9. (10pts) Answer true or false: (a) The domain of f(x,y) = In(1-z?-уг) + Vi-z?-уг is the unit ball {(z, y): x2 + y2 1} . (b) The direction of the maximum rate of increase of g(x, y, 2yz at the point (1,1,1) is 2,1,1 (c) For F2y,2r3y1>, F-dr is independent of path in the plane. (d) × (▽ . F) makes sense. (e) ▽f.dr =4 where f(x, y, z) = zyz and C is the line segment starting at...
Let f(x, y, z) = xeyz – cos(x2 – y2 + 22) a) Find the directional derivative of f at the point (0,0,0) toward the point (1,2,0). b) Find the maximum rate of change of f at point (0,0,0). In which direction does the max rate of change at (0,0,0) does occur? (two questions here!)
which of the following is a potential function for F(x,y,z)= < y2 +y?ex?,x2 + 2ye*?,xy + xy?e *V> f(x,y,z) = xyz + y2exyz f(x,y,z) = xyz + y2e*+2 b. F(x,y,z) has a potential function but it is not one of the other choices. F(x,y,z) does not have a potential function. d. f(x,y,z) = xyz + y2exZ e.
This diagram is a simplification of function : -(x+2)+(7+2)+(X+Y+Z) O F-X2)*(Y2)+(XYZ) F-(X+ZXY +2XX+Y+Z) P-XZXYZXX *Y+2)
Use Lagrange multiplier to determine the maximum and minimum values of (f,x,y,z) = x^2 +y^2 +z^2 subject to xyz=4 Detailed solution please. Thank you! 20. Use Lagrange Multiplier to determine the maximum and minimum values of f(x, y, z)-x2 + y2 +12 subject to 20. Use Lagrange Multiplier to determine the maximum and minimum values of f(x, y, z)-x2 + y2 +12 subject to
17. Given f(x, y, z) = x^yz -- xyz', P(2,-1,1) and vector v =<1,0,1 >. Find i. the directional derivative of the function at the point P in the direction of v. ii. the maximum rate of change of f.
Find the directional derivative of the function f(x,y,z) = z4−x3y2 at P(1,-1,1) in the direction of the vector from P(1,-1,1) to the point Q(2,1,0). What is the maximum rate of change of f at the point P(1,-1,1) and in which direction
Please help me solve this problem. Thanks! Problem 1 (weight 25%) Consider the problem Maximise f(x, y, z) = x + y +2z when g(x,y,z) = x2 +y2 +2z2 = 4. (*) (a) Explain why the problem (*) does have a solution (b) Suppose that ( has a solution, and use Lagrange's method to set up the necessary conditions for solving the problem. (c)Find all the triplets (r. y, 2) that satisfy the necessary conditions for solving the problem (*),...
Evaluatef(x, y, z) dS. f(x, y, z) = x2 + y2 +z2 Evaluatef(x, y, z) dS. f(x, y, z) = x2 + y2 +z2
a) A concentration of a carbon monoxide in a tank is described by f(X,y,z) X2 + y2 + Z2. Based on Fick's Law, the diffusion happens in the direction of maximum decrease of concentration Point P is at (1, -2, 3) in the respective tank. Find a vector field to describe diffusion field that happens in the tank. 1. Determine a unit vector in the direction of diffusion at P. ii. Determine unit vector(s) in the direction of zero change...