Question

use the graph of the function ( w(x) ) shown below to complete each task.
(a) ( w(2) = )
(b) ( w(0) )
(c) ( w(-2) )
(d) list all points on ( (-3, 4) ) at which ( w(x) ) is not differentiable:

Explanation:

Step1: Find $W'(2)$ (slope at $x=2$)

The segment through $x=2$ has endpoints $(2, -2)$ and $(4, -4)$. Slope formula: $\frac{y_2-y_1}{x_2-x_1} = \frac{-4 - (-2)}{4 - 2} = \frac{-2}{2} = -1$

Step2: Find $W'(0)$ (slope at $x=0$)

At $x=0$, the graph has a sharp corner (discontinuity in slope), so derivative does not exist.

Step3: Find $W'(-2)$ (slope at $x=-2$)

The curve at $x=-2$ is increasing, so its slope is positive, meaning $W'(-2) > W'(2)$ (since $W'(2)=-1$ is negative).

Step4: Identify non-differentiable points

A function is non-differentiable at: 1) sharp corners, 2) discontinuities, 3) vertical tangents. Here:

• $x=0$: sharp corner (slope changes abruptly)
• $x=2$: discontinuity (jump in function value)

Answer:

(a) $-1$
(b) dne
(c) $> W'(2)$
(d) $x=0$, $x=2$