Solve it step by step and please with clear handwritten. Be ready to follow the comment

Integrability Topic

Question1. Let g : [0, 1] —> R be defined by if x=0, g(x)=1; if x=m/n (m and n are positive integer with no common factor), g(x)=1/n; if x doesn't belong to rational number, g(x)=0

Use the sequential criterion for continuity to prove that g is discontinuous at every rational number in[0,1]

Question.2 g is continuous at any irrational point in[0,1]. Explain why g is Riemann integrable on[0,1] based on the following fact that

Suppose h:[a,b]→R is continuous everywhere except at a countable number of points in[a,b]. Then h is Riemann integrable on[a,b]

Question.3

Letf:[0,1]→R be defined by f(x)=0 if x=0; f(x)=1 if 0<x<=1 we know that f is integrable on [0,1] Suppose c is a rational number in [0,1]. Compute(f◦g)(c). Now suppose c is an irrational number in[0,1]. Compute(f◦g)(c). Can you recognize the function f◦g:[0,1]→R?

Use R to solve:

Find the approximate solution x' to :

0.89x1 + 0.53x2 = 0.36

0.47x1 + 0.28x2 = 0.19

Find the error x'-x* between the computed solution and the true solution.

Compare the size of this error with the size of the residual r=b-Ax'

Rudin Ch 4, p. 99 #7. If E ⊂ X and if f is a function defined on X, the restriction of f to E is the function g whose domain of definition is E, such that g(p) = f (p) for p ∈ E. Define f and g on 2 by: f (0, 0) = g(0, 0) = 0, f (x, y) = € xy 2 x 2 + y 4 , g(x, y) = € xy 2 x 2 + y 6 if (x, y) ≠ (0, 0). Prove that f is bounded on 2 , that g is unbounded in every neighborhood of (0, 0) and that f is not continuous at (0, 0); nevertheless, the restrictions of both f and g to every straight line in 2 are continuous! .Explain step by step in detal

Find the subgroup of d4 and the normal and non normal subgroups of d3 and d4 using u and v, u being the flips and v being the rotations.

Give 2 different examples of an infinite dimensional vector space and provide an explanation in possible.

This is a review question for linear algebra and I am trying to better understand the concept.

Thank you!

A semi trailer of 9.0% wb moisture soybeans sits outside overnight without a tarp and it rains 1.5 in. The trailer had been level, filled to a depth of 4.5 feet. Assume trailer dimensions of 52 ft × 99 in. Assume the 1.5 in. rain came straight down and was uniformly distributed over the entire volume of soybeans and that nothing leaked from the trailer. Also assume that a bushel of soybeans (1.25 ft3) weighs 60 lb before it rains. Calculate the new moisture content of the soybeans. Assume that 1 bu occupies 1.25 ft3, water = 62.4 lb/ft3. Note: all information is accurate but not necessarily useful in solving the problem.

a) In your own words, explain the concept of variable scope. Include one or more code fragments that you write for this discussion post to help illustrate your case. Clearly mark as appropriate any code fragments as either "script" code or "function" code, and use comments in your code to note the locality of each variable at least the first time the variable appears in the script or function. If a variable is local in scope, be sure to notate what the variable is local to (i.e. the script, a particular function, etc.). Your code for this discussion post should include at least two different local variables, and at least one global variable.

b) Compare and contrast separate function files with anonymous functions. In your opinion, when may it be best to use one over the other? Can you use them interchangeably in all circumstances?

Prove the case involving ∨E(or elimination) of the inductive step of the (strong) soundness theorem for natural deduction in classical propositional logic. Hint: you need to simultaneously consider 3 different instances of entailment, 1 regular and 2 featuring the transformation of an assumption into a premise.

Find the maximum and minimum values, and at what point they occur.

A. F(x,y)= 25-16x-2x²+8y+4y² over the region bounded by x=5, y= -2, and y=x

B. F(x,y)= x²y+3xy-4y+15x over the region bound by x=0, x=3, y= -4, and y=4

Solve the given differential equation by undetermined coefficients.

y''' − 3y'' + 3y' − y = x − 9^x

For each of the following equations, find the general solution:

(x^2)y′′+xy′+ 4y= sin(lnx) + sin(2 lnx)

y′′+y=tcost

Thank you!

Answer the following questions: (a) Show that the following is a tautology by using truth table and using list of equivalences. (This problem should be solved using 2 different methods mentioned above). ((¬p −→ q) ∧ (¬p −→ ¬q)) −→ p (b) Show that the compound propositions are logically equivalent, by using truth table and using list of equivalences. ¬p ∨ (r −→ ¬q) and (¬p ∨ ¬q) ∨ ¬r (c) Show that the propositions ¬p ∨ (¬r ∨ q) and r −→ (p −→ q) are logically equivalent using the method of your choice.

problem 10 :

Explain why ∃xP(x) ∨ ∃xQ(x) ≡ ∃x(P(x) ∨ Q(x)). You need not formally prove it, but you should give a convincing explanation for why it is true?

Use the fact established in the problem that ∃xP(x) ∨ ∃xQ(x) ≡ ∃x(P(x) ∨ Q(x)) to prove that ∀xP(x) ∧ ∀xQ(x) ≡ ∀x(P(x) ∧ Q(x)).
Use Problem 10 above to prove that ∀xP(x) → ∃xP(x) ≡ T

How would you do the work to find which PDEs below can be separated into ODEs by assuming a product of unknown functions x,y and z?

1) Uxx+Uyy+Uzz=0

2) Uxx+Uyy+Ux+Uy=0

3) Uxx+3Uxy+Uyy=0

4) Uxx+3Uxy+7Uy=0

5) x^2Uxx+yUyy=0

6) aUxy+bu=0