Theorems BC Sem 1
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Created by:
jrutheiser on January 16, 2012
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Terms | Definitions |
|---|---|
 Intermediate Value Theorem (IVT) | If f is continuous on [a,b] and f(a)<N<f(b) with f(a)≠f(b) then there is a c S.T. a≤c≤b with f(c) = N |
| Extreme Value Theorem | If f is continuous on [a,b] then f has an absolute max and an absolute min |
| Mean Value Theorem | If f is a function S.T. 1. f is cont on [a,b] 2. f is differentiable on (a,b) then there is a c∈(a,b) S.T. f'(c) = (f(b)-f(a))/(b-a) |
| Mean Value Theorem for Integrals | If f is cont on [a,b] then there exists a c∈[a,b] S.T. f(c) = favg = 1/(b-a) ∫(a,b) f(x)dx |
| Fermat's Theorem | If f has local extrema at x=c, and f'(c) exists then f'(c)=0 |
| Rolle's Theorem | If f is a function S.T. 1. f is cont. on [a,b] 2. f is differentaible on (a,b) 3. f(a)=f(b) then there is a c∈(a,b) S.T. f'(c)=0 |
| Fundamental Theorem of Calculus Part 1 | If g(x) = ∫(a,x) f(t)dt where g(x) is continuos on [a,b] and differentiable on (a,b) then g'(x) = f(x) |
| Fundamental Theorem of Calculus Part 2 | If f is continuous on [a,b] then ∫(a,b) f(x)dx = F(b) - F(a) where F'(x)=f(x) |
| L'Hôpital's Rule | if f and g are differentiable on an open interval I that contains a except at a and lim(x->a) f(x)/g(x)= 0/±∞ then lim(x->a) f(x)/g(x) = lim (x->a) f'(x)/g'(x) |
| Squeeze Theorem | If f(x) ≤ g(x) ≤ h(x) for all x in an interval containing a, and lim (x->a) f(x) = lim (x->a) h(x) = L then lim (x->a) g(x) = L |
| First Derivative Test | Given f is a continuous function If f'(c)=0 or DNE then: 1. If f'(x) switches from + to - at x=c, f(c) is a local max 2. If f'(x) switches from - to + at x=c, f(c) is a local min 3. If f'(x) goes + to + or - to - or else it's a saddle point |
| Second Derivative Test | If f''(x) is continuous near x=c f''(x) exists ->f' diff -> f' cont -> f diff -> f cont and f'(c) = 0 then 1. If f''(c)>0 then f(c) is local min 2. If f''(c)<0 then f(c) is local max 3. If f''(c) = 0 then the test fails and use 1st derivative test |
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