consider the graph of y = f(x) shown below in blue. estimate the value of the derivative at x = - 1. then…

Explanation:

Step1: Recall derivative definition

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

Step2: Draw tangent at $x = - 1$

Draw the tangent line to the curve $y=f(x)$ at $x=-1$ on the given graph.

Step3: Estimate slope of tangent

Pick two points on the tangent line. Let's say the points are $(x_1,y_1)$ and $(x_2,y_2)$. The slope $m=\frac{y_2 - y_1}{x_2 - x_1}$. From the graph, if we assume two points on the tangent line at $x=-1$, we can estimate the slope. The curve is decreasing at $x = - 1$, and by visual - inspection, if we consider a small interval around $x=-1$, the slope of the tangent line seems to be approximately $-\frac{1}{2}$.

Answer:

$-0.5$