2. (2 points) use the figure to find the values of a, b, c, and d about f at p, where f(a)=b and f(c)=d. a =…

Explanation:

Step1: Recall function - value meaning

The equation $f(a)=b$ means that when $x = a$, the value of the function $y = f(x)$ is $b$. Looking at the point $P=(4,1)$ on the curve $y = f(x)$, when $x=a = 4$, $y=b = 1$.

Step2: Recall derivative - meaning

The equation $f^{\prime}(c)=d$ means that the slope of the tangent line to the curve $y = f(x)$ at $x = c$ is $d$. The slope of the tangent - line passing through two points $(x_1,y_1)=(2,5)$ and $(x_2,y_2)=(4,1)$ is given by the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $c$ is the $x$ - value at the point of tangency (which is the $x$ - value of the point $P$), so $c = 4$, and the slope $d=\frac{1 - 5}{4 - 2}=\frac{-4}{2}=-2$.

Answer:

$a = 4$ $b = 1$ $c = 4$ $d=-2$