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For *70 we set r(x) = 5 txi etat =(n-1)! on and "(n) = (n-1)! for...
1. Write R' = {(x, y) |X, Y ER} to represent the set of all 1x2...
1. Write R' = {(x, y) |X, Y ER} to represent the set of all 1x2 row vectors of real numbers. This is the standard Euclidean plane you all know and love. If such a row vector is multiplied on the right by a 2x2 matrix, then the output is again in R"; such matrices are called linear transformations. 1. Find a linear transformation that rotates the plane R by a radians. That is, find a matrix T such that...
Problem 10.13. Recal that a polynomial p over R is an expression of the form p(x) an"+an--+..+ar +ao where each aj E R and n E N. The largest integer j such that a/ 0 is the degree of p. We d...
Problem 10.13. Recal that a polynomial p over R is an expression of the form p(x) an"+an--+..+ar +ao where each aj E R and n E N. The largest integer j such that a/ 0 is the degree of p. We define the degree of the constant polynomial p0 to be -. (A polynomial over R defines a function p : R R.) (a) Define a relation on the set of polynomials by p if and only if p(0) (0)...
(5 pts) Consider the function f(x) = 8e7r. We want to find the Taylor series of...
(5 pts) Consider the function f(x) = 8e7r. We want to find the Taylor series of f(x) at x = x = -5. (a) The nth derivative of f(x)is f(n)(x) = 8(7)^ne^(7x) At = -5, we get f(n)(-5) = 8(7)^ne^-35 (c) The Taylor series at x = -5 is too T(x) = (3/7^n](^-35)n!/(n+ (x + 5)” n=0 (d) To find the radius of convergence, we use the ratio test. an+1 L= lim n+oo 1/(x+1) |x + 51 an and so...
Please do exercise 129: Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N...
Please do exercise 129:
Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N → N be the unique function that satisfies f(0) = 2 and f(next(n)) =r(f(n)) for all n E N. 102 1. Prove that f(3) = 8. 2. Prove that 2 <f(n) for all n E N. Exercise 129: Define r and f as in Exercise 128. Assume that x + y. Define r' = {(x,y),(y,x)}. Let g:N + {x,y} be the unique function that...
Given that H is a set and that x is an interior point of the set R∖H, prove that x is not close t...
Given that H is a set and that x is an interior point of the set R∖H, prove that x is not close to H. So, I have an example where they are opposite of the question, I just don't know how to spin it around. Suppose that S is a set of real numbers and that x is a given number. Then the following two conditions are equivalent to one another: 1.The number x is close to the set...
Problem 5. Indicator variables S points possible (graded) Consider a sequence of n 1 independent tosses...
Problem 5. Indicator variables S points possible (graded) Consider a sequence of n 1 independent tosses of a biased coin, at times k = 0,1,2,...,n On each toss, the probability of Heads is p, and the probability of Tails is 1 -p {1,2,.., at time for E resulted in Tails and the toss at time - 1 resulted in A reward of one unit is given if the toss at time Heads. Otherwise, no reward is given at time Let...
7 70 (d) n 0, r 100% 4. Suppose you are a farmer that stores grain...
7 70 (d) n 0, r 100% 4. Suppose you are a farmer that stores grain in a leaky silo so that you lose 2% of your crop each year as a result of dampness and rot. To obtain the future value of X, you still employ the same formula XHX(1). tt n If r=-2%, what is the value of X,+n as n approaches infinity? That is, what is lim X,(1 +r)"2 Will the value of X+n ever equal zero?...
Let the universal set be R, the set of all real numbers, and let A {xE...
Let the universal set be R, the set of all real numbers, and let A {xE R I -3 sxs 0, B {xER -1< x 2}, and C xE R | 5<xs 7}. Find each of the following: (a) AUB {xR-3 < x2} s -3orx > 과 xs. (b) AnB xR-12 {*E찌-1 <xs마 frER< -1 orx {*ER|x s -1 or*> 아 (c) A {*ER-3 <x< 아} {*ER|-3 < 아} s-3 orx> 아 frER< 3 orx x s 0 (d) AUC...
(a) Let A be a fixed mx n matrix. Let W := {x ER" : Ax...
