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Mastering Curve Sketching: Derivatives Unveiled

Learn how to relate a graph to its derivative using derivatives instead of plotting points. Discover critical points, curvature, concavity, and more in this comprehensive guide. Dive into slope, inflection points, and absolute/relative extrema on graphs.

odette-rasmussen
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Mastering Curve Sketching: Derivatives Unveiled

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  1. Chapter 4 Curve Sketching Using Derivatives instead of Plotting Points

  2. Lets relate a graph to its derivative Find the first derivative. Put zeroes on a number line Find the second derivative. Put zeroes on a number line Lets look at the graph here:

  3. Slope y′ = slope, correct? Logic says (as did the graph): y′ > 0 m+ Graph rises y′ < 0 m– Graph falls y ′ = 0 m = 0 Graph hits peak or valley

  4. Curvature y′′ = curvature We saw on the graph y ′′ > 0 concave up y ′′ < 0 concave down y ′′ = 0 inflection point

  5. Some vocabulary Critical Point where y′ = f′ = 0 or DNE Absolute Maximum f(c) ≥ f(x) for all x in domain (highest point overall) Absolute Minimum f(c) ≤ f(x) for all x in domain (lowest point overall) Relative Maximum/Minimum f(c) ≥ f(x)/f(c) ≤ f(x) for x near c (high or low somewhere in a small area)

  6. How to graph this way? Lets see if we can figure it out, with an example: Please may I have another? Of course you may. 

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