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Applications of Differentiation

Applications of Differentiation. 3. 3.1. Maximum and Minimum Values. Maximum and Minimum Values. Some of the most important applications of calculus are optimization problems , which find the optimal way of doing something. Meaning: find the maximum or minimum values of some function.

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Applications of Differentiation

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  1. Applications of • Differentiation • 3

  2. 3.1 • Maximum and Minimum Values

  3. Maximum and Minimum Values • Some of the most important applications of calculus are optimization problems, which find the optimal way of doing something. Meaning: find the maximum or minimum values of some function. • So what are maximum and minimum values? • Example: • highest point on the graph of the function f shown is the point (3, 5) so the largest value off is f(3) = 5. • Likewise, the smallest value is f(6) = 2. • Figure 1

  4. Absolute Maximum and Minimum • The maximum and minimum values of f are called extreme values of f.

  5. Local Maximum and Minimum • Something is true near c means that it is true on some open interval containing c.

  6. Example 1

  7. Example 2 • Figure 2 shows the graph of a function f with several extrema: • At a : absolute minimum is f(a) • At b : local maximum is f(b)At c : local minimum is f(c)At d : absolute (also local) maximum is f(d) • At e : local minimum is f(e) • Abs min f(a), abs max f(d) • loc min f(c) , f(e), loc max f(b), f(d) • Figure 2

  8. Practice! • Define all local and absolute extrema of the graph below

  9. Maximum and Minimum Values • The following theorem gives conditions under which a function is guaranteed to have extreme values.

  10. Maximum and Minimum Values • The Extreme Value Theorem is illustrated in Figure 7. • Note that an extreme value can be taken on more than once. • Figure 7

  11. Maximum and Minimum Values • The Extreme Value Theorem says that a continuous function on a closed interval has a maximum value and a minimum value, but it does not tell us how to find these extreme values. We start by looking for local extreme values. • Figure 10 shows the graph of a function f with a local maximum at c and a local minimum at d. • Figure 10

  12. Counter examples: f’ exists but there is no local min or max f’ doesn’t exist but there is a local min or max

  13. Maximum and Minimum Values • To find an absolute maximum or minimum of a continuous function on a closed interval, we note that either it is local or it occurs at an endpoint of the interval. • Thus the following three-step procedure always works.

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