Find the Upper and Lower Darboux sums for the following functions.
(i) f(x) = −x − 1 on [0, 3], n = 3. [10]
(ii) f(x) = 1 + 2x 0n [0, 1] , n = 3

Find the Upper and Lower Darboux sums for the following functions.
(i) f(x) = −x − 1 on [0, 3], n = 3. [10]
(ii) f(x) = 1 + 2x 0n [0, 1] , n = 3

(e) Find the first 4 terms of the Taylor series for the following functions
(i) ln x centered at x0 = 1 . [8]
(ii) sin x centered at x0 =
π
4

Solve the following initial value problems:

a) ut+xux= -tu, x is in R, t>0; u(x,0) =f(x), x is in R.

c) ut+ux=-tu, x is in R, t>0; u(x,0)=f(x), x is in R

d)2ut+ux = -2u, x,t in R, t>0; u(x,t)=f(x,t) on the straight line x = t, where f is a given function.

Use Newton-Raphson to find the real root to five significant figures 64x^3+6x^2+12-1=0

Prove that x^8 - 14x^4 + 25 is irreducible

This problem is also a Monte Carlo simulation, but this time in the continuous domain: must use the following fact: a circle inscribed in a unit square

has as radius of 0.5 and an area of ?∗(0.52)=?4.π∗(0.52)=π4.

Therefore, if you generate num_trials random points in the unit square, and count how many land inside the circle, you can calculate an approximation of ?

For this problem, you must create code in python

(A) Draw the diagram of the unit square with inscribed circle and 500 random points, and calculate the value of ?

Prove the following statements!

1. There is a bijection from the positive odd numbers to the integers divisible by 3.

2. There is an injection f : Q→N.

3. If f : N→R is a function, then it is not surjective.

Prove the following statements!

1. If A and B are sets then

(a) |A ∪ B| = |A| + |B| − |A ∩ B| and

(b) |A × B| = |A||B|.

2. If the function f : A→B is

(a) injective then |A| ≤ |B|.

(b) surjective then |A| ≥ |B|.

3. For each part below, there is a function f : R→R that is

(a) injective and surjective.

(b) injective but not surjective.

(c) surjective but not injective.

(d) neither injective nor surjective

Prove the following statements!

1. Let S = {0, 1, . . . , 23} and define f : Z→S by f(k) = r when 24|(k−r). If g : S→S is defined by

(a) g(m) = f(7m) then g is injective and

(b) g(m) = f(15m) then g is not injective.

2. Let f : A→B and g : B→C be injective. Then g ◦f : A→C is injective.

3. Let f : A→B and g : B→C be surjective. Then g ◦ f : A→C is surjective.

4. There is a surjection f : A→B such that f −1 : B→A is not a function.

Define a function from N to Z that is both one to one and onto. Explain why it is a bijection?

Find a function from Q to Z that is one to one.

Please help me with these two questions. Thank you!

Find the set of ALL optimal solutions to the following LP: min z= 3x1−2x2 subject to 3x1+x2≤12 3x1−2x2−x3= 12 x1≥2 x1, x2, x3≥0

4. Translate each of the following statements into a symbolic logic and tell if each of the following is true or false, with a full justification (you do not have to justify your answer to (ii), which was done before) : (i) Every integer has an additive inverse. (ii) If a and b are any integers such that b > 0, then there exist integers q and r such that a = bq + r, where 0 ≤ r < b. (Note that this sentence does not have a uniqueness part of q or r.) (iii) Every integer has a unique multiplicative inverse. (Answer this question without using the symbol ∃! that we have not used much in class.) (iv) Any two real numbers x and y satisfy x < y. (v) Every real number has a greater real number. (vi) There exists a real number that is less than any real number. (vii) There are two real numbers x and y satisfy x < y. (viii) Given any two real numbers one of them is bigger than the other.

Write each vector as a linear combination of the vectors in S. (Use

s₁ and s₂,

respectively, for the vectors in the set. If not possible, enter IMPOSSIBLE.)

S = {(1, 2, −2), (2, −1, 1)}

(a)    z = (−13, −1, 1)

z =  

(b)    v = (−1, −5, 5)

v =  

(c)    w = (−2, −14, 14)

w =  

(d)    u = (1, −4, −4)

u =