Question

shown above is the graph of the differentiable function f, along with the line tangent to the graph of f at x = 2. what is the value of f(2)?

Explanation:

Step1: Recall derivative - tangent relationship

The derivative of a function $f(x)$ at a point $x = a$, $f^{\prime}(a)$, is equal to the slope of the tangent line to the graph of $y = f(x)$ at $x=a$.

Step2: Identify two points on the tangent line

From the graph, two points on the tangent - line at $x = 2$ can be chosen. Let's take the points $(0,2)$ and $(2,3)$.

Step3: Calculate the slope of the tangent line

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $x_1 = 0,y_1 = 2,x_2 = 2,y_2 = 3$. So, $m=\frac{3 - 2}{2 - 0}=\frac{1}{2}$.

Answer:

$\frac{1}{2}$