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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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Concave Down
f ' x is decreasing, f '' x is negative, f (x) will shed water
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Second Derivative ITSC
Find where the 2nd Derivative is 0 or undefined. Use those numbers to make intervals
- Intervals
- Test Value
- Sign of f '' x
- Conclusio
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If f '' x is positive
interval is concave up
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If f '' x is negative,
interval is concave down
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Inflection Point
- Where the function is switching from concave up to concave down or
- from concave down to concave up.
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2nd Derivative Test
- 1. Plug your critical numbers into the 2nd Derivative
- 2. If the result is positive, then the there is a relative min at the critical
- number
- 3. If the result is negative, then there is a relative max at the critical
- number.
- 4. If the result is 0, the test fails
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Definition of a Differential
dy= f(x)dx
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Definition of Δy
Δy= f(x+Δx) − f(x)
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Optimization Problem
- 1. Write your two equations. The equation that you are minimizing or maximizing is the Primary Equation.
- 2. Get your Primary equation in terms of one variable by substituting in the other equation.
- 3. Differentiate your primary equation and find the critical numbers.
- 4. Show that the critical numbers are a max or min by the 2nd Derivative Test or a First Derivative ITSC
- 5. Answer the question.
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