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The Derivative

The Derivative. Integrated Math 4 Mrs. Tyrpak. Rate of Change: Slope. Line: constant rate of change Curves: the rate at which the graph rises or falls changes from point to point Visual:. Rate of Change: Slope of Tangent Line.

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The Derivative

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  1. The Derivative Integrated Math 4 Mrs. Tyrpak

  2. Rate of Change: Slope Line: constant rate of change Curves: the rate at which the graph rises or falls changes from point to point Visual:

  3. Rate of Change: Slope of Tangent Line The tangent line to a graph of a function at a point (x, y) is the line that best approximates the slope of the graph at that point. Examples:

  4. How do we calculate this? We will combine two previous ideas: ___________ and ___________

  5. Formal Slope Definition The slope, m, of the graph of f at the point (x, f(x)) is equal to the slope of its tangent line at (x, f(x)), and is given by Provided this limit exists!

  6. Example #1: Find the slope of the graph of at the point (-2, 4).

  7. Example #2: Find the slope of f(x) = -2x + 4

  8. Example #3: Find a formula for the slope of the graph of What is the slope at (-1, 2) and (2, 5)?

  9. Derivative The derivative of f at x is Provided this limit exists.

  10. Example #4: Find the derivative of

  11. Example #5: Find the derivative of . Then, find the slope of f at points (1,1) and (4,2).

  12. You are almost done  Persevere a little longer! This unit will prepare you for future math courses whether they be high school calculus or collegiate math. Don’t forget to complete your worksheets!

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