In Exercises 11-14, match the table with a graph. x f^'(x) a does not exist

step 1 Okay, so we want to match the function where at A, the derivative does not exist, B, the derivat

In Exercises 11-14, match the table with a graph. x ...

In Exercises $11-14,$ match the table with a graph.
$$
\begin{array}{cc}{x} & {f^{\prime}(x)} \\ \hline a & {\text { does not exist }} \\ {b} & {\text { does not exist }} \\ {c} & {-1.7} \\ \hline\end{array}
$$

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Key Concepts

Non-Differentiable Points

These are points on a graph where the function does not have a well-defined tangent line because the derivative does not exist. Common reasons include corners, cusps, or vertical tangents. In a table of derivative values, a note that the derivative 'does not exist' highlights these special points which require careful graphical interpretation.

Differentiability

This concept refers to whether a function has a derivative at a particular point. A function is differentiable at a point if its derivative exists there, meaning the limit defining the derivative is finite and the function is locally linear. When the derivative does not exist at a point, it often indicates a corner, cusp, vertical tangent, or discontinuity in the graph.

Graphical Interpretation of Derivatives

The derivative of a function at a given point represents the slope of the tangent line to the graph at that point. When matching a derivative table to a graph, one needs to identify points where the graph has specific slopes and where the tangent line behavior might be abrupt or undefined, signaling non-differentiability.

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