Question

if the tangent line to y = f(x) at (4, 3) passes through the point (0, 2), find f(4) and f(4).
f(4) =
f(4) =

Explanation:

Step1: Recall function - value meaning

The point $(4,3)$ lies on the curve $y = f(x)$. By the definition of a function, if the point $(x_0,y_0)$ lies on the graph of $y = f(x)$, then $f(x_0)=y_0$. Here $x_0 = 4$ and $y_0 = 3$, so $f(4)=3$.

Step2: Recall derivative - as - slope meaning

The derivative $f^{\prime}(x_0)$ is the slope of the tangent line to the curve $y = f(x)$ at the point $(x_0,f(x_0))$. The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. The tangent line to $y = f(x)$ at $(4,3)$ passes through $(4,3)$ and $(0,2)$. Using the slope formula with $(x_1,y_1)=(0,2)$ and $(x_2,y_2)=(4,3)$, we have $m=\frac{3 - 2}{4 - 0}=\frac{1}{4}$. So $f^{\prime}(4)=\frac{1}{4}$.

Answer:

$f(4)=3$
$f^{\prime}(4)=\frac{1}{4}$