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Critical Numbers
Values of x in the domain of f (x) where f 'x= 0 or f 'x is undefined
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How do you find absolute minimums and maximums on a closed interval?
- 1. Find critical numbers
- 2. Find values for the function at each endpoint and critical number
- 3. Determine absolute max and absolute min for the interval by your results from step 2.
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Rolle’s Theorem
- If f is differentiable on the interval (a , b) and f(a) = f(b) then there is at least one place in the interval
- where f 'x= 0
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Mean Value Theorem (MVT)
- If f (x) is continuous and differentiable on the interval [a , b], there is someplace in the interval where f ' x
- equals the slope of the line going through the endpoints of the interval.
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First Derivative ITSC
1) Use critical numbers to determine intervals.
- Interval
- Test Value
- Sign of f ' x
- Conclusion
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If sign of f ' x is positive what is happening to f(x)?
f(x) is increasing
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If sign of f ' x is negative then f(x) is _______.
f(x) is decreasing
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First Derivative Test
Do a first derivative ITSC,
- If f ' x is changing from pos to neg, then relative max
- If f ' x is changing from neg to pos, then relative min
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Concave Up
f'(x) is increasing, f''(x)is positive, f(x) will hold water
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