Analysis

Question 1

Lemma; Prove that f(x) = sqrt(x) is continous

Answer 1

Question 2

Answer 2

Question 3

Lemma; If f,g: E to R are continuous at c then f+g is continuous at c

Answer 3

Question 4

Answer 4

Question 5

Lemma; Suppose that f,g: E to R are continuous at c in E. The the function |f| is continuous

Answer 5

Question 6

Answer 6

Question 7

Show that the function

f(x) = 1/q for rationals when x = p/q

0 for irrationals

is discontinous at every rational

Answer 7

Question 8

Answer 8

Question 9

Theorem: The function f: R to [-1,1] given by f(x)=sinx is continuous

Answer 9

Question 10

Show that x^2 = 2 has a root in the interval (0,2)

Answer 10

Consider f(x) = x^2 -2

f(0)= -2

f(2) = 2

so by IVT there is a root

Question 11

Proposition; Any odd degree polynomial has at least one real root

Answer 11

Correction: Take the x* and x_* to be 2A/a_2n+1 and -2A/a_2n+1

Question 12

Lemma; Any continuous function f: [a,b] to [a,b] has a fixed point ie there is an x* in [a,b] such that f(x*) = x*

Answer 12

Question 13

Prove the Extreme Value Theorem

Answer 13

Question 14

Lemma; The interval (a,b) is open

Answer 14

Question 15

Lemma; Suppose that A and B are open subsets. Then A u B and A n B are open

Answer 15

Question 16

Lemma; The interval R[a,b] is closed

Answer 16

We have R[a,b] = (-inf,a) u (b,inf). This is a union of two open sets, so its open

Question 17

Lemma; A subset A of R is closed if and only if (a_n) in A with a_n tends to a implies that a is in A

Answer 17

Question 18

Lemma; A function f:R to R is continuous if and only if f^-1U is open for every open set U in R

Answer 18

Question 19

Answer 19

Question 20

Answer 20

Question 21

Answer 21

Question 22

State and prove the sandwich rule for continuous limits

Answer 22

Question 23

State and prove the continuous limits and composition proposition.

Answer 23

Question 24

Show that if f(x) = sinx then f’(x) = cosx

Answer 24

Question 25

Lemma; If f is differentiable at x₀ then f is continuous at x₀

Answer 25

Question 26

Lemma; Suppose that f,g: (a,b) to R are differentiable at x₀ then f + g is differentiable at x₀

Answer 26

Question 27

Lemma; Suppose that f,g: (a,b) to R are differentiable at x₀ then fg is differentiable at x₀

Answer 27

Question 28

Lemma; If g is differentiable at x₀ and g(x₀) doesnt equal 0 then 1/g is differentiable at x₀

Answer 28

Question 29

Prove the Caratheodory formulation of differentiation

Answer 29

Question 30

State and prove the chain rule

Answer 30

Question 31

Prove the derivative of inverses theorem

Answer 31

Question 32

State and prove L’Hopitals rule

Answer 32

Question 33

Lemma; Let f:(a,b) to R. Suppose that x₀ in [a,b] is a local maximum or minimum of f, and that f is differentiable. Then either

i) x₀ is an endpoint
ii) f’(x₀) = 0

Answer 33

Question 34

Prove Rolles Theorem

Answer 34

Question 35

Lemma; An nth degree polynomial can have at leat n distinct real roots

Answer 35

Let P be a polynomial of degree n with at least n+1 distinct roots. Then by noting that P is infinitely differentiable and that all derivatives are continuous we can repeatedly apply Rolles Theorem, Then P⁽¹⁾(x) has at least n distinct roots,….., P⁽ⁿ⁾(x) has at least 1 root. But P⁽ⁿ⁾ is a non-zero constant, so this is impossible

Question 36

Prove Taylors Theorem

Answer 36

Question 37

Lemma; Any nth degree polynomial P is completely determined by its first n derivatives at any point a in R

Answer 37