Let pn = (an+bn)/2 , p = limn→∞pn, and en = p−pn. Here [an,bn], with n...
Let pn = (an+bn)/2 , p = limn→∞pn, and en = p−pn. Here [an,bn], with n ≥ 1, denotes the successive intervals that arise in the Bisection method when it is applied to a continuous function f.
Show that |pn −pn+1| = .
(b1-a1)
Homework Answers
![Prantben, keranston ant t I bisection method, Either, [ann, bron] = [an, Pn] or, [and, but] = [en, bu] Case I [anos, buri] =](http://img.homeworklib.com/questions/d8c20390-925a-11eb-8400-d540e0b1595e.png?x-oss-process=image/resize,w_560)
![11Pn-Pasil = Ibn-an) 4 Case I [ant, bad]=[Pn, br] > ant=Pn= anton e bur= bn • Ano-an-an-bn, bir-bn ao an- anos + on-tomon lan](http://img.homeworklib.com/questions/d9aa4470-925a-11eb-91d0-ed7355508431.png?x-oss-process=image/resize,w_560)
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Let pn = (an+bn)/2 , p = limn→∞pn, and en = p−pn. Here [an,bn],
with n...
(2) Let Pn [x] = {p € P[x] : degp <n}, where P[x] is the set...
(2) Let Pn [x] = {p € P[x] : degp <n}, where P[x] is the set of all polynomials. Let the polynomials li() defined by II;tilt - a;) i=0,1,...11 bi(T) = 11: a; - aj) where aj, j = 0,1,..., are distinct real numbers and aia . Show that (d) The change of basis transformation from the standard basis ', j = 0,1,...,n to l; () is given by the Vandermonde matrix (1 00 ... am 1 01 .01 1...
Prove the following Definition 6.6.1 (Subsequences). Let (an) =) and (bn), m=0 be sequences of real...
Prove the following
Definition 6.6.1 (Subsequences). Let (an) =) and (bn), m=0 be sequences of real numbers. We say that (bn)is a subsequence of (an) a=iff there exists a function f :N + N which is strictly increasing (i.e., f(n + 1) > f(n) for all n EN) such that bn = f(n) for all n E N.
Problem 6. Consider the n independent trails in Problem 5. Let On be the probability that...
Problem 6. Consider the n independent trails in Problem 5. Let On be the probability that there is no three consecutive successes in n trails. (1). Show that limn+cQn = 0 (2). Show that Qn = (1 - pQn-1 + p(1 - pQn-2 + p (1 - p)Qn-3 for n 3 (Hint: condition on the first failure). Problem 5. Suppose we do n independent trails that each has a probability P E (0,1) to result in success. Let Pn be...
Please answer d,e,f and g, thank you! roblem 1. Let (U common p.d.f. i 1 be...
Please answer d,e,f and g, thank you!
roblem 1. Let (U common p.d.f. i 1 be a sequence of ii.d. discrete random variables with f(k) for k = 1, 2, 3 and for n 21 let Sn = Σ,u. (a) Find the probability that S2 is even. (b) Find the probability that Sn is even given that S,-1 is even. (e) Find the probability that Sn is even given that S-1 is odd. (d) Let pn P(Sn is even). Find...
For some n > 1, let T E End(Pn) be given by T(p) = p'. Show...
For some n > 1, let T E End(Pn) be given by T(p) = p'. Show that T is not diagonalizable.
Exercise 5.17. Let {en}nez be the set of trigonometric functions. Suppose that {an}nez, {bn}nez are sequences...
Exercise 5.17. Let {en}nez be the set of trigonometric functions. Suppose that {an}nez, {bn}nez are sequences of complex numbers, f = Enez anen, and g = Enez bnen, where the equalities are in the Lº sense. Show that (f, g) = Enez anbr. In particular, show that || F || 22 (1) - Enez lan?
2. Let {An}n>1 and {Bn}n>ı be two sequences of measurable sets in the measurable space (12,F)....
2. Let {An}n>1 and {Bn}n>ı be two sequences of measurable sets in the measurable space (12,F). Set Cn = An ñ Bn, Dn = An U Bn: (1) Show that (Tim An) ^ ( lim Bm) – lim Cn (lim An) ( lim Bu) C lim Dm and 100 noo (2) Show by example the two inclusions in (1) can be strict.
1. Let T : Pn(R) + Pn+1(R) be defined: T(P(x)) = (x + 1)p(x + 2)...
1. Let T : Pn(R) + Pn+1(R) be defined: T(P(x)) = (x + 1)p(x + 2) bases {1, X, ..., (a) (2 marks) Show that T is a linear transformation. (b) (3 marks) Is T one-to-one? Describe ker(T). What is the rank of T? (c) (8 marks) Find a matrix representation for T with respect to the standard xn} for Pn and {1, 2, ..., xn+1} for Pn+1 if n = 4. (d) (5 marks) Let D : Pn+1(R) +...
