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3. (a) (6 points) Expand S(x, y, z) around (20,40, zo) to the second order in...
please answer both parts - (a) (10 points) Find the maximum value of f(x, y, z) = xy + yz+xz on the plane x+y+z= 6 using Lagrange multipliers. No credit given for any other method. (10 points) Explain why the extremum found in part (a) is a maximum. Hint: turn the problem in part (a) into one involving 2 variables.
4. (10 points) Calculate the maximum and minimum values of the function f(x, y, z) = xyz in the first octant subject to the constraint x + 4y + 2z = 1..
(1 point) Let F = xi+ (x + y) 3+ (x – y+z) k. Let the line l be x = 4t – 3, y = — (5 + 4t), z = 2 + 4t. = (20, Yo, zo) where F is parallel to l. (a) Find a point P P= Find a point Q = (x1, Yı, z1) at which F and I are perpendicular. Q - Give an equation for the set of all points at which F...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
Use Stokes' Theorem to evaluate S (double integral) curl F · dS. F(x, y, z) = x^2*y^3*z i + sin(xyz) j + xyz k, S is the part of the cone y^2 = x^2 + z^2 that lies between the planes y = 0 and y = 3, oriented in the direction of the positive y-axis.
Given the constraint, find the stationary points for the following function and evaluate the second order conditions. Use the language technique. Subject to X2ザ=6 f(x,y)=ln x +2 lny Subject to X2ザ=6 f(x,y)=ln x +2 lny
Need only parts 5 and 6 Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0, 1) 1. Define Z = max (X, Y) as the larger of the two. Derive the CD. F. and density function for Z 2. Define W- min (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W. 3. Derive the joint density of the pair (W, Z). Specify...
3. (30 points) Let f(x) = 1/x and data points Zo = 2, x,-3 and x2 = 4. Note that you can use the abscissae to find the corresponding ordinates (a) (8 points) Find by hand the Lagrange form, the standard form, and the Newton form of the interpolating polynomial p2(x) of f(x) at the given points. State which is which! Then, expand out the Newton and Lagrange form to verify that they agree with the standard form of p2...
3. (3 Points) Intuitively describe what the slope of the budget line tells us about the consumer's ability to trade goods X and Y. 4. (6 Points) For each description of a change in the budget line given below, state which of the following increases or decreases for the budget constraint variables would cause it: † Px + PxPy. Pytw, and I W. (a) The budget line rotates inward around its horizontal intercept (meaning that it's horizontal intercept does NOT...
6. (10 points) (a) (6 points) The gradient of the function o(x, y, z) at (1,2,3) is the vector (2, 1, 1) and g(1,2,3) = 1 (1) (2 points) Find the equation of the tangent plane of the level surface g(r, y, z) = 1 at (1,2,3) (ii) (2 points) Find the maximum rate of change of g(x, y, z) at (1, 2, 3). hax. rarte ot change: 23 14 (iii) (2 points) Find the rate of change of g...