Advanced Calculus-2 proof: Suppose that we have two numbers a,b belong to R with a<=b. Suppose...
Advanced Calculus-2 proof:
Suppose that we have two numbers a,b belong to R with a<=b.
Suppose we have a continuous
function f : [a,b] -> R. Then there are two numbers c,d belong
to R with c<=d such that f([a,b]) = [c,d].
Homework Answers
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Advanced Calculus-2 proof:
Suppose that we have two numbers a,b belong to R with a<=b.
Suppose...
please solve 2 to 6 with details Advanced Calculus: HW 3 (1) Suppose that a E...
please solve 2 to 6 with details
Advanced Calculus: HW 3 (1) Suppose that a E R has the following property: for all n e N, a < Prove that a<0. (2) Prove that the set of integers Z is not dense in R (3) Let A = {xeQ: >0}. Determine whether A is dense in R, and justify your answer with a proof. (4) Find the supremum of the set A= {a e Q: <5} (5) Let a >...
(Advanced Calculus and Real Analysis) - Lebesgue integral, Convergence properties of the Integral for Non-negative functions...
(Advanced Calculus and Real Analysis) - Lebesgue
integral, Convergence properties of the Integral for Non-negative
functions
*
Supposef is a nonnegative M-measurable function with Soofd) <0. Then we define the Laplace transform of f, denoted, F, by F(t) = -f(x) dx(r), t> 0. J[0,00) Show that a) F is real valued. b) F is continuous on (0,0). Hint: First establish that F is nonincreasing, c) lim- F(t) = 0.
10. Use the Fundamental Theorem of Calculus to provide a proof of Theorem 8.4 under the additiona...
10. Use the Fundamental Theorem of Calculus to provide a proof of Theorem 8.4 under the additional assumption that each fis continuous on I la, b).(Hint: For x in la, b.o)If f g uniformly on [a, b], then Theorem 8.3 implies that im f.(x) f (x8. It follows that frpuintwise on la, b), where F(x) -lim frCro) + .By Theorem 6.12, F()-x) on la,b). Now show that f uniformly on la, b].] F heorem 8.4 Suppose that neN is a...
This is for an advanced calculus/advanced math course. Please be as detailed as possible in your...
This is for an advanced calculus/advanced math course.
Please be as detailed as possible in your answer. Thank you so much
in advance.
PLEASE DO NOT USE CALCULATORS OR SOFTWARE TO SOLVE THESE
PROBLEMS. PLEASE DO EVERYTHING BY HAND. THANK YOU!!
You can use the theorem below to solve the
problem:
16. Apply the Divergence Theorem to compute I = SS. F.dS, where F(x, y, z) = (xz2 + cos(y + 2), šv* +e”,z²z+y+ 2) 1 and S is the...
Need help in proof There are two functions f(x) and g(x) and two real numbers a,...
Need help in proof There are two functions f(x) and g(x) and two real numbers a, b. the period of the function f(x) is T1 and the period of the function g(x) is T2. How do I prove that if T1 and T2 have common multiple, the function y = a*f(x) ± b*g(x) is periodic function and her period is equal to the lowest common multiple of T1 and T2?
6 Proof of the dual theorem Proof: We will assume that the primal LP is in canonical form Maximiz...
please explain it to me clearly
6 Proof of the dual theorem Proof: We will assume that the primal LP is in canonical form Maximize Zr, such that Arb 20 12 Its dual is Minimize W·ry, such that ATy c (no sign constraints on y). Step 1: Suppose xB is the basic variables in the optimal BFS (say r*) f follows from the above discussion that Row (0) of the optimal tableau will be the Prianal LP. It Basic VariableRow2...
(Limit of functions) Let f : 2-» C be a function, and assume that D(a, r)...
(Limit of functions) Let f : 2-» C be a function, and assume that D(a, r) C Q. We say that lim f(z) L Ď(a, 6) we have |f(z) Ll < e. if for any e > 0 there exists 6 > 0, such that for any z e (a) State the negation of the assertion "lim^-,a f(z) = L". (b) Show that lim- f(z) L if and only if for any sequence zn -» a, with zn a for...
You're the grader. To each "Proof", assign one of the following grades: A (correct), if the claim and proof are correct, even if the proof is not the simplest, or the proof you would...
You're the grader. To each "Proof", assign one of the following grades: A (correct), if the claim and proof are correct, even if the proof is not the simplest, or the proof you would have given. C (partially correct), if the claim is correct and the proof is largely a correct claim, but contains one or two incorrect statements or justifications. . F (failure), if the claim is incorrect, the main idea of the proof is incorrect, or most of...
Lemma. If two vector spaces have the same dimension then they are isomorphic Proof. To show that ...
Lemma. If two vector spaces have the same dimension then they are isomorphic Proof. To show that any two spaces of dimension n are isomorphic, we can simply show that any one is isomorphic to R. Then we will have shown that they are isomorphic to each other, by the transitivity of isomorphism (which was established in the first Theorem of this section) Theorem 1 Isomorphism is an equivalence relation among ctor spaces Let v be n--dimensional. Fix a basis...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and conti...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
please solve 2 to 6 with details
Advanced Calculus: HW 3 (1) Suppose that a E R has the following property: for all n e N, a < Prove that a<0. (2) Prove that the set of integers Z is not dense in R (3) Let A = {xeQ: >0}. Determine whether A is dense in R, and justify your answer with a proof. (4) Find the supremum of the set A= {a e Q: <5} (5) Let a >...
(Advanced Calculus and Real Analysis) - Lebesgue
integral, Convergence properties of the Integral for Non-negative
functions
*
Supposef is a nonnegative M-measurable function with Soofd) <0. Then we define the Laplace transform of f, denoted, F, by F(t) = -f(x) dx(r), t> 0. J[0,00) Show that a) F is real valued. b) F is continuous on (0,0). Hint: First establish that F is nonincreasing, c) lim- F(t) = 0.
10. Use the Fundamental Theorem of Calculus to provide a proof of Theorem 8.4 under the additional assumption that each fis continuous on I la, b).(Hint: For x in la, b.o)If f g uniformly on [a, b], then Theorem 8.3 implies that im f.(x) f (x8. It follows that frpuintwise on la, b), where F(x) -lim frCro) + .By Theorem 6.12, F()-x) on la,b). Now show that f uniformly on la, b].] F heorem 8.4 Suppose that neN is a...
This is for an advanced calculus/advanced math course.
Please be as detailed as possible in your answer. Thank you so much
in advance.
PLEASE DO NOT USE CALCULATORS OR SOFTWARE TO SOLVE THESE
PROBLEMS. PLEASE DO EVERYTHING BY HAND. THANK YOU!!
You can use the theorem below to solve the
problem:
16. Apply the Divergence Theorem to compute I = SS. F.dS, where F(x, y, z) = (xz2 + cos(y + 2), šv* +e”,z²z+y+ 2) 1 and S is the...
please explain it to me clearly
6 Proof of the dual theorem Proof: We will assume that the primal LP is in canonical form Maximize Zr, such that Arb 20 12 Its dual is Minimize W·ry, such that ATy c (no sign constraints on y). Step 1: Suppose xB is the basic variables in the optimal BFS (say r*) f follows from the above discussion that Row (0) of the optimal tableau will be the Prianal LP. It Basic VariableRow2...
(Limit of functions) Let f : 2-» C be a function, and assume that D(a, r) C Q. We say that lim f(z) L Ď(a, 6) we have |f(z) Ll < e. if for any e > 0 there exists 6 > 0, such that for any z e (a) State the negation of the assertion "lim^-,a f(z) = L". (b) Show that lim- f(z) L if and only if for any sequence zn -» a, with zn a for...
You're the grader. To each "Proof", assign one of the following grades: A (correct), if the claim and proof are correct, even if the proof is not the simplest, or the proof you would have given. C (partially correct), if the claim is correct and the proof is largely a correct claim, but contains one or two incorrect statements or justifications. . F (failure), if the claim is incorrect, the main idea of the proof is incorrect, or most of...
Lemma. If two vector spaces have the same dimension then they are isomorphic Proof. To show that any two spaces of dimension n are isomorphic, we can simply show that any one is isomorphic to R. Then we will have shown that they are isomorphic to each other, by the transitivity of isomorphism (which was established in the first Theorem of this section) Theorem 1 Isomorphism is an equivalence relation among ctor spaces Let v be n--dimensional. Fix a basis...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
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