Let Pn be the set of real polynomials of degree at most n. Show that s (pEP5 :p(-7) (9)) is a sub...
Let Pn be the set of real polynomials of degree at most n. Show that s (pEP5 :p(-7) (9)) is a subspace of P5
Let Pn be the set of real polynomials of degree at most n. Show that s (pEP5 :p(-7) (9)) is a subspace of P5
1. Let T : P (R) Pn+1(R) be defined: T(p()) = (x + 1)p(x + 2)...
1. Let T : P (R) Pn+1(R) be defined: T(p()) = (x + 1)p(x + 2) (a) (2 marks) Show that T is a linear transformation. (b) (3 marks) Is T one-to-one? Describe ker(T). What is the rank of T? (c) (8 marks) Find a matrix representation for T with respect to the standard bases {1, 2, ..., 2"} for Pn and {1, 2, ..., xn+1} for Pn+1 if n = 4. (d) (5 marks) Let D : Pn+1(R) +...
(2) Let Pn [x] = {p € P[x] : degp <n}, where P[x] is the set of all polynomials. Let the polynomials li() defined by II;tilt - a;) i=0,1,...11 bi(T) = 11: a; - aj) where aj, j = 0,1,..., are distinct real numbers and aia . Show that (d) The change of basis transformation from the standard basis ', j = 0,1,...,n to l; () is given by the Vandermonde matrix (1 00 ... am 1 01 .01 1...
Prove the following
Definition 6.6.1 (Subsequences). Let (an) =) and (bn), m=0 be sequences of real numbers. We say that (bn)is a subsequence of (an) a=iff there exists a function f :N + N which is strictly increasing (i.e., f(n + 1) > f(n) for all n EN) such that bn = f(n) for all n E N.
Problem 6. Consider the n independent trails in Problem 5. Let On be the probability that there is no three consecutive successes in n trails. (1). Show that limn+cQn = 0 (2). Show that Qn = (1 - pQn-1 + p(1 - pQn-2 + p (1 - p)Qn-3 for n 3 (Hint: condition on the first failure). Problem 5. Suppose we do n independent trails that each has a probability P E (0,1) to result in success. Let Pn be...
Please answer d,e,f and g, thank you!
roblem 1. Let (U common p.d.f. i 1 be a sequence of ii.d. discrete random variables with f(k) for k = 1, 2, 3 and for n 21 let Sn = Σ,u. (a) Find the probability that S2 is even. (b) Find the probability that Sn is even given that S,-1 is even. (e) Find the probability that Sn is even given that S-1 is odd. (d) Let pn P(Sn is even). Find...
For some n > 1, let T E End(Pn) be given by T(p) = p'. Show that T is not diagonalizable.
Exercise 5.17. Let {en}nez be the set of trigonometric functions. Suppose that {an}nez, {bn}nez are sequences of complex numbers, f = Enez anen, and g = Enez bnen, where the equalities are in the Lº sense. Show that (f, g) = Enez anbr. In particular, show that || F || 22 (1) - Enez lan?
2. Let {An}n>1 and {Bn}n>ı be two sequences of measurable sets in the measurable space (12,F). Set Cn = An ñ Bn, Dn = An U Bn: (1) Show that (Tim An) ^ ( lim Bm) – lim Cn (lim An) ( lim Bu) C lim Dm and 100 noo (2) Show by example the two inclusions in (1) can be strict.
1. Let T : Pn(R) + Pn+1(R) be defined: T(P(x)) = (x + 1)p(x + 2) bases {1, X, ..., (a) (2 marks) Show that T is a linear transformation. (b) (3 marks) Is T one-to-one? Describe ker(T). What is the rank of T? (c) (8 marks) Find a matrix representation for T with respect to the standard xn} for Pn and {1, 2, ..., xn+1} for Pn+1 if n = 4. (d) (5 marks) Let D : Pn+1(R) +...
Let Pn be the set of real polynomials of degree at most n. Show that s (pEP5 :p(-7) (9)) is a subspace of P5
Let Pn be the set of real polynomials of degree at most n. Show that s (pEP5 :p(-7) (9)) is a subspace of P5
1. Let T : P (R) Pn+1(R) be defined: T(p()) = (x + 1)p(x + 2) (a) (2 marks) Show that T is a linear transformation. (b) (3 marks) Is T one-to-one? Describe ker(T). What is the rank of T? (c) (8 marks) Find a matrix representation for T with respect to the standard bases {1, 2, ..., 2"} for Pn and {1, 2, ..., xn+1} for Pn+1 if n = 4. (d) (5 marks) Let D : Pn+1(R) +...
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