use the graph to estimate the derivative at $x_1 = 0$ and $x_2 = 2$.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Graphical Derivatives", "Tangent Line Estimation" ], "new_concepts": [], "current_concepts": [ "Graphical Derivatives", "Tangent Line Estimation" ] } ## Estimate the derivative at $x_1 = 0$ $$ \begin{aligned} &\text{At } x_1 = 0, \text{ the graph of } y = g(x) \text{ has a local maximum (vertex) at } (0, -1).\ &\text{The tangent line at a local maximum is horizontal.}\ &g'(0) = 0 \end{aligned} $$ ## Estimate the derivative at $x_2 = 2$ $$ \begin{aligned} &\text{At } x_2 = 2, \text{ the tangent line is drawn in red.}\ &\text{Identify two points on the red tangent line: } (0, 3) \text{ and } (2, -5).\ &g'(2) \approx \frac{-5 - 3}{2 - 0} = \frac{-8}{2} = -4 \end{aligned} $$ At $x_1 = 0$, the estimated derivative is $0$. At $x_2 = 2$, the estimated derivative is $-4$. { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Calculus", "Graphical Derivatives" ] }

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Estimate the derivative at \(x_1 = 0\)

Estimate the derivative at \(x_2 = 2\)

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