ You have two points on the tangent line: one is the point of tangency $(2, 7)$ and the other is the point the line also passes through $(-3, -3)$ and so you can compute the slope of the tangent line:
$$\text{slope} - \dfrac{7-(-3)}{2-(-3)} = \frac {10}{5} = 2.$$ So $f'(2) = 2.$
$\endgroup$ $\begingroup$Hint: What does the value of $f'(2)$ express?
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