Please help with the following problems:

Q1:

1 What is the direction of fastest increase at (3, 4, 4) for the function f (x, y, z) = . ++ (Use symbolic notation and fractions where needed. Give your answer in the form (ac, *, *).) direction: Let f (x, y, z) = