Intermediate Analysis Test 3

Front 1

Definition of Derivative

Back 1

f: (a,b)->R and c in (a,b). If the limit as ->c (f(x)-f(c))/(x-c) exists than f is said to be differentiable at c.

Front 2

Theorem 6.1

Back 2

If f: (a,b)->R is differentiable at a point c in (a,b), then f is continuous at c.

Front 3

Theorem 6.3 (Chain Rule)

Back 3

If f:(a,b)->R is differentiable at a point c in (a,b) and if g:I->R is differentiable at f(c), where I is an open interval containing f((a,b)), then the composition g o f is differentiable at c and (g o f)'(c)=g'(f(c))f'(c).

Front 4

Extremum

Back 4

a point c is called a local minimum point for a function f is there exists a neighborhood U of c s.t. f(x)>=f(c) for all x in U

Front 5

Critical point

Back 5

a point c s.t. f'(c) does not exist or is 0

Front 6

Theorem 6.4 (First Derivative Test)

Back 6

If f: (a,b)->R is differentiable at a local extremum point c in (a,b), then f'(c)=0

Front 7

Theorem 6.5 (Rolles Theorem)

Back 7

If f: [a,b]->R is continuous and f is differentiable on (a,b), and f(a)=f(b)=0, then there exists c in (a,b) with f'(c)=0

Front 8

Theorem 6.6 (Mean Value Theorem)

Back 8

If f: [a,b]->R is continuous and f is differentiable on (a,b), then there exists c in (a,b) with f'(c)=(f(b)-f(a))/(b-a).

Front 9

Corollary 6.7

Back 9

If f: (a,b)->R has f'(c)=0 for all x in (a,b), then f is constant on (a,b).

Front 10

Corrolary 6.8

Back 10

If f: (a,b)-> R has f'(x)>0 then f(x) is strictly increasing on (a,b). Similarly for >=0, <0, <=0.

Front 11

Corollary 6.9

Back 11

If f'(x)=g'(x) for all x in (a,b) then there exists a constant C s.t. f(x)=g(x)+c for all x in (a,b).

Front 12

Theorem 6.10 (Taylor's theorem)

Back 12

If f: (a,b)->R is n+1 times differentiable, and the first n of these derivatives are continuous on (a,b), and if x0 in (a,b), then for every x in (a,b) there is a number c between x0 and x s.t. f(x)=taylor polynomial formula

Front 13

Theorem 6.11 (Second Derivative Test)

Back 13

Suppose that f''' is continuous on a neighborhood of c. If f'(c)=0 and f''(c)>0 then f has a strict local minimum at x=c. If f'(c)=0 and f''(c)<0 then f has a strict local maximum at x=c.

Front 14

Theorem 6.12

Back 14

Suppose that f: (a,b)->R is injective and continuous. Then f is either strictly increasing on (a,b), or it is strictly decreasing on (a,b).

Front 15

Proposition 6.13

Back 15

Suppose that f: (a,b)->R is injective and continuous. Then f((a,b)) is an open interval.

Front 16

Theorem 6.14

Back 16

Suppose that f: (a,b)->R is injective and continuous. Then f^-1 is continuous.

Front 17

Theorem 6.15 (The inverse function theorem)

Back 17

Suppose that f: (a,b)->R is a differentiable function with f'(x)!=0 for all x in (a,b). Then f is injective, its range f((a,b)) is an open interval (c,d), and for any y in (c,d), we have
(f^-1)'(y)=1/f'(x), f(x)=y.

Front 18

Partition

Back 18

A partition P is an ordered set {x0,x1,...,xn} where a=x0<x1<...<xn=b.

Front 19

Mk
mk

Back 19

sup {f(t) : xk-1<=t<=xk}
inf {f(t) : xk-1<=t<=xk}
for each k=1,2,...,n

Front 20

upper sum U(f,P)
lower sum L(f,P)

Back 20

Mk*deltaxk summed over k from 1 to n
mk*deltaxk summed over k from 1 to n

Front 21

upper integral U(f)
lower integral L(f)

Back 21

inf {U(f,P):P in p}
sup {L(f,P):P in p}

Front 22

integrable

Back 22

if L(f)=U(f)=integral

Front 23

Observation 1

Back 23

If m<=f(x)<=M for all x in [a,b], then m(b-a)<=L(f,P)<=U(f,P)<=M(b-a), for any partition P of [a,b].

Front 24

Observation 2

Back 24

If P refines Q then L(f,Q)<=L(f,P)<=U(f,P)<=U(f,Q)

Front 25

Observation 3

Back 25

If P,Q are two partitions of [a,b] then L(f,P)<=U(f,Q).

Front 26

Observation 4

Back 26

L(f)<=U(f)

Front 27

Observation 5

Back 27

L(f,P)<=L(f)<=U(f)<=U(f,P) for all partitions P of [a,b].

Front 28

Theorem 7.1 (Riemann Condition)

Back 28

A bounded function f:[a,b]->R is integrable iff for all e>0, there exists a partition of [a,b] s.t. U(f,P)-L(f,P)<e.

Front 29

Uniformly Cotninuous

Back 29

A function f:D-> R is called uniformly continuous if for all e>0 there exists d>0 s.t. |f(x)-f(y)|<e whenever x,y in D, and |x-y|<d

Front 30

Theorem 7.2

Back 30

If D is compact, and f: D->R is continuous, then f is uniformly continuous.

Front 31

Theorem 7.3

Back 31

If f : [a,b] -> R is continuous, then f is integrable.

Front 32

Fact I2

Back 32

If f:[a,b]-> is integrable, and if m<=f(x)<=M for all x in [a,b], then
m(b-a)<=integral<=M(b-a).

Front 33

Fact I4

Back 33

If f and g are integrable on [a,b], and if f(x)<=g(x) for all x in [a,b] then integral f<=integral g

Front 34

Fact I6

Back 34

If f is integrable on [a,b], then so is |f(x)|, and |integral f|<=integral |f|

Front 35

Fact I8

Back 35

If f is monotone on [a,b] then f is integrable on [a,b].

Front 36

Fact I9 (first fundamental theorem of Calculus)

Back 36

If f is integrable on [a,b], define F(x)=integral from a to x of f(t), for x in [a,b]. If f is continuous at a point c in (a,b), then F'(c)=f(c).

Front 37

Fact I10 (second fundamental theorem of Calculus)

Back 37

If f:[a,b]-> R is continuous, and is differentiable on (a,b), and if f' is integrable on [a,b], then integral f'= f(b)-f(a)

Front 38

Fact I11 (Integration by parts)

Back 38

If f' and g' are continuous on an open interval containing [a,b] then integral fg'=f(b)g(b)-f(a)g(a)-integral f'g

Front 39

Fact I12 (change of variable)

Back 39

If g is a differentiable function defined on an open interval containing numbers c<d, with g' integrable on [c,d], and if f is a continuous function on an open interval I containing the range of g, then integral c-d f(g(x))g'(x)=integral g(c)-g(d) f(x).