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| f is continuous at c if... |
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Definition
| limit as x goes to c of f(x) = f(c). Which also means that for each epsilon > 0, there is some delta > 0 so that if x is an element of A and x-c < delta, then f(x)-f(c) < epsilon. |
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| Boundedness of Continuous Functions |
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Definition
| If f : I maps to R is continuous on the closed interval I = [a,b], then f is bounded on I. |
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Definition
| Let f : I goes to R be continuous on the closed interval I = [a,b] and let m = inf { f(x):x element of I } and M = sup { f(x):x element of I }. Then there must be values of c and d in I so that f(c)=m and f(d)=M. |
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| Intermediate Value Theorem |
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Definition
| Let f : I goes to R be continuous on the closed interval I = [a,b] and let f(a) != f(b). For any value t between f(a) and f(b), there is a value c element (a,b) so that f(c) = t. |
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Definition
| If f is continuous on the closed interval [a,b], and f(a)f(b) < 0, then there is some point c element of (a,b), so that f(c)=0. |
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| The function f is uniformly continuous on the set S if... |
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Definition
| for every epsilon > 0, there is a delta > 0 so that for all x,u where x-u < delta, we have f(x)-f(u) < epsilon. |
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Term
| I is an interval of real numbers, and let c be an interior point of I. f is differentiable at c if... |
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Definition
| lim as x goes to c of [f(x)-f(c)/(x-c)] exists. If it does exist, then the value of the limit is called the derivative of f at c and is denoted by the symbol f'(c). |
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| f has a relative minimum at c element of I if... |
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Definition
| there is a neighborhood U of c such that f(x) >= f(c) for all x element of U intersect I. |
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Term
| f has a relative maximum at c element of I if... |
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Definition
| there is a neighborhood U of c such that f(x) <= f(c) for all x element of U intersect I. |
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Definition
| If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b) = 0, then there is some value c element of (a,b) so that f'(c) = 0. |
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Definition
| If f is continuous on [a,b] and differentiable on (a,b) then there is at least one value c element of (a,b) such that f'(c) = [(f(b)-f(a)/(b-a)]. |
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Corollary 20.6
If f has a bounded derivative on an interval I, then... |
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Definition
| f is uniformly continuous on I. |
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Term
| If f(x) is integrable on [a,b], then a theorem states that |
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Definition
| For all epsilon > 0, there exists a partition of [a,b] s.t. U(f,P)-L(f,P) < epsilon |
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Term
| If f'(x) is bounded for all x element of I, then |
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Definition
| f is uniformly continuous on I |
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| If for all x element of S, f(x) does not equal the supremum of all the function values, then |
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Definition
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| To see if somethings integrable: (4 steps) |
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Definition
1. FInd partitions
2. Find U and L
3. Find inf U and sup L
4. Does inf U = sup L?
if it does, integrable! |
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Term
| If F(x) equals (integral)(a to x) f(t) dt, assuming that f(t) is integrable on [a,b], then |
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Definition
F(x) is a continuous function
F'(x) = f(x) at any point where f is continuous. |
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Term
| If f is bounded and continuous, |
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Definition
| then f is uniformly continuous on S |
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Term
| If f is unbounded on S, then |
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Definition
| f is not uniformly continuous on S, unless S = [a,infinity), then f(x)=x would be a counterexample |
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Term
| If f has an unbounded derivative on S, |
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Definition
| then f is not uniformly continuous on S |
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Term
| If f has a bounded derivative on S, |
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Definition
| then f is uniformly continuous on S. |
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Term
| If f is unbounded and f' is unbounded on S, then |
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Definition
| f is not uniformly continuous on S. |
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Term
| If f is continuous on [a,b], then |
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Definition
| f is uniformly continuous on [a,b] |
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Term
