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PROBLEM 5.3. The function f: R3 R3 given by a bH (3b-a 3a +c 3b +...
The multivariable function f(a, b) = 12 + 3a? + 3b? - 12a + 18b has...
The multivariable function f(a, b) = 12 + 3a? + 3b? - 12a + 18b has the following first partial derivatives: fa = 6a - 12 and fo = 6b + 18. Find the location of the critical point. O(-3,2) O (3,-2) o(-2,3) O (2, -3)
problem 3b Problem 3a Assume the states(ln), n = 0,1,2, ) are mutually orthonormal: (nlm)-δυǐn ....
problem 3b
Problem 3a Assume the states(ln), n = 0,1,2, ) are mutually orthonormal: (nlm)-δυǐn . It is known that the operators a, and a. have the following properties: a,In) = vn + 11n + 1),n20 a-10)-0 The system's Hamiltonian is given by H-h Now, assume the system is prepared in a state described by the (unnormalized) superposition: V 1o) +11) a) Normalize this wavefunction. b) Compute the commutator of operators a, and a c) Compute the average energy (expectation...
3a. Give an example of a function f(x) such that • f(x) < 0 • f'(x)...
3a. Give an example of a function f(x) such that • f(x) < 0 • f'(x) > 0 • f''(x) < 0 3b. Give an example of a function h(t) such that • h'(t) > 0 when t < −1 and t > 1 • h'(t) < 0 when −1 < t < 1 • h'(−1) = 0 • h'(1) is undefined
Problem 2: Consider the following 2-dimensional linear subspace of R3: X = {(a,b,c) ER’: a+b+c=0}. Define...
Problem 2: Consider the following 2-dimensional linear subspace of R3: X = {(a,b,c) ER’: a+b+c=0}. Define a linear map F: X X by setting F(a,b,c) = (20 – 3b+c, -3a + 2b+c, a +b – 2c). (a) Find the matrix A representing F with respect to the basis 21 = (1,0, -1), 22 = (0,1, -1). (b) Find the matrix A representing F with respect to the basis î1 = (3,1,-4), f2 = (1, -2,1). (c) Find an invertible matrix...
Question 8 (9 marks) For the function given by f(x) = are arctan(r3), find (a) dom(f)...
Question 8 (9 marks) For the function given by f(x) = are arctan(r3), find (a) dom(f) (b) dom(f'). (c) dom(f"). (d) Any stationary points of f. (e) The interval on which f is concave up. (f) The interval on which f is concave down. (g) Use parts (e) and (f) to determine whether f has an inflection point.
Question 8 (9 marks) For the function given by f(x) = are arctan(r3), find (a) dom(f) (b) dom(f'). (c) dom(f"). (d) Any...
(i) Find an orthonormal basis {~u1, ~u2} for S (ii) Consider the function f : R3...
(i) Find an orthonormal basis {~u1, ~u2} for S
(ii) Consider the function f : R3 -> R3 that to each vector ~v
assigns the vector of S given by
f(~v) = <~u1, ~v>~u1 + <~u2; ~v>~u2. Show that it is a
linear function.
(iii) What is the matrix of f in the standard basis of R3?
(iv) What are the null space and the column space of the matrix
that you computed in the
previous point?
Exercise 1. In...
PROBLEM 7.2. Let f:R3 - 10 6 3) R3 be given by f(x) = Ax, where...
PROBLEM 7.2. Let f:R3 - 10 6 3) R3 be given by f(x) = Ax, where A = 6 3 1 a li -2 1] Find the matrix representations of f:R3 the domain and the codomain. R3 and of f-1:R3 → R3 with respect to the standard bases for
Problem 2 [10pts] Let f : R3 + R2 be a linear transformation given by f((x,...
Problem 2 [10pts] Let f : R3 + R2 be a linear transformation given by f((x, y, z) = (–2x + 2y +z, -x +2y). Find the matrix that corresponds to f with respect to the canonical bases of R3 and R2.
Problem 5. Let f be the function defined in the previous problem, so f(t) dr C...
Problem 5. Let f be the function defined in the previous problem, so f(t) dr C Show that the inverse of this function is a solution of the differential equation y+y 1. That is, let g(t) function g and its derivative. It says that the parametric curve y(t) the solution set of the equation g equation. This is one of a family of curves known as elliptic curves. The connection with ellipses f(t). Show that g(t)2-1-g(t)4. This is a kind...
Given vector space R3 and the function defined
Given: vector space R3 and the function: 〈x, y〉= 2x1y1 + x2y2 + 5x3y3 a) Show that the given function defines an inner product on R3.b)c)
The multivariable function f(a, b) = 12 + 3a? + 3b? - 12a + 18b has the following first partial derivatives: fa = 6a - 12 and fo = 6b + 18. Find the location of the critical point. O(-3,2) O (3,-2) o(-2,3) O (2, -3)
problem 3b
Problem 3a Assume the states(ln), n = 0,1,2, ) are mutually orthonormal: (nlm)-δυǐn . It is known that the operators a, and a. have the following properties: a,In) = vn + 11n + 1),n20 a-10)-0 The system's Hamiltonian is given by H-h Now, assume the system is prepared in a state described by the (unnormalized) superposition: V 1o) +11) a) Normalize this wavefunction. b) Compute the commutator of operators a, and a c) Compute the average energy (expectation...
Problem 2: Consider the following 2-dimensional linear subspace of R3: X = {(a,b,c) ER’: a+b+c=0}. Define a linear map F: X X by setting F(a,b,c) = (20 – 3b+c, -3a + 2b+c, a +b – 2c). (a) Find the matrix A representing F with respect to the basis 21 = (1,0, -1), 22 = (0,1, -1). (b) Find the matrix A representing F with respect to the basis î1 = (3,1,-4), f2 = (1, -2,1). (c) Find an invertible matrix...
Question 8 (9 marks) For the function given by f(x) = are arctan(r3), find (a) dom(f) (b) dom(f'). (c) dom(f"). (d) Any stationary points of f. (e) The interval on which f is concave up. (f) The interval on which f is concave down. (g) Use parts (e) and (f) to determine whether f has an inflection point.
Question 8 (9 marks) For the function given by f(x) = are arctan(r3), find (a) dom(f) (b) dom(f'). (c) dom(f"). (d) Any...
(i) Find an orthonormal basis {~u1, ~u2} for S
(ii) Consider the function f : R3 -> R3 that to each vector ~v
assigns the vector of S given by
f(~v) = <~u1, ~v>~u1 + <~u2; ~v>~u2. Show that it is a
linear function.
(iii) What is the matrix of f in the standard basis of R3?
(iv) What are the null space and the column space of the matrix
that you computed in the
previous point?
Exercise 1. In...
PROBLEM 7.2. Let f:R3 - 10 6 3) R3 be given by f(x) = Ax, where A = 6 3 1 a li -2 1] Find the matrix representations of f:R3 the domain and the codomain. R3 and of f-1:R3 → R3 with respect to the standard bases for
Problem 2 [10pts] Let f : R3 + R2 be a linear transformation given by f((x, y, z) = (–2x + 2y +z, -x +2y). Find the matrix that corresponds to f with respect to the canonical bases of R3 and R2.
Problem 5. Let f be the function defined in the previous problem, so f(t) dr C Show that the inverse of this function is a solution of the differential equation y+y 1. That is, let g(t) function g and its derivative. It says that the parametric curve y(t) the solution set of the equation g equation. This is one of a family of curves known as elliptic curves. The connection with ellipses f(t). Show that g(t)2-1-g(t)4. This is a kind...
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