Calculus 3: Show work please

14.5Gradientanddirectionalderivatives:

1.

Let f(x, y) = x2 y' and c(t) = (t2, 2+3 ) (a) Calculate: Vf . c(t ) = (b) Use the Chain Rule for Paths to evaluate d dt f(c(t) ) att = 1. d P f(c(1)) =Calculate the gradient of f(x, y) = cos 2x2 + 5y V\fCalculate the directional derivative of x, 3;} 2 may\" in the direction of v 2 2i + 3j at the point P = (2, 2}. Remember to normalize the direction vector. Duf(729 2) = Calculate the directional derivative of x, 3;} : cos1(ry} in the direction of v : 4i 2j at the point P : (0.5, 0.5}. Remember to normalize the direction vector. Burma's, 0.5) = Determine the direction in which t, y] = 1'2 83y + g;2 has maximum rate of increase from P = [5, 8}, and give the rate of change in that direction. (Use symbolic notation and fractions where needed.) Rate of change 2 Let f(:,y) : 19329 and P = (8, 64). (a) Calculate ||pr||_ (b) Find the rate of change of f in the direction V1}. (c) Find the rate of change of f in the direction of a vector making an angle of 45 with pr. Answers : (a) (b) (C)