| If f is continuous on (a,b), then |
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Definition
| f is not necessarily uniformly continuous on (a,b), for example f(x) = 1/x |
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Term
| If f is continuous on [a,infinity), then |
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Definition
| f is not necessarily uniformly continuous, f(x) = x2 on [1,infinity) is a counterexample. |
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Term
| If if is bounded and continuous on [a,b] |
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Definition
| then f is uniformly continuous on [a,b] |
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Term
| if f is bounded and continuous on (a,b) |
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Definition
| then f is not necessarily uniformly continuous on (a,b), f(x) = sin(1/x) on (0,1) is a counterexample. |
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Term
| If f is bounded and continuous on [a,infinity) |
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Definition
| then f is not necessarily uniformly continuous on [a, infinity), f(x) = sin(x2) from [1,infinity) is a good counterexample |
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Term
| If f is unbounded on [a,b], |
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Definition
| then f is not uniformly continuous on [a,b] |
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Term
| If f is unbounded on (a,b), |
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Definition
| then f is not uniformly continuous on (a,b). |
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Term
| If f is unbounded on [a,infinity), |
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Definition
| then f could be uniformly continuous, like with f(x) = x. |
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Term
| If f' is unbounded on [a,b], then |
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Definition
| f could be uniformly continuous on [a,b], like with f(x) equals square root of x on [0,1]. |
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Term
| If f' is unbounded on (a,b), then |
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Definition
| f could be uniformly continuous on (a,b), like f(x)=square root of x on (0,1) |
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Term
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Definition
| then f is certainly uniformly continuous on S, (any type of interval). |
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Term
| If f and f' are unbounded on [a,b] or (a,b), |
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Definition
| then f is not uniformly continuous on S. |
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Term
| If f is integrable on [a,b], and F is continuous on [a,b] with F'=f on (a,b), then |
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Definition
| integral from a to b of f(x) dx = F(b)-F(a). |
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Term
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Definition
1. Denial
2. Construct sequence and subsequence
3. Say lim xnk = x0
4. Conclude f(xnk) converges to f(x0). |
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Definition
1. Say {yn} goes to M
2. Say f(xn) = yn
3. Say {xnk} goes to d.
4. So {f(xnk)} goes to M and f(d) = M. |
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| Continuous function is uniformly continuous |
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Definition
1. Denial
2. Construct {xnk} that goes to x0
3. Assert that {unk} goes to x0
4. Then |unk - xnk + xnk - x0| < e/2 + e/2 = e
5. And |f(xnk) - f(unk)| < e0 |
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Term
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Definition
1. There exists an x so f(x) > 0
2. Form a bounded set S.
3. Let M = sup of S
4. By Lemma 20.1, there is a c s.t. f(c) = M
5. Therefore f'(c) = 0. |
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Term
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Definition
1. Apply Rolle's Theorem to weird function.
2. Check for cont. and diff.
3. Does f(a) = f(b) = 0?
4. Then differentiate weird function.
5. Show that f'(c) = f(b)-f(a)/(b-a). |
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Term
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Definition
1. L(f,P) < sup L(f,P) < inf U(f,P) < U(f,P) for any partition.
2. Since there is a partition so U-L < e, it is also true that 0<sup-inf<e.
3. Since e is arbitrary, lower sum equals upper sum. |
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Term
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Definition
1. Apply MVT to F on [xi-1,xi]
2. Rearrange and take summation.
3. Show that left equals F(xn)-F(x0) = F(b)-F(a) |
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| Fundamental Theorem of Calculus (Part 1) |
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Definition
1. L < integral < U
2. Then L < summation (Lemma 27.1) < U
3. Then L < F(b)-F(a) (Lemma 27.1) < U
4. Since U-L is less than epsilon, integral - (F(b)-F(a)) < epsilon.
5. So, integral = F(b)-F(a). |
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| Fundamental Theorem of Calculus (Part 2.1) |
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Definition
1. There is a B s.t. f(x) < B.
2. Assume x > c
3. Do some work and F(x)-F(c) < B(x-c)
4. Let delta = epsilon/B
5. So F(x) is cont. at c. |
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| Fundamental Theorem of Calculus (Part 2.2) |
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Definition
1. MVT with F,x,x0
2. Do simplification (add in a -f(x0) + f(x0))
3. Show that left limit thing is zero.
4. So F'(x) = f(x). |
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