(3) Let f(x, y) = x²y + esiny. Compute the gradient vector Vƒ(1, 0). In what direction is the function f decreasing most rapidly at (1,0)?
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- Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii a1=0.1 11. Consider the function f(x)=4x2(1x) a. Find any equilibrium points where f(x)=x. b. Determine the derivative at each of the equilibrium points found in part a. c. What does the theorem on the Stability of Equilibrium points tell us about each of the equilibrium points found in part a? d. Find the next four iterations of the function for the following starting values. i. a1=0.4. ii. a2=0.7 e. Describe the behavior of successive iteration found in part d. f. Discuss how the behavior found in part d relates to the results from part c.Find derivatives of the functions defined as follows. fz=2z+e-z22Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x, y) =8xy2, (5,-7) maximum rate of change direction vector
- Ex.: Find the directional derivative of the function f(x, y) = x'y* + x*y* at the point (1,1) in the direction of the vector (4,3).Find the gradient of the function w = xy2z2, and the maximum value of the directional derivative at the point (2, 1, 1).Consider the function defined by f(x,y) = 4xy - 2x2. Evaluate the directional derivative of f at point(1,-1) in direction of the vector which subtends an angle of pi/6 with the positive x-axis.
- 5. (a) Find the directional derivative of f(x, y) = √√√x² - y² at P(5,3) in the direction of v = (1, -1). (b) Find the unit vector u in the direction of which ƒ has the maximum rate of change at (5,3). What is this maximum rate of change?Find the directional derivative of ø = x² + y² + z² at point (1, 2, 1) for a direction determined by dx = 2dy = -2dz.Let f(x, y) = xeY, P = (2, -1), and v = (2, 3). Calculate the directional derivative in the direction of v.