Test 2

Question 1

Define Cauchy Sequence

Answer 1

(an) is said to be Cauchy if given any ε>0, there exists N (natural) so that for all m, n > N we have |am-an| < ε.

Question 2

Every convergent sequence is…

Answer 2

cauchy.

Question 3

Every cauchy sequence is…

Answer 3

bounded.

Question 4

Define complete subset.

Answer 4

A non-empty subset A in Real is said tobe complete if every Cauchy sequences taking values in A converges to a limit in A.

Question 5

If (an) is a Cauchy sequence in Real, then it converges to…

Answer 5

a limit in Real.

Question 6

Algebra of Limits

Answer 6

If lim(x to a)f(x) = l and lim(x to a)g(x) = m,
lim(x to a)(f+g)(x) = l + m
lim(x to a)(fg)(x) = lm
lim(x to a)(kf) = kl
lim(x to a)(f/g) = l/m.

Question 7

Define continuous

Answer 7

A function f:R to R is continuous at a in Df, if lim(x to a)f(x) exists and equals f(a).

Question 8

A function f: R to R is continuous at a in Df if:

2 things

Answer 8

1. Given any sequence (xn) with xn in Df for all n natural, s.t. lim(x to inf)(xn) = a, we have
   lim(x to inf)f(xn) = f(a).
2. Given any ε>0, there exists δ>0, s.t. whenever x in Df with |x-a|

Question 9

Suppose f, g: R to Rare both continuous at a in (Df and Dg). Then the following (4) are also continuous at a:

Answer 9

1. f + g
2. fg
3. αf
4. f/g

Question 10

f, g: R to R. What conditions are needed for f(g(x)) to be continuous at a?

Answer 10

g in continuous at a.
g(a) is in Df.
f is continuous at g(a).

Question 11

Intermediate Value Theorem

Answer 11

Let f be continuous on [a,b] with f(a)>0 and f(b)<0, or f(a)<0 and f(b)>0.
Then there exists c in (a, b) such that f(c)=0.

Question 12

The Boundedness Theorem

Answer 12

If f is continuous on [a,b], then it is bounded on [a,b] and it attains both of its bounds there.

Question 13

Inverse Function Theorem

Answer 13

If f: R to R is continuous and strictly increasing(respectively, strictly decreasing), then f is invertible and f(-1) is strictly increasing on [f(a), f(b)] (respectively, strictly decreasing on [f(b), f(a)]), and continuous on (f(a), f(b)) (respectively, on (f(b), f(a))).

Question 14

Define when a function is differentiable.

Answer 14

Let f: R to R then f is differentiable at a in Df if
lim(x to a)( f(x)-f(a) / x-a ) exists and is finite.
Then we write,
f’(a)=lim(h to 0)( f(a+h)-f(a) / h )

Question 15

Define Infinitely Differentiable / Smooth.

Answer 15

f is smooth is f^(n)(a) exists and is finite for all natural n.

Question 16

The mapping f: R to R is differentiable at a in Df iff…

Answer 16

both left and right derivatives exist and are equal.

Question 17

Triangle Inequality

Answer 17

|a+b| <= |a| + |b|

| |a| - |b| | <= |a-b|