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1007slides_Part4

# 1007slides_Part4 - Math 1007 Fall 2006(These slides replace...

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Math 1007, Fall 2006 (These slides replace neither the text book nor the lectures.) Part IV: Applications of Derivatives 27. Maximum and Minimum Values (p. 76-82) 28. Critical Numbers (p. 83-84) 29. Finding Absolute Extrema of a continuous function on a closed interval (85-87) 30. Intervals of Increase and Decrease (88-89) 31. The First Derivative Test (90-91) 32. Intervals of Concavity (92) 33. Points of In°ection (93) 34. The Second Derivative Test (94) 35. L'H^opital's Rule (95-96) 36. Summary of Curve Sketching (97-108) 75

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27. Maximum and Minimum Values (4.1) De¯nition: A function f has an absolute maximum (or global maximum ) at c if f ( c ) f ( x ) for all x in the domain ( D ) of f . The number f ( c ) is called the maximum value value of f on D . De¯nition: A function f has an absolute mimimum (or global minimum ) at c if f ( c ) f ( x ) for all x in the domain ( D ) of f . The number f ( c ) is called the minimum value value of f on D . The maximum and minimum values of f are called the extreme values (or extrema ) of f . 76
De¯nition: A function f has a relative maximum or ( local maximum ) at c if f ( c ) f ( x ) when x is near c . [This means that f ( c ) f ( x ) for all x in some open intervale containing c .] De¯nition: A function f has an relative minimum (or local mimimum ) at c if f ( c ) f ( x ) when x is near c . The set of all local maximum and minimum values of f is called the relative extrema of f . 77

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Example 1: The function f ( x ) = x 2 has global (and local) mini- mum at x = 0 and this minimum value is 0 (since f (0) = 0). There is no local or global maximum. Example 2: The function f ( x ) = sin x has values that range from +1 to 1. f occurs when x = π 2 . In fact, it occurs in¯nitely often when x = π 2 + 2 πn . f occurs when x = 3 π 2 . In fact, it occurs in¯nitely often when x = 3 π 2 + 2 πn . (Note that x = π 2 can be written as 3 π 2 2 π ) 78
Example 3: Consider the graph of the function f ( x ) = x 4 2 x 2 + 3 2 x 3 Local Minimum: f ( 1) = 0 Local Maximum: f (1) = 4 Global Minimum: f (2) = 0 and f ( 1) = 0 Global Maximum: f ( 3) = 20 79

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Example 4: The function f ( x ) = x 3 has neither a maximum nor a minimum. Example 5: For each of the numbers a, b, c, d, e, r, s and t , state whether the function whose graph is shown below has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum or minimum. x = a x = b x = c x = d x = e x = r x = s x = t 80
For each of the numbers a, b, c, d, e, r, s and t , state whether the function whose graph is shown below has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum or minimum. x

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1007slides_Part4 - Math 1007 Fall 2006(These slides replace...

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