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inflection point
- any value of x where concavity changes
- more formally, point P on y = f(x) if f(x) is continuous @ that point if curve changes from CU to CD or from CD to CU @ P
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increasing/decreasing test
- f'(x) is +: f(x) is increasing
- f'(x) is - : f(x) is decreasing
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first dx test
- local max @ c: if f'(x) changes from + to - @ c
- local min @ c: if f'(x) changes from - to + @ c
- none @ c: if f'(x) does not change sign @ c
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concave upward
if graph of f(x) lies above all its tangents on an interval I
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concave downward
if graph of f(x) lies below all its tangents on an interval I
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second dx test helps determine
- * intervals of concavity
- * max & min values
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concavity test
- concave up on I: f"(x) + for all x in I
- concave down on I: f"(x) - for all x in I
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second dx test
- f"(x) is continuous near c
- local max @ c: if f'(c) = 0 & f"(c) +
- local min @ c: if f'(c) = 0 & f"(c) -
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inconclusive second dx test
- when f"(c) = 0 or when f"(c) dne
- max, min, or neither
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T/F:
if f'(x) is increasing, f(x) is +
if f'(x) is decreasing, f(x) is -
F: inverse/backwards
- correct:
- if f(x) is increasing, f'(x) is +
- if f(x) is decreasing, f'(x) is -
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T/F:
if f(x) is increasing, f'(x) is +
if f(x) is decreasing, f'(x) is -
T
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What does f'(x) say about f(x)?
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What does f"(x) say about f(x)?
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