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| Fundamental Theorem of Calculus Part 1 |
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Definition
| If f is continuous on [a,b] & g(x)=∫[(a to x) f(x)dt] for x in [a,b], then g(x) is continuous and differentiable and g'(x)=f(x). |
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| Fundamental Theorem of Calculus Part 2 |
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Definition
| If f is continuous on [a,b] then ∫[(a to b) f(t)dt]=F(b)-F(a), where F is any anti-derivative of f. |
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Definition
| If a & b are the legs of a triangle, and c is the hypotenuse, then (a^2)+(b^2)=(c^2). |
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Definition
| The area between a curve f(t) & the x-axis over an interval [a,b] is related to F(t), an anti-derivative of f(t). |
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| Absolute (Global) Maximum |
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Definition
| A function f has an absolute max at (c,f(c)) if f(c)≥f(x) for all x in the domain of f. f(c) is the absolute maximum value of x. |
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| Absolute (global) minimum |
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Definition
| A function f has an absolute min at (c,f(c)) if f(c)≤f(x) for all x in the domain of f. f(c) is the absolute minimum value of x. |
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Definition
| A function f has a relative max at (a,f(a)) if f(a)≥f(x) for all x near a. |
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Definition
| A function f has a relative min at (a,f(a)) if f(a)≤f(x) for all x near a. |
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Definition
| If f is continuous on a closed interval [a,b] then f obtains an absolute max (c,f(c)) and absolute min (d,f(d)) for some c & d on the closed interval [a,b]. |
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Definition
| If f has a local (relative) max or min at (c,f(c)) & if f'(c) exists then f'(c)=0. |
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Definition
| If f'(c)≠0, then either f'(c) does not exist or (c,f(c)) is not a local max or min. |
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| Closed Interval Method - how to find the absolute max/min of f(x) on [a,b]. |
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Definition
1)Find Critical numbers of f on [a,b].
2)Find f(c) for each critical number c.
3)Find f(a) & f(b).
4)Order the output values and find the max and min. |
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Definition
| A number c of the function f in the domain of f such that either f'(c)=0 or f'(c) DNE. |
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| Intermediate Value Theorem |
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Definition
| If f is a continuous on [a,b] & N is any # between f(a) & f(b) then there is a c between a & b such that f(c)=N. |
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Definition
Let f be a function that satisfies the following three hypotheses:
1)f is continuous on [a,b]
2)f is differentiable on (a,b)
3)f(a)=f(b)
Then, there is a number c between (a,b) such that f'(c)=0. |
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Definition
Let f be a function that satisfies the following two hypotheses:
1)f is continuous on closed interval [a,b]
2)f is differentiable on open interval (a,b)
Then there is a c in (a,b) such that f'(c)=[(f(a)-f(b))/(a-b)]. |
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