130B Test 1 Flashcards
A set A as a subset of reals is compact iff
Every open cover of A has a finite subcover.
A function f:D-Reals is uniformly continuous on E (subset of D) iff
For every €>0 given, there exists a d>0 such that if x,y€E and
|x-y|<€
If f is uniformly continuous on D, then f is uniformly continuous.
What does it mean that a set E (subset of R) is closed?
E contains all of its accumulation points.
What does it mean that a set E (subset of R) is open?
For each x€E, there exists a neighborhood Q of x such that Q is a subset of E.
A set E is compact iff
For every family {Ga}a€A of open sets such that E is a subset of
U(a€A)Ga, there exists a finite set {a1,…,an} as a subset of A such that E is a subset of U(i=1to n)Gai
•In other words, E is compact iff every open cover of E has a fnite subcover.
Heine-Borel Theorem
A set E(subset of R) is compact iff E is closed and bounded.
Let f:D-R be continuous with D compact (closed and bounded), then f is
Uniformly continuous
If f:D-R is uniformly continuous and D is a bounded set, then
f(E) is a bounded set.
Let f:E-R be continuous with E compact, then
f(E) is compact.
Bolzano Intermediate Value Theorem: Let f:[a,b]-R be continuous with f(a)<f(a)], then
There exists a c€(a,b) such that f(c)=y
If f:[a,b]-R be continuous and 1-1, then
F is monotone
Let x. be an accumulation point of a set D(subset of R) and x.€D, then f is differentiable at x. iff
•For x€D{x.}
•the limit(x->x.) of T(x)=[f(x)-f(x.)]/(x-x.) exists. If this limit exists,
then it is the derivative of f at x..
•the limit(t->0) of [f(x+t)-f(x.)]/t exists. If this limit exists, then it is
the derivative of f at x..
•For every sequence {Xn}(n=1->infinity) of points of D{x.} converging to x., the sequence {[f(Xn)-f(X.)]/(Xn-X.)}(n=1->infinity) converges.
Let f:D->R be differentiable at x., then
x.€D, x. Is an accumulation point of D, and f is continuous at x..
Let f,g:D->R be differentiable at x.. What can you say about f+g, fg, and f/g?
- (f+g)’(x.)=f’(x.)+g’(x.) exists
- (fg)’(x.)=f(x.)g’(x.)+f’(x.)g(x.) exists
- (f/g)’(x.)=[g(x.)f’(x.)-f(x.)g’(x.)]/[g(x.)]^2 exists given that g(x.) doesn’t = 0
Chain rule: If f:D->R and g:D’->R with f(D) a subset of D’
If f is differentiable at x. and g is differentiable at f(x.), then what can be said about g•f (g composed of f)?
(g•f)’(x.)=g’(f(x.))f’(x.) exists
If n€Z and f(x)=x^n, then what can be said about f’(x)?
f’(x)=nx^(n-1) exists for all x if n>0 and for all x not= 0 if n(x)=0 for all x.
Let f:D->R what does it mean that a point x.€D is a relative maximum (minimum)?
There exists a neighborhood Q of x. such that if x€Q(int)D, then f(x)f(x.) or f(x)=f(x.))
Let f:[a,b]->R and suppose f has either a relative maximum or relative minimum at x.€(a,b). If f is differentiable at x. what is f’(x.)?
0
Rolle’s Theorem: Let f:[a,b]->R be continuous on [a,b] and differentiable on (a,b). Then if f(a)=f(b)=0
there exists a c€(a,b) such that f’(c)=0
Mean-Value Theorem: If f:[a,b]->R is continuous on [a,b] and differentiable on (a,b), then
there exists a c€(a,b) such that f’(c)=[f(b)-f(a)]/(b-a)
Suppose f is continuous on [a,b] and differentiable on (a,b), then
1• If f’(x) not= 0 for all x€(a,b), then ?
2• If f’(x)=0 for all x€(a,b), then ?
3• If f’(x)>0 for all x€(a,b), then ?
4• If f’(x)<0 for all x€(a,b), then ?
1• f is 1-1
2• f is constant
3• xf(y) (strictly decreasing)
Suppose f,g continuous on [a,b] and differentiable on (a,b) and that f’(x)=g’(x) for all x€(a,b), then
There exists a c€R such that f(x)=g(x)+c for all x€[a,b]
In other words, f and g only differ by a constant.
If f is differentiable on [a,b] and f’(x) not= 0 for all x€(a,b), then
either f’(x)>or=0 for all x€[a,b] or f’(x)<or=0 for all x€[a,b]
L’hopital’s Rule: suppose f,g are continuous on [a,b] and differentiable on (a,b). If x.€[a,b],
1• g’(x) not= 0 for all x€[a,b], x not= x.
2• f(x.)=g(x.)=0
3• f’/g’ has a limit at x.
then,
f/g has a limit at x. and lim(x->x.) of (f/g)(x)=lim(x->x.) of (f’/g’)(x)
Inverse Funtion Theorem: suppose f:[a,b]->R is continuous and differentiable with f’(x) not= 0 for all x€[a,b]. Then f is 1-1, f(inv) is continuous and differentiable on f([a,b]), and
[f(inv)]’(f(x))= 1/f’(x)
