Lecture 3

Question 1

Monotonic sequence theorem

Answer 1

a bounded monotonic sequence is always divergent

Question 2

infimum

Answer 2

greatest lower bound

Question 3

supremum

Answer 3

least upper bound

Question 4

continuity at a point

Answer 4

lim(x->a)f(x)=f(a)

Question 5

one-sided continuity at a point

Answer 5

lim(x->a+)f(x)=f(a)

Question 6

Intermediate Value Theorem

Answer 6

suppose that f is continuous on (a;b) and N a number such that f(a)<N<f(b), then there is c e (a;b) such that f(c)=N

Question 7

differentiability property(+,-)

Answer 7

the func is derivative if and only if lim(x->a+)(f’(x))=lim(x->a-)(f’(x))=

Question 8

Extreme value theorem

Answer 8

if f is continuous on an interval [a,b], then f has an absolute maximum f(c) for some c e [a;b]

Question 9

Fermat’s theorem

Answer 9

If f has a maximum or minimum at c, and if f’(c), then f’(c)=0

Question 10

Rolle’s theorem

Answer 10

let f satisfy the following:
1) continuous on [a,b]
2) f is differentiable on (a,b)
3) f(a)=f(b)
then there exists c e (a,b) such that f’(c)=0

Question 11

Mean Value Theorem (slope)

Answer 11

let f satisfy:
1) f is continuous on [a,b]
2) f is differentiable (a,b)
there exists c e (a,b) s.t
f’(c)=(f(b)-f(a))/(b-a)