2. A lot of effort goes into determining the behavior of a function f on an interval I . Does it have a maximum in the interval? Where is it increasing, decreasing? p. 164
3. A function need not have a maximum or minimum in an interval.
4. You can see that continuity or discontinuity can affect the existence of an extremum on an interval. This suggests this theorem: Notice that this theorem guarantees a minimum and maximum, but doesn’t help you find them! p. 164
6. Example 1, p. 165 Find the value of the derivative at each of the relative extrema shown: a. At the point (3, 2), f’(3) = 0
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9. We can see that at relative extrema, the derivative is either zero or does not exist. The x-values of these extrema are called critical numbers. There are two types. p. 166
12. Notice that all critical numbers don’t have to produce extrema. Converse of Thm 3.2 is not necessarily true! In other words, if x=c is a critical number, f doesn’t have to have a relative max or min there.
13. Ex 3 p. 168 Find extrema of on [-1, 3] 1. Find critical numbers. Two critical numbers, x = 0 because that makes f’ undefined and it doesn’t exist, and x = 1 because that makes f’ = 0 2&3. Evaluate f at critical numbers and endpoints 4. Determine max and min for interval. Left endpoint Critical number Critical number Right endpoint f(-1) = -5 Minimum f(0) = 0 Maximum f(1) = -1 f(3) ≈-0.24
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15. Ex 4 p. 168 Finding Extrema on a closed interval Find the extrema of f(x) = 2sin x – cos 2x on [0, 2 π ] 1. Critical numbers: 2&3. Plug in endpoints and critical numbers 4. Determine maximum and minimum Left endpt Critical # Critical # Critical # Critical # Right endpt f(0) = -1 f( π /2) = 3 Maximum f(7 π /6) = -1.5 minimum f(3 π /2)=-1 f(11 π /6) =-1.5 mimumum f(2 π ) = -1
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17. Assignment 3.1 p. 169/ 1-45 every other odd, 53-59 odd, 63-66 due Wednesday, Oct 19