(a) Let A be a fixed mx n matrix. Let W := {x ER" : Ax = 0}. Prove that W is a subspace of R". (b) Consider the differential equation ty" – 3ty' + 4y = 0, t> 0. i. Let S represent the solution space of the differential equation. Is S a subspace of the vector space C?((0.00)), the set of all functions on the interval (0,0) having two continuous derivatives? Justify ii. Is the set {tº, Int}...
POINT ESTIMATION We have a sample of size n=1 of random variable with probability function , If the initial densit...
POINT ESTIMATION We have a sample of size n=1 of random variable with probability function , If the initial density for the parameter is , with , write the final distribution for and the point bayesian estimator if x=2 and m=2 Thank you for your explanations. f(r | θ) = θ (1-0)1-1 r 1,2, .7> PE(0, 1) We were unable to transcribe this imageπ (0) = k (1-0)K- PE(0, 1) We were unable to transcribe this image f(r | θ)...
1. Write R' = {(x, y) |X, Y ER} to represent the set of all 1x2 row vectors of real numbers. This is the standard Euclidean plane you all know and love. If such a row vector is multiplied on the right by a 2x2 matrix, then the output is again in R"; such matrices are called linear transformations. 1. Find a linear transformation that rotates the plane R by a radians. That is, find a matrix T such that...
Problem 10.13. Recal that a polynomial p over R is an expression of the form p(x) an"+an--+..+ar +ao where each aj E R and n E N. The largest integer j such that a/ 0 is the degree of p. We define the degree of the constant polynomial p0 to be -. (A polynomial over R defines a function p : R R.) (a) Define a relation on the set of polynomials by p if and only if p(0) (0)...
(5 pts) Consider the function f(x) = 8e7r. We want to find the Taylor series of f(x) at x = x = -5. (a) The nth derivative of f(x)is f(n)(x) = 8(7)^ne^(7x) At = -5, we get f(n)(-5) = 8(7)^ne^-35 (c) The Taylor series at x = -5 is too T(x) = (3/7^n](^-35)n!/(n+ (x + 5)” n=0 (d) To find the radius of convergence, we use the ratio test. an+1 L= lim n+oo 1/(x+1) |x + 51 an and so...
Please do exercise 129:
Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N → N be the unique function that satisfies f(0) = 2 and f(next(n)) =r(f(n)) for all n E N. 102 1. Prove that f(3) = 8. 2. Prove that 2 <f(n) for all n E N. Exercise 129: Define r and f as in Exercise 128. Assume that x + y. Define r' = {(x,y),(y,x)}. Let g:N + {x,y} be the unique function that...
Problem 5. Indicator variables S points possible (graded) Consider a sequence of n 1 independent tosses of a biased coin, at times k = 0,1,2,...,n On each toss, the probability of Heads is p, and the probability of Tails is 1 -p {1,2,.., at time for E resulted in Tails and the toss at time - 1 resulted in A reward of one unit is given if the toss at time Heads. Otherwise, no reward is given at time Let...
7 70 (d) n 0, r 100% 4. Suppose you are a farmer that stores grain in a leaky silo so that you lose 2% of your crop each year as a result of dampness and rot. To obtain the future value of X, you still employ the same formula XHX(1). tt n If r=-2%, what is the value of X,+n as n approaches infinity? That is, what is lim X,(1 +r)"2 Will the value of X+n ever equal zero?...
Let the universal set be R, the set of all real numbers, and let A {xE R I -3 sxs 0, B {xER -1< x 2}, and C xE R | 5<xs 7}. Find each of the following: (a) AUB {xR-3 < x2} s -3orx > 과 xs. (b) AnB xR-12 {*E찌-1 <xs마 frER< -1 orx {*ER|x s -1 or*> 아 (c) A {*ER-3 <x< 아} {*ER|-3 < 아} s-3 orx> 아 frER< 3 orx x s 0 (d) AUC...
(a) Let A be a fixed mx n matrix. Let W := {x ER" : Ax = 0}. Prove that W is a subspace of R". (b) Consider the differential equation ty" – 3ty' + 4y = 0, t> 0. i. Let S represent the solution space of the differential equation. Is S a subspace of the vector space C?((0.00)), the set of all functions on the interval (0,0) having two continuous derivatives? Justify ii. Is the set {tº, Int}...
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