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)|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.
1. Let Xn ER be a sequence of real numbers. (a) Prove that if Xn is an increasing sequence bounded above, that is, if for all n, xn < Xn+1 and there exists M E R such that for all n E N, Xn < M, then limny Xn = sup{Xnin EN}. (b) Prove that if Xn is a decreasing sequence bounded below, that is, if for all n, Xn+1 < xn and there exists M ER such that for...
1. Let A be a nonempty subset of R such that every number in A is greater than 2 (NOTE: This doesn’t necessarily mean that A = (2,∞)). (a) Explain why A must have an infimum. (b) Let c = inf(A). Prove that a∈A INTERSECTION (−∞,a] = (−∞,c]. CAN SOMEONE PLZ HELP ME WITH THIS QUESTION. 1. Let A be a nonempty subset of R such that every number in A is greater than 2 (NOTE: This doesn't necessarily mean...
Let the function f: (a, b) → R is continuous in (a, b). If sup {f(x): x ∈ (a, b)} = L> 0 and inf {f(x): x ∈ (a, b)} = M <0, then prove that there is a c ∈ (a , b) such that f (c) = 0.
Problem 3. (1) Let H be a Hilbert space and S, TE B(HH). Then, prove that ||ST|| ||||||||| (2) Let X, Y be Hilbert spaces and Te B(X,Y). Then, prove that ||1||| sup ||T3|1 TEX=1 Let X, Y be Banach spaces. Definition (review) We denote by B(X, Y) a set of all bounded linear operators T:X + Y with D(T) = X. B(X, Y) is a vector space. Definition (review) A linear operator T:X + Y is said to be...
Let D P3P3 be the function that sends a polynomial of degree 3 to its derivative (a) Find an eigenvector for D or explain why no eigenvector exists Write your solution here (b) Let B 1 x, x + x2, x2 + x3,x3}. B is a basis for P3. Find MDB-B Here, MD.- is the unique matrix such that MD-xs = [D(x)]s Write your solution here Recall that D: P is polynomial differentiation. 1x, x +x2, x2 +x3,x3} and C...
4. Let f()VI+ x. (a) Compute P2(x), the degree 2 Taylor polynomial for f at ro 0. (b) Use P2 to approximate f(0.5) required to evaluate a real polynomial of degree 5. How many multiplications number? Explain n at a real are 6. Show that if x, y and ry are real mumbers in the range of our floating point system, then ay-f(ry3 + O(*) ay
be a coordinate function for 1.) (Coordinate functions) Let f: R a (a) (Exercise 3A) Prove that -f is a coordinate function as well. (5 (b) For which real numbers a, b e R is a f+b a coordinate function? (c) Let g:E → R be a coordinate function. Prove that there exists a line ( points) Justify your answer with a proof. (10 points) real number b E R, such that g-f+b or gf+b. (10 points)
Question 1: Let R be the set of real numbers and let 2R be the set of all subsets of the real numbers. Prove that 2 cannot be in one-to-one correspondence with R. Proof: Suppose 2 is in one-to-one correspondence with R. Then by definition of one- to-one correspondence there is a 1-to-1 and onto function B:R 2. Therefore, for each x in R, ?(x) is a function from R to {0, 1]. Moreover, since ? is onto, for every...
Let f : [a, b] → R and xo e (a,b). Assume that f is continuous on [a,b] \{x0} and lim x approaches too x0 f(x) = L (L is finite) exists. Show that f is Riemann integrable. 1. (20 pts) Let f : [a, b] R and to € (a,b). Assume that f is continuous on [a, b]\{ro} and limz-ro f (x) = L (L is finite) exists. Show that f is Riemann integrable. Hint: We split it into...