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Chapter 3

Chapter 3. Applications of Differentiation. Definition of Extrema. Figure 3.1. Theorem 3.1 The Extreme Value Theorem. Extreme Value Theorem. If f is continuous on [ a,b ] then f must have both an absolute maximum and absolute minimum value in the interval. Examples:.

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Chapter 3

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  1. Chapter 3 Applications of Differentiation

  2. Definition of Extrema

  3. Figure 3.1

  4. Theorem 3.1 The Extreme Value Theorem

  5. Extreme Value Theorem If f is continuous on [a,b] then f must have both an absolute maximum and absolute minimum value in the interval. Examples:

  6. Extreme Value Theorem If f is continuous on [a,b] then f must have both an absolute maximum and absolute minimum value in the interval. Examples: Absolute Maximum Abs Max Abs Max Abs Min Abs Min Absolute Minimum

  7. Extreme values can be in the interior or the end points of a function. No Absolute Maximum Absolute Minimum

  8. No Maximum No Minimum

  9. Definition of Relative Extrema

  10. Local Extreme Values: A local maximum is the maximum value within some open interval. A local minimum is the minimum value within some open interval.

  11. Figure 3.2

  12. Local extremes are also called relative extremes. Absolute maximum (also local maximum) Local maximum Local minimum Local minimum Absolute minimum (also local minimum)

  13. Local extremes are also called relative extremes. Absolute maximum (also local maximum) Local maximum Local minimum Local minimum Absolute minimum (also local minimum)

  14. Notice that local extremes in the interior of the function occur where is zero or is undefined. Absolute maximum (also local maximum) Local maximum Local minimum

  15. Definition of a Critical Number and Figure 3.4

  16. Theorem 3.2 Relative Extrema Occur Only at Critical Numbers Note: Maximum and minimum points in the interior of a function always occur at critical points, but critical points are not always maximum or minimum values.

  17. Guidelines for Finding Extrema on a Closed Interval

  18. EXAMPLE 3FINDING ABSOLUTE EXTREMA Find the absolute maximum and minimum values of on the interval . There are no values of x that will make the first derivative equal to zero. The first derivative is undefined at x=0, so (0,0) is a critical point. Because the function is defined over a closed interval, we also must check the endpoints.

  19. At: At: At: To determine if this critical point is actually a maximum or minimum, we try points on either side, without passing other critical points. Since 0<1, this must be at least a local minimum, and possibly a global minimum.

  20. At: Absolute minimum: Absolute maximum: At: At: To determine if this critical point is actually a maximum or minimum, we try points on either side, without passing other critical points. Since 0<1, this must be at least a local minimum, and possibly a global minimum.

  21. Critical points are not always extremes! (not an extreme)

  22. (not an extreme) p

  23. Locate the extrema of the function on [-5, 10] 1. Find f’(x) = 0 2. Test the critical points and the endpoints in the original function. Absolute max at (3, 40.5) Absolute min at (8, -181.3) 3. Identify the highest and lowest points.

  24. max min

  25. Find the extrema on [-5, -2] 1. Find f’(x) = 0 2. Test the critical points and the endpoints in the original function. We don’t use x = .737 since it’s not in the interval. f(-5) = 0 f(-4.07) = 7.035 f(.737) = -48.52 f(-2) = -15 3. Identify the highest and lowest points. Absolute max (-4.07. 7.035) Absolute min (-2, -15)

  26. max min

  27. Find the extrema of

  28. Find the extrema of the function. 1. Find f’(x) = 0 2. Test the critical points and the endpoints in the original function. 3. Identify the highest and lowest points. Absolute max (-2, 2) Absolute min (1, -7)

  29. max min

  30. Or you can set the equation equal to 0 and look for the critical number!!!!!!

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