Theoretical Final

Question 1

fundamental theorem of calculus (no variable)

Answer 1

f:[a.b] be continuous, diff of (a,b). and F is integrable such that f=F’. then integral f = F(b)-F(a)

Question 2

fundamental theorem of calculus (variables)

Answer 2

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)

Question 3

pointwise convergence

Answer 3

fn:s->r is a function. {fn} converges pointwise to f if for every x, f(x) = lim fn

Question 4

uniform convergence

Answer 4

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|

Question 5

uniform convergence

Answer 5

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|

Question 6

uniform norm

Answer 6

f:s->r be bounded. then ||f|| = sup{f}

Question 7

uniform norm uniform convergence theorem

Answer 7

fn converges uniformly iff lim||fn-f||=0

Question 8

integratable fcns and uniform convergence

Answer 8

if its uniformly convergent, then integral of limit function = lim integral of fn

Question 9

m-test

Answer 9

{fn} converges uniformly if:

1. |fn|

Question 10

power series

Answer 10

series(0 to infinity) an(x-x0)^n centered at x0

Question 11

Definition of Convergent Sequence

Answer 11

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}.

Question 12

cluster point

Answer 12

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.

Question 13

limit

Answer 13

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|

Question 14

continuity of function

Answer 14

for every e > 0 there is a delta > 0 such that whenever x in S and |x-c|

Question 15

ivt

Answer 15

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.

Question 16

uniform continuity

Answer 16

for any e > 0 there exists a delta>0 such that whenever x,y in S and |x-y|

Question 17

lipschitz continuous

Answer 17

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)|

Question 18

derivative

Answer 18

lim [f(x)-f(c)]/(x-c)

Question 19

taylor

Answer 19

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

Question 20

relative min and max

Answer 20

there exists a delta>0 s.t. for all x when |x-c|

Question 21

rolles

Answer 21

f:[a,b} continous and diff such that f(a)=f(b)=0. then there exists a c such that f’(c)=0

Question 22

cauchy criterion

Answer 22

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)

Question 23

mvt

Answer 23

f:[a,b] be continuous and diff. then there exists a c such that f(b)-f(a)=f’(c)(b-a)

Question 24

cauchy criterion

Answer 24

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)

Question 25

boundedness, continuity, integrability of uniform convergence

Answer 25

if bounded fn converges to unbounded/not continuous/not integrable limit function f, then fn does not confirm uniformly

Question 26

boundedness, continuity, integrability of uniform convergence

Answer 26

if bounded fn converges to unbounded/not continuous/not integrable limit function f, then fn does not confirm uniformly

Question 27

differentiability and uniform convergence

Answer 27

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