Theoretical Final Flashcards
fundamental theorem of calculus (no variable)
f:[a.b] be continuous, diff of (a,b). and F is integrable such that f=F’. then integral f = F(b)-F(a)
fundamental theorem of calculus (variables)
less f:[a,b] be integrable and F = integral (a, x) of f. Then F is continous. if f is continuous at c, then F is diff at c and F’(c)=f(c)
pointwise convergence
fn:s->r is a function. {fn} converges pointwise to f if for every x, f(x) = lim fn
uniform convergence
let fn:s->r be a function. {fn} converges uniformly to f iff for all e>0, there exists an N such that for all n>=N, |fn-f|
uniform convergence
let fn:s->r be a function. {fn} converges uniformly to f iff for all e>0, there exists an N such that for all n>=N, |fn-f|
uniform norm
f:s->r be bounded. then ||f|| = sup{f}
uniform norm uniform convergence theorem
fn converges uniformly iff lim||fn-f||=0
integratable fcns and uniform convergence
if its uniformly convergent, then integral of limit function = lim integral of fn
m-test
{fn} converges uniformly if:
1. |fn|
power series
series(0 to infinity) an(x-x0)^n centered at x0
Definition of Convergent Sequence
A sequence {xn} is said to converge to a number x in R, if for every e > 0, there exists an M in N such that |xn - x| = M. The number x is said to be the limit of {xn}.
cluster point
Let S ⇢ R be a set. A number x in R is called a cluster point of S if for every e > 0, the set (x - e , x + e ) intersect S \ {x} is not empty.
limit
Let f : S -> R be a function and c a cluster point of S. Suppose that there exists an L in R and for every e > 0, there exists a delta > 0 such that whenever x in S \ {c} and |x -c|
continuity of function
for every e > 0 there is a delta > 0 such that whenever x in S and |x-c|
ivt
Let f : [a, b]->R be a continuous function. Suppose that there exists a y such that f(a) y > f(b). Then there exists a c in [a,b] such that f(c) = y.
uniform continuity
for any e > 0 there exists a delta>0 such that whenever x,y in S and |x-y|
lipschitz continuous
Let f : S->R be a function such that there exists a number K such that for all x and y in S we have |f(x)-f(y)|
derivative
lim [f(x)-f(c)]/(x-c)
taylor
Suppose f : [a, b] -> R is a function with n continuous derivatives on [a, b] such that f(n+1) exists on (a, b). Given distinct points x0 and x in [a, b], we can find a point c between x0 and x such that f(x)=Pn(x)+f^(n+1)(c)/(n+1)!^n+1
relative min and max
there exists a delta>0 s.t. for all x when |x-c|
rolles
f:[a,b} continous and diff such that f(a)=f(b)=0. then there exists a c such that f’(c)=0
cauchy criterion
let f:[a,b]->R be bounded. It is integrable if and only if for all e>0, there exists a partition such that U(P,f)-L(P,f)
mvt
f:[a,b] be continuous and diff. then there exists a c such that f(b)-f(a)=f’(c)(b-a)
cauchy criterion
let f:[a,b]->R be bounded. It is integrable if and only if for all e>0, there exists a partition such that U(P,f)-L(P,f)
boundedness, continuity, integrability of uniform convergence
if bounded fn converges to unbounded/not continuous/not integrable limit function f, then fn does not confirm uniformly
boundedness, continuity, integrability of uniform convergence
if bounded fn converges to unbounded/not continuous/not integrable limit function f, then fn does not confirm uniformly
differentiability and uniform convergence
if fn is diff on [a,b] such that fn’ converges to g uniformly and fn converges to f pointwise, then f is diff with f’=g
