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Math_137_Winter_2010_Week_9_Notes

Math_137_Winter_2010_Week_9_Notes - Math 137 Week 9 Notes...

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Math 137 Week 9 Notes – Dr. Paula Smith 4.3 Derivatives of a function give us information about the shape of the graph of the function. If f (x) > 0 on an interval, then f(x) is increasing on that interval. If f (x) < 0 on an interval, then f(x) is decreasing on that interval. If f (x) > 0 on an interval, then f(x) is concave upward on that interval. If f (x) < 0 on an interval, then f(x) is concave downward on that interval. We recall that if f (c) = 0, then c is a critical number. If f (c) = 0, then (c,f(c)) is an inflection point. First derivative test: If c is a critical number for a continuous function f, then a) if f changes from positive to negative at c, then f has a local maximum at c b) if f changes from negative to positive at c, then f has a local minimum at c c) if f does not change sign across c, then f has no local extremum at c Second derivative test: If f is continuous near critical point c, then a) if f (c) = 0 and f (c) > 0, then f has a local minimum at c b) if f (c) = 0 and f (c) < 0, then f has a local maximum at c 4.5 From the material in 4.3 it is clearly useful to determine zeros of the function, of its first
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