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Using Derivatives to Analyze Graphs

Using Derivatives to Analyze Graphs. Lesson 7.5. Theorems. Asdf Parabola Vertex Theorem: vertex at. Example 1 . Use the derivative of f(x) = –8x 2 + 4x + 7 to determine the intervals on which f is increasing and the intervals on which f is decreasing. f’(x) = -16x + 4

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Using Derivatives to Analyze Graphs

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  1. Using Derivatives to Analyze Graphs Lesson 7.5

  2. Theorems Asdf • Parabola Vertex Theorem: vertex at

  3. Example 1 Use the derivative of f(x) = –8x2 + 4x + 7 to determine the intervals on which f is increasing and the intervals on which f is decreasing. f’(x) = -16x + 4 -4 (4x – 1) x = ¼  Inflection point (-∞, ¼) increasing (¼, ∞) decreasing

  4. Example 2 What is the maximum height reached by the baseball with height h(t) = 55t - 16 t2 + 4, t seconds after being hit? Vertex : h(1.73) = 55(1.73) – 16(1.73)2 + 4 51.3 feet

  5. Example 3 A family wishes to enclose a rectangular yard abutting their house with 100 feet of fencing. Their house will be one side of the yard. • a. What is the maximum area that can be fenced in this way? A = lw A = 2l(100-l) A = 200l – 2l2 vertex = So, 50 * 25 = 1250 ft2

  6. Example 4 When , is f increasing, decreasing, or both on the interval from 0 to 4. F’(0) = -6 F’(4) = 62 – 144 + 16 – 6 = -72 If you graph, you will notice all values between 0 and 4 are negative.

  7. Homework Pages 448 – 449 3, 5 – 7, 9 - 14

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