Show that the best possible Lipschitz constant in the second variable for the function
Show that the best possible Lipschitz constant in the second variable for the function 3(a) Show that the best possible...
Please solve Q 7 & 8 7. 14+6 marks] Consider the initial value problem y_y2, 2,y(1) = 1 y'= 1-t (a) Assuming y(t) is bounded on [1, 2], Show that f(t,v)--satisfies Lipschitz condition with respect to y. (b) Use second order Taylor method with h 0.2 to approximate y(1.2), then use the Runge- Kutta method: to compute an approximation of y(1.4). 8. [4 marks) Assuming that a1, o2 are non negative constants, determine the parameters o and β1 of the...
Suppose the random variable X has probability density function (pdf) - { -1 < x<1 otherwise C fx (x) C0 : where c is a constant. (a) Show that c = 1/7; (b) Graph fx (х); (c) Given that all of the moments exist, why are all the odd moments of X zero? (d) What is the median of the distribution of X? (e) Find E (X2) and hence var X; (f) Let X1, fx (x) What is the limiting...
4. a) Find a second order Taylor ser XN-[ 1]Tand then use it to find an approximate value of fX) at Note: the superscript "T" denotes the transpose of the vector) ies approximation of the following function about the point, a X - [0.8 1.2]. b) Consider the following multivariable cost function: f(X) (a x)(bx) where "a" and "b" denote nx1 constant vectors and X is a nx1 vector of decision variables. Find the gradient vector and Hessian matrix of...
Question Let X be a continuous random variable with the following probability density function (pdf) 0.5e fx (x) = { 0.5e-1 x < 0. <>0.. (a) Show that fx (x) is a valid pdf. (b) Find the cumulative distribution function Fx (.x). (e) Find F='(X). (d) Write an algorithm to generate a sample of size 1000 from the distribution of X using the inverse-transform method. Be as precise as possible.
problem 34 Equations with the Independent Variable Missing. If a second order differential equation has the form y"f(y, y), then the independent variable t does not appear explicitly, but only through the dependent variable y. If we let y', then we obtain dv/dt-f(y, v). Since the right side of this equation depends on y and v, rather than on and v, this equation is not of the form of the first order equations discussed in Chapter 2. However, if we...
(1 point) For partial derivatives of a function use the subscript notation, so for the second partial derivative of the function u(x,t) with respect to x use uxx. For ordinary differential equations use the prime notation, so the second derivative of the function f(x) is f" Solve the heat equation 14 1502,>0 a(0,t) = 92,21(2, t) 87, t > 0 using a steady-state and transient solution: ie write u(z, 1) _ u(z,t) + S(z) with u a solution of the...
1. For each function in question 1 of section 6.1 exercises, now use second- order conditions to determine whether each stationary value you found is a maximum, minimum, or point of inflection. y function in the neighborhood of the s (a) y = x3 – 3x2 + 1 (b) y = x4 - 4x3 + 16x - 2 (C) y = 3x3 – 3x - 2 (d) y = 3x4 - 10x3 + 6x2 +1 (e) y = 2x/(x +1)...
Differential Equations for Engineers II Page 2 of 6 2. Consider the nonhomogeneous ordinary differential equation XY" + 2(x – B)y' + (x – 2B)y = e-1, x > 0, (2) 5 marks where ß > 0 is a given constant. (a) A solution of the associated homogeneous equation is yı = e-*. Use the formula for the method of reduction of order, as described in the lecture notes / record- ings, to find a second solution, y2, of the...
2.34. Probability integral transformation. Consider a random variable X with cumulative function Fx(x), 0-x-00, Now define a new random variable U to be a particular function of X, namely, U = Fx(X) For example, if FX(x)-1-e-Ax, then U = 1-e-Ax = g(X). Show [at least for reasonably smooth Fx(x)] that the random variable U has a constant density function on the interval O to 1 and is zero elsewhere. Hint: Con vince yourself graphically thatgg (u)- u and assume that...
The function u(x, t) satisfies the partial differential equation with the boundary conditions u(0,t) = 0 , u(1,t) = 0 and the initial condition u(x,0) = f(x) = 2x if 0<x<} 2(1 – x) if}<x< 1 . The initial velocity is zero. Answer the following questions. (1) Obtain two ODES (Ordinary Differential Equations) by the method of separation of variables and separating variable -k? (2) Find u(x, t) as an infinite series satisfying the boundary condition and the initial condition.