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- September 13, 2020
- By menge

Answer Q2-Q6 in PDF, need the a brief justification.

MTH299 – Homework 7 Name: XXXXXXXX Homework 7; Due Wednesday, 10/19/2016

Quick Answer Questions. No work needed.

Question 1. Come up with a simpler expression for the following sets

(a) P = ?

[ Nn , where Nn = {1, 2, 3, . . . , n}. n=1 (b) Q = ?

n=1

1

1

1 ? ,2 +

.

n

n

?

[

1

2

(c) R =

1 + ,5 ?

.

n

n

n=1

(d) S = ?

[

1

n=1 (e) T = ?

[ n

,1 . (?n, n). n=1 See section 1 of the additional notes for this notation on union and intersection.

Question 2. For any real number r ? [1, ?), define

1

1

Ar = x ? R | ? ? x ? 2 ?

.

r

r

[

For the following choice of index sets I, compute

Ar and

Ar : (See section 1 of the additional notes

r?I r?I for these notations.)

(a) I = {1, 2, 3, 8}.

(b) I = {x ? R : x > 1}.

Medium Justification Questions. Provide brief justifications for your responses.

Question 3. For each real number r, define the subset Ar of R2 by

Ar = {(x, y) ? R2 | y = rx}.

(a) Describe geometrically the sets A?1 , A0 , and A1 .

MSU 1 Due: 10/19/2016 MTH299 – Homework 7 Name: XXXXXXXX (b) Bonus Question: Let B = {(x, y) ? R2 | if x = 0 then y = 0}. Prove that [ Ar = B. r?R (Note that B consists of all points in R2 except those on the positive and negative y-axis.)

Complete Justification Questions. Provide complete justifications for your responses.

Question 4. Define the function, f : R ? R via the assignment

f (x) = ?x2 ? 4x + 1,

and define the set E as

E := {x ? R : f (x) > 0}.

Prove that the set E is bounded.

Question 5. Define sets

A = {(?, ?) ? R2 : ?1 ? ? ? 1, ?1 ? ? < 1}

p

B = {(x, y) ? R2 : 1 ? x2 + y 2 ? 3, 0 < x ? 3, 0 ? y ? 3}

and the function f : A ? B given by

f ((?, ?)) = ((2 + ?) cos(?(? + 1)/4), (2 + ?) sin(?(? + 1)/4)) .

Answer the following questions.

p

(a) For (?, ?) ? A, let (x,p

y) = f ((?, ?)). Can you express x2 + y 2 and tan?1 (y/x) in terms of ? and ??

(That is, write down x2 + y 2 and tan?1 (y/x) as an expression in ? and ?)

p

(b) Let r = x2 + y 2 and ? = tan?1 (y/x). Define a new set, C, such that B and C are the same subsets

of R2 when expressed in the x, y variables (but not in the r,? variables), but the set C is expressed in

terms of r and ?.

(That is, write out the set C in the form C = {(r, ?) ? X : description of r and ?}. You have to

determine the set X and the correct description of (r, ?))

Associated with this re-definition of B, rewrite the assignment rule for f such that it can be realized

as a function, f : C ? B. That is to say, f should now output ordered pairs in these new (r, ?)

?co-ordinates?.

(c) Prove that f : C ? B is a bijection (i.e., it is both injective and surjective).

(d) Bonus question: Can you picture what is this function doing geometrically?

Question 6. Define the functions h : R ? R via h(x) = x6 and i : R ? R via i(x) = x7 . Use only the

following three items, plus basic algebraic manipulations, to identify and justify the formulas for both h0 (x)

and i0 (x) (you will provide two separate proofs, one for each). Here are your items you are allowed to use:

(i) for f : R ? R, with f (x) = x3 , it is known that f 0 (x) = 3×2 .

(ii) for g : R ? R, with g(x) = x4 , it is known that g 0 (x) = 4×3 .

(iii) the product rule for differentiable functions.

(Note, the point of this question is to test your ability to write clear, coherent, and correct justifications of

the given statements ? it is not really meant to challenge your thinking abilities.)

MSU 2 Due: 10/19/2016