# Shared Flashcard Set

## Details

Analysis
Test 2
46
Mathematics
11/12/2011

Term
 f is continuous at c if...
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.
Term
 Boundedness of Continuous Functions
Definition
 If f : I maps to R is continuous on the closed interval I = [a,b], then f is bounded on I.
Term
 Extreme Value Theorem
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.
Term
 Intermediate Value Theorem
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.
Term
 Location of Roots
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.
Term
 The function f is uniformly continuous on the set S if...
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.
Term
 I is an interval of real numbers, and let c be an interior point of I. f is differentiable at c if...
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).
Term
 f has a relative minimum at c element of I if...
Definition
 there is a neighborhood U of c such that f(x) >= f(c) for all x element of U intersect I.
Term
 f has a relative maximum at c element of I if...
Definition
 there is a neighborhood U of c such that f(x) <= f(c) for all x element of U intersect I.
Term
 Rolle's Theorem
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.
Term
 Mean Value Theorem
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)].
Term
 Corollary 20.6If f has a bounded derivative on an interval I, then...
Definition
 f is uniformly continuous on I.
Term
 If f(x) is integrable on [a,b], then a theorem states that
Definition
 For all epsilon > 0, there exists a partition of [a,b] s.t. U(f,P)-L(f,P) < epsilon
Term
 If f'(x) is bounded for all x element of I, then
Definition
 f is uniformly continuous on I
Term
 If for all x element of S, f(x) does not equal the supremum of all the function values, then
Definition
 f is not continuous on S
Term
 To see if somethings integrable: (4 steps)
Definition
 1. FInd partitions2. Find U and L3. Find inf U and sup L4. Does inf U = sup L? if it does, integrable!
Term
 If F(x) equals (integral)(a to x) f(t) dt, assuming that f(t) is integrable on [a,b], then
Definition
 F(x) is a continuous functionF'(x) = f(x) at any point where f is continuous.
Term
 If f is bounded and continuous,
Definition
 then f is uniformly continuous on S
Term
 If f is unbounded on S, then
Definition
 f is not uniformly continuous on S, unless S = [a,infinity), then f(x)=x would be a counterexample
Term
 If f has an unbounded derivative on S,
Definition
 then f is not uniformly continuous on S
Term
 If f has a bounded derivative on S,
Definition
 then f is uniformly continuous on S.
Term
 If f is unbounded and f' is unbounded on S, then
Definition
 f is not uniformly continuous on S.
Term
 If f is continuous on [a,b], then
Definition
 f is uniformly continuous on [a,b]
Term
 If f is continuous on (a,b), then
Definition
 f is not necessarily uniformly continuous on (a,b), for example f(x) = 1/x
Term
 If f is continuous on [a,infinity), then
Definition
 f is not necessarily uniformly continuous, f(x) = x2 on [1,infinity) is a counterexample.
Term
 If if is bounded and continuous on [a,b]
Definition
 then f is uniformly continuous on [a,b]
Term
 if f is bounded and continuous on (a,b)
Definition
 then f is not necessarily uniformly continuous on (a,b), f(x) = sin(1/x) on (0,1) is a counterexample.
Term
 If f is bounded and continuous on [a,infinity)
Definition
 then f is not necessarily uniformly continuous on [a, infinity), f(x) = sin(x2) from [1,infinity) is a good counterexample
Term
 If f is unbounded on [a,b],
Definition
 then f is not uniformly continuous on [a,b]
Term
 If f is unbounded on (a,b),
Definition
 then f is not uniformly continuous on (a,b).
Term
 If f is unbounded on [a,infinity),
Definition
 then f could be uniformly continuous, like with f(x) = x.
Term
 If f' is unbounded on [a,b], then
Definition
 f could be uniformly continuous on [a,b], like with f(x) equals square root of x on [0,1].
Term
 If f' is unbounded on (a,b), then
Definition
 f could be uniformly continuous on (a,b), like f(x)=square root of x on (0,1)
Term
 If f' is bounded on S,
Definition
 then f is certainly uniformly continuous on S, (any type of interval).
Term
 If f and f' are unbounded on [a,b] or (a,b),
Definition
 then f is not uniformly continuous on S.
Term
 If f is integrable on [a,b], and F is continuous on [a,b] with F'=f on (a,b), then
Definition
 integral from a to b of f(x) dx = F(b)-F(a).
Term
 Theorem 7.4
Definition
 1. Denial 2. Construct sequence and subsequence 3. Say lim xnk = x0 4. Conclude f(xnk) converges to f(x0).
Term
 EVT
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.
Term
 Continuous function is uniformly continuous
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
Term
 Rolle's Theorem
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.
Term
 MVT
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).
Term
 RIC
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
Term
 Lemma 27.1
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)
Term
 Fundamental Theorem of Calculus (Part 1)
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).
Term
 Fundamental Theorem of Calculus (Part 2.1)
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.
Term
 Fundamental Theorem of Calculus (Part 2.2)
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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