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Definition
| Evaluating by direct substitution and getting a result that may or may not exist |
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| Examples of Indeterminate Forms |
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Definition
0/0
∞/∞
∞ * 0
∞^0
1^∞
0^0
∞-∞ |
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Definition
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| What does it mean for a function to increase and/or decrease? |
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Definition
-If f(x2)>f(x1), when x1<x2, then f is increasing
-If f(x1)>f(x2), when x1<x2, then f is decreasing
-If f(x1)=f(x2), when x1<x2, then f is constant
A function can switch directions as x values change |
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| The Increase/Decrease Test |
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Definition
A) If f'(x)>0 on an interval I, then f(x) is increasing on that interval
B) If f'(x)<0 on an interval I, then f(x) is decreasing on the interval
A function only changes DIRECTION at critical points, thus these INTERVALS are broken up by the location of critical points |
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| The First Derivative Test |
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Definition
A) If f(x) switches from increasing to decreasing at a critical point C, then f(C) is a relative max
B) If f(x) switches from decreasing to increasing at a critical point D, then f(D) is a relative min
C) If f(x) does NOT switch direction at a critical point P, then f(P) is neither a max nor min |
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Definition
| The shape of cereal bowl sitting correctly on a table. All tangent lines lie below the graph. |
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Definition
| The shape of a cereal bowl when turned upside down. All tangent lines lie above the graph |
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Definition
- If f" (x)<0 on an interval I, then f(x) is concave DOWN on that interval
- If f" (x)>0 on an interval I, then f(x) is concave UP on that interval |
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Definition
| Points where the second derivative equals zero or doesn't exist (where f" switches from + to - or vice versa) |
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