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Analyzing Graphs Using Derivatives Lesson 7.5 Theorems

Learn to apply derivatives to analyze graphs using examples like finding intervals of increase and decrease, maximizing area with fencing, and determining maximum height of objects. Homework exercises provided.

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Analyzing Graphs Using Derivatives Lesson 7.5 Theorems

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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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