Question

use the graph to estimate the derivative at \\(x_1 = -2\\) and \\(x_2 = 0\\).

Explanation:

Estimate the derivative at \(x_1 = -2\)

The derivative \(g'(-2)\) is the slope of the tangent line to the curve \(y = g(x)\) at \(x = -2\).
Looking at the graph, a tangent line (colored red) is drawn at the point \((-2, 3)\).
We find two points on this red line to calculate its slope:

• Point 1: \((-2, 3)\)
• Point 2: \((0, -1)\)

Using the slope formula:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 3}{0 - (-2)} = \frac{-4}{2} = -2 $$

Thus, \(g'(-2) \approx -2\).

Estimate the derivative at \(x_2 = 0\)

The derivative \(g'(0)\) is the slope of the tangent line to the curve \(y = g(x)\) at \(x = 0\).
Looking at the graph, the vertex of the parabola is at \((0, 1)\).
The tangent line at the vertex of a parabola is a horizontal line.
The slope of a horizontal line is \(0\).
Thus, \(g'(0) \approx 0\).

Answer:

At \(x_1 = -2\), the estimated derivative is \(-2\).

At \(x_2 = 0\), the estimated derivative is \(0\).