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# MATH1050

### University of Southampton MATH1050

Contrapositive of P implies Q. ~Q implies ~P.
Converse of P implies Q. Q implies P.
Modus Ponens. If we know x and we know x implies y we can infer y.
A is a subset of B iff for all x: x in A implies x in B
A is equal to B iff for all x: x in A iff x in B
union x in A or x in B
intersection x in A and x in B
tautology a statement which is always true
contradiction a statement which is always false
interval for all x,y,z in R: x in I, z in I, x<y, y<z implies y in I
open interval for all y in I there exists x,z in I with y<z and x<y
injective function for all x,y in X: f(x)=f(y) implies x=y
surjective function for all y in Y there exists x in X such that f(x)=y
bijective function injective and surjective
range of f(x) {y in Y: there exists x in X s.t. f(x)=y}
infinite set S is countable if there exists a bijection f: N to S
preimage of B {x in X: f(x) in B}
image of A f(A)={y in Y: there exists x in A s.t. f(x)=y}
composition g@f(x) g(f(x))
limit (x to infin) f(x) to L as x to infin if (for all E>0)(there exists k>0)(for all x in X) x>k implies |f(x)-L|<E
cauchy criterion (x to infin) f(x) to L as x to infin if (for all E>0)(there exists k>0)(for all x1,x2 in X) x1,x2>k implies |f(x1)-f(x2)|<E
limit (x to a) f(x) to L as x to a if (for all E>0)(there exists d>0)(for all x in X) 0<|x-a|<d implies |f(x)-L|<E
cauchy criterion (x to a) f(x) to L as x to a if (for all E>0)(there exists d>0)(for all x1,x2 in X) 0<|x-a|<d and 0<|x-a|<d implies |f(x1)-f(x2)|<E
continuous (simple) f is continuous at a if f(x) to f(a) as x to a
continuous (exact) f is continuous at a if a in X and (for all E>0)(there exists d>0)(for all x in X) |x-a|<d implies |f(x)-f(a)|<E
Intermediate Value Theorem if f is continuous on the closed interval [a,b] and f(a),f(b) have opposite signs then there exists c in (a,b) such that f(c)=0
x in interior of S if there exists an open interval (a,b) in S with x in (a,b)
min and max if f is a continuous function on a closed interval [a,b] then f achieves its min and max (Cmin, Cmax in [a,b]) such that f(Cmin)<eq f(x)<eq f(Cmax) for all x in [a,b]
differentiable at a if a in X and [f(a+h)-f(a)]/h to a limit as h to 0 continuous
a in domain of f then f is diff. at a with derivative m iff there exists e(x) continuous at a with e(a)=0 such that f(x)=f(a)+(m+e(x))(x-a)
f'(c)=0 if f has local min/max at c
Mean Value Theorem if f is continuous on [a,b] and diff on (a,b) then there exists c in (a,b) s.t. f'(c)=(f(b)-f(a))/(b-a)
increasing for all a,b if a<b then f(a)<eq f(b)
strictly increasing for all a,b if a<b then f(a)<f(b)
e as x to infin lim (1+(1/x))^x
f is twice diff on (a,b) and f(x),f'(x) continuous on [a,b] there exists c in (a,b) s.t. f(b)=f(c)+f'(c)(b-a)+f''(c)(.5)(b-a)^2
nth Taylor polynomial is f is n times diff at a then Pn(x)= sum from m=o to n {[f^m(a)]/m!}(x-a)^m
partition of [a,b] a list a0,a1,..,an where a0=a, a0<a1<..<an and an=b
area under f f(x1)(a1-a0)+f(x2)(a2-a1)+..+f(xn)(an-a{n-1}) for some points x1,x2,..,xn with x1 in (a0,a1), x2 in (a1,a2),..,xn in (a{n-1},an)
f has integral A on [a,b] if (for all E>0)(there exists partition a0,a1,..,an of [a,b])(for all x1,x2,..,xn) if x1 in (a0,a1), x2 in (a1,a2),.., xn in (a{n-1},an) then mod sum from m=1 to m=n of f(xm)(am-a{m-1}) - A is less than E
f integrable on [a,b] integrated from b to a: f(x)dx = A continuous
fundamental theorem of calculus (i) if f continuous on [a,b] then F(x)= integrated from x to a: f(t)dt is diff on [a.b] and F'(x)=f(x)
fundamental theorem of caluculus (ii) if F diff on [a,b] and inverseF(x) continuous on [a,b] then integrated from b to a: inverseF(X)dx = F(b)-F(a)
Created by: meggles