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Omniguides by NOX
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Graph shape and local approximation

What students will learn

Use derivative evidence and theorem hypotheses to explain extrema, concavity, graphs, and nearby estimates.

Before this topic

  • Differentiation tools
  • Continuity and interval domains

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Unit 5: Analytical Applications of DifferentiationUnit 4: Contextual Applications of Differentiation

Read and connect

  1. Read: Using the Mean Value Theorem →

    Verify the Mean Value Theorem hypotheses before linking an average rate to an interior instantaneous rate.

  2. Read: Extreme Value Theorem, Global Versus Local Extrema, and Critical Points →

    Distinguish attained absolute extrema, local extrema, and interior critical numbers from endpoints.

  3. Read: Determining Intervals on Which a Function Is Increasing or Decreasing →

    Use a complete derivative sign chart to justify increasing and decreasing intervals.

  4. Read: Using the First Derivative Test to Determine Relative (Local) Extrema →

    Use one-sided derivative sign changes to classify a local extremum, including a continuous corner.

  5. Read: Using the Candidates Test to Determine Absolute (Global) Extrema →

    Compare original-function values at all endpoints and interior critical numbers on a closed interval.

  6. Read: Determining Concavity of Functions over Their Domains →

    Use changing slope to determine concavity and verify an actual concavity change at a point of the graph.

  7. Read: Using the Second Derivative Test to Determine Extrema →

    Apply the second derivative test at stationary points and use a different argument when it is inconclusive.

  8. Read: Sketching Graphs of Functions and Their Derivatives →

    Build a labeled calculus sketch from slope signs, concavity, domain behavior, and a function anchor.

  9. Read: Connecting a Function, Its First Derivative, and Its Second Derivative →

    Connect heights and slopes across a function and its first two derivatives without conflating their outputs.

  10. Read: Approximating Values of a Function Using Local Linearity and Linearization →

    Construct a nearby tangent approximation and use interval concavity to justify its error direction.

  11. Read: Exploring Behaviors of Implicit Relations →

    Analyze local implicit branches, tangent candidates, and second derivatives with point and domain checks.

Check understanding

Use the complete prompt to make student reasoning visible, then consult the separate worked reasoning and criteria. A reading completion tick records reading history.

Standards and earlier reading lists

These topic placements organize reading. Edition-qualified Standards mappings and the existing complete unit task sets retain their reviewed scope.

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