Let g: R→R be a polynomial function of even degree and let B={g(x)|x ∈R} be the range of g. Define g such that it has AT LEAST TWO TERMS G(x) - 1 - 3x^2 1. Using the properties and definitions of the...
Let g: R→R be a polynomial function of even degree and let B={g(x)|x ∈R} be the range of g. Define g such that it has AT LEAST TWO TERMS
G(x) - 1 - 3x^2
1. Using the properties and definitions of the real number system, and in particular the definition of supremum, construct a formal proof showing inf(B) exists OR explain why B does not have an supremum.
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Let g: R→R be a polynomial function of even degree and let B={g(x)|x ∈R} be the range of g. Define g such that it has AT LEAST TWO TERMS G(x) - 1 - 3x^2 1. Using the properties and definitions of the...
Let g: R→R be a polynomial function of even degree and let B={g(x) ER) be the...
Let g: R→R be a polynomial function of even degree and let B={g(x) ER) be the range of g. Define g such that it has at least two terms. 1. Using the properties and definitions of the real number system, and in particular the definition of infimum, construct a formal proof showing inf(B) exists OR explain why B does not have an infimum. 2. Using the properties and definitions of the real number system, and in particular the definition of...
3. (a) Let f be an infinitely differentiable function on R and define х F(x) =...
3. (a) Let f be an infinitely differentiable function on R and define х F(x) = e-y f(y) dy. Find and prove a formula for F(n), the nth derivative of F. (b) Show that if f is a polynomial then there exists a constant C such that F(n)(x) = Cem for sufficiently large n. Find the least n for which it is true.
3. (a) Let f be an infinitely differentiable function on R and define F(x) = [-vf(u)...
3. (a) Let f be an infinitely differentiable function on R and define F(x) = [-vf(u) dy. Find and prove a formula for F(n), the nth derivative of F. (b) Show that if f is a polynomial then there exists a constant C such that F(n)(x) = Cea for sufficiently large n. Find the least n for which it is true.
(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,...
(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,... , cn be distinct elements in R. Furthermore, let (ii) Let f(c)g(c) for all i = 0,1,2,...n. g(x) Prove that f(x - where r, s E Z, 8 ± 0 and gcd(r, s) =1. Prove that if x is a root of (iii) Let f(x) . an^" E Z[x], then s divides an. aoa1
(i)...
Q9 6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a...
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
1- What are the units in Z? What are the units in F[x]? Don’t write out a formal proof, but discuss why. 2- What is the analogy between Z and F[x]? 3- Let p(x) = x^3 + 3x + 1 = (x+3)^2 * (x+4) in Z5[x...
1- What are the units in Z? What are the units in F[x]? Don’t write out a formal proof, but discuss why. 2- What is the analogy between Z and F[x]? 3- Let p(x) = x^3 + 3x + 1 = (x+3)^2 * (x+4) in Z5[x]. (a) Perform the following computation in Z5[x]/(p(x)). Give your answers in the form [r(x)] where r(x) has degree as small as possible. i. [4x] + [3x^2 + x + 2] ii. [x^2][2x^2+1] (b) Show...
Real analysis 10 11 12 13 please (r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and li...
Real analysis
10 11 12 13 please
(r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....
Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be...
Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be the usual “less than or equal to” total order on the set A=(−2,0)∪(0,2). Consider the subset B={−1/n | n∈N,n≥1}⊆A. Determine an upper bound for B in A.. Then formally prove that B has no least upper bound in A by arguing that every element of A fails the criteria in the definition of least upper bound. Note: make sure you are addressing the technical...
Let f be a function of two variables x,y. Define r(t) Yobt y(t) Хо + at,...
Let f be a function of two variables x,y. Define r(t) Yobt y(t) Хо + at, Let g(t) f(x(t), y(t)) (a) Explain what does (x(t), y(t)) represent in the plane (b) Explain how the graph of g can be viewed as a part of the graph of f. dg (c) Find dt \t=0 in terms of partial derivatives of f. What does this repre- sent? (d) What does your answer in part c become if a 0 or b=0? (e)...
Please prove the following theorems using the provided axioms and definitions, using terms like s...
Please prove the following
theorems using the provided axioms and definitions, using terms
like suppose, let..ect. Please WRITE CLEARLY AND TYPE IF YOU
CAN.
1 Order Properties Undefined Terms: The word "point" and the expression "the point x precedes the point y" will not be defined. This undefined expression will be written x 〈 y. Its negation, "x does not precede y," will be written X y. There is a set of all points, called the universal set, which is...
Let g: R→R be a polynomial function of even degree and let B={g(x) ER) be the range of g. Define g such that it has at least two terms. 1. Using the properties and definitions of the real number system, and in particular the definition of infimum, construct a formal proof showing inf(B) exists OR explain why B does not have an infimum. 2. Using the properties and definitions of the real number system, and in particular the definition of...
3. (a) Let f be an infinitely differentiable function on R and define х F(x) = e-y f(y) dy. Find and prove a formula for F(n), the nth derivative of F. (b) Show that if f is a polynomial then there exists a constant C such that F(n)(x) = Cem for sufficiently large n. Find the least n for which it is true.
3. (a) Let f be an infinitely differentiable function on R and define F(x) = [-vf(u) dy. Find and prove a formula for F(n), the nth derivative of F. (b) Show that if f is a polynomial then there exists a constant C such that F(n)(x) = Cea for sufficiently large n. Find the least n for which it is true.
(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,... , cn be distinct elements in R. Furthermore, let (ii) Let f(c)g(c) for all i = 0,1,2,...n. g(x) Prove that f(x - where r, s E Z, 8 ± 0 and gcd(r, s) =1. Prove that if x is a root of (iii) Let f(x) . an^" E Z[x], then s divides an. aoa1
(i)...
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
Real analysis
10 11 12 13 please
(r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....
Let f be a function of two variables x,y. Define r(t) Yobt y(t) Хо + at, Let g(t) f(x(t), y(t)) (a) Explain what does (x(t), y(t)) represent in the plane (b) Explain how the graph of g can be viewed as a part of the graph of f. dg (c) Find dt \t=0 in terms of partial derivatives of f. What does this repre- sent? (d) What does your answer in part c become if a 0 or b=0? (e)...
Please prove the following
theorems using the provided axioms and definitions, using terms
like suppose, let..ect. Please WRITE CLEARLY AND TYPE IF YOU
CAN.
1 Order Properties Undefined Terms: The word "point" and the expression "the point x precedes the point y" will not be defined. This undefined expression will be written x 〈 y. Its negation, "x does not precede y," will be written X y. There is a set of all points, called the universal set, which is...
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