question 2\nif the tangent line to y = f(x) at (8, 8) passes through the point (6, -32), find f(8).\no f…

Explanation:

Step1: Recall slope - formula for a line

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Identify the two points

The tangent line to $y = f(x)$ at the point $(8,8)$ passes through $(8,8)$ and $(6,-32)$. Here $x_1 = 8,y_1 = 8,x_2=6,y_2=-32$.

Step3: Calculate the slope

$m=\frac{-32 - 8}{6 - 8}=\frac{-40}{-2}=20$.

Step4: Recall the relationship between the slope of the tangent line and the derivative

The slope of the tangent line to the curve $y = f(x)$ at $x = a$ is equal to $f^{\prime}(a)$. Here $a = 8$, so $f^{\prime}(8)$ is the slope of the tangent line.

Answer:

$f^{\prime}(8)=20$