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Analyzing Tangents, Critical Points, and Continuity in Function Graphs

This guide explores the properties of functions through their derivatives and graphs. It focuses on finding the slope of the tangent line at various points on the curve, identifying critical numbers, and examining intervals where the function is increasing or decreasing. The analysis includes interpreting the positions of local maxima and minima, as well as recognizing where horizontal tangents occur. Additionally, it discusses the relationship between differentiability and continuity, highlighting scenarios where derivatives may not exist.

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Analyzing Tangents, Critical Points, and Continuity in Function Graphs

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Presentation Transcript

  1. Calculator Review

  2. Find the slope of the tangent line to the curve at

  3. Interpreting Graphs

  4. The graph of g is shown. Where does g have critical numbers? On what intervals is g’ negative? Positive? Where does the graph have a relative maximum? Minimum?

  5. The graph of h’ is shown. Where is the graph of h increasing? Where is the graph of h decreasing? Where does the graph have a local maximum? Where does the graph have a local minimum? Where does the graph have a horizontal tangent?

  6. Differentiability 3.2

  7. If f has a derivative at x = a, then f is continuous at x = a. Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

  8. To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: corner cusp discontinuity vertical tangent

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