Analysis Definitions

Question 1

What are the triangle inequalities for

i) |a+b|
ii) |a-b|

Answer 1

Question 2

Give the definition of convergence for a sequence

Answer 2

Question 3

When is a sequence cauchy

Answer 3

Question 4

Give the definition of continuity

Answer 4

Question 5

Give the definition of discontinuity

Answer 5

Question 6

Give the definition of sequential continuity

Answer 6

Question 7

Give the identity connecting x,sinx and tanx for an interval (0, pi/2)

Answer 7

0 < sinx < x < tanx

Question 8

State the Intermediate Value Theorem

Answer 8

Question 9

State the Extreme Value Theorem

Answer 9

If f: [a,b] to R is continuous then f is bounded above and below on [a,b]. There exists a x* and an y* in [a,b] such that f(x*) = inf(f(x)) and f(y*)=sup(f(x))

Question 10

When is a subset A of R open

Answer 10

if for every x in A there exists an epsilon e such that (x-e,x+e) is a subset of A

Question 11

When is a subset A of R closed?

Answer 11

if R \ A is open

Question 12

Show the preimage of U, f^-1U

Answer 12

Question 13

When is a function f: A to B injective

Answer 13

f(x)=f(y) implies x=y

Question 14

When is a function f:A to B surjective

Answer 14

For every y in B there exists an a in A such that f(a)=b

Question 15

When is a function f:E to R increasing

Answer 15

x >= y implies f(x) >= f(y)

Question 16

State the Inverse function theorem for continuous functions

Answer 16

Let I be an interval and suppose that f: I to R is continuous and strictly monotonic. Then J=f(I) is an interval and f^-1: J to I is continuous and strictly monotonic

Question 17

When is a function f:I to R injective if its continuous

Answer 17

If its strictly monotonic

Question 18

Give the definition of a continuous limit

Answer 18

Question 19

State and prove the sandwich rule for continuous limits

Answer 19

Question 20

State and prove the continuous limits and composition proposition.

Answer 20

Question 21

Give the definition of one-sided limits for a continuous limit

Answer 21

Question 22

Define the conditions for f(x) tending to infinity as x tends to c

Answer 22

Question 23

Give the two limits that must exist for a function f:(a,b) to R to be differentiable at x₀ in (a,b)

Answer 23

Question 24

State the Caratheodory formulation of differentiation

Answer 24

then f(x) = f(x₀) = phi(x)(x-x₀)

Question 25

State and prove the chain rule

Answer 25

Question 26

State the derivative of inverses theorem

Answer 26

Question 27

State and prove L’Hopitals rule

Answer 27

Question 28

When is a function right differentiable, left differentiable and differentiable

Answer 28

Question 29

When is a function f continuously differentiable on an interval I

Answer 29

When f is differentiable on I and f’ is continuous on I

Question 30

When is a function n times continuously differentiable

Answer 30

When f⁽ⁿ⁾ exists and is continuous on I

Question 31

When does a function f have

i) local maximum
ii) local minimum

Answer 31

Question 32

State Rolles Theorem

Answer 32

Question 33

State Taylor’s theorem

Answer 33

Question 34

When f is continuously differentiable what is the taylor expansion of f about a

Answer 34

Question 35

How do we show a taylor series expansion is convergent

Answer 35

Show that the remainder terms tend to zero

Question 36

What are the series expansions for sinx and cosx

Answer 36