Answer Happy

.
The surface of a hill is modeled by the equation z = (60 - 3x² - 5y²) m shown in the figure. If a freshwater spring is located at the point (x, y, z) = (1,2,37), in what direction will the water flow? Find the unit vector u in this direction. (Use symbolic notation and fractions where needed. Give your answer in vector form.) u= Incorrect -(x,y) 6 20 j /437 √437 +√437 k

(a) Find the direction for which the directional derivative of the function f(x, y, z) = at P = (7,5,3). (Use symbolic notation and fractions where needed. Give your answer in vector form.) direction: (b) Find the maximum value of the directional derivative. ||V f(7,5,3)|| = z² is a maximum

Find a unit vector u that is normal to the level curve of f(x, y) = 9x²y through P = (x, y) = (5,-3) at P. The figure shows the level curves of f(x, y) = ax² y. (x,y) (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) U=

(a) Find the direction for which the directional derivative of the function f(x, y) = 9xy + 2y² is a maximum at P = (2, 1). (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) direction: 9i+ 4j Incorrect (b) Find the maximum value of the directional derivative. (Give an exact answer. Use symbolic notation and fractions where needed.) ||V ƒ(2, 1) || = Incorrect 52

(a) Find the gradient of the function f(x, y) = 4xy + 2y² at the point P = (2, 1). (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) Vf(2, 1) = 24i + 16j Incorrect (b) Use the gradient to find the directional derivative Du f(x, y) of f(x, y) = 4xy + 2y² at P = (2, 1) in the direction from P = (2, 1) to Q = (4, 1). (Give an exact answer. Use symbolic notation and fractions where needed.) Du f(2, 1) = Incorrect 24

Suppose that the electrical potential (voltage V) at each point in space is V(x, y, z) = xyz volts and that electric charges move in the direction of greatest potential drop (most rapid decrease of potential). (a) In what direction does a charge at the point (2, -3, 3) move? Find the unit vector u in this direction. (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) u= (b) How fast does the potential change as the charge leaves this point? (Express numbers in exact form. Use symbolic notation and fractions where needed.) U =