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(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.) Incorrect (b) Use the gradient to find the directional derivative Duf(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.)
Suppose that the electrical potential (voltage V ) at each point in space is V(x,y,z)=exyz 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.) 11 (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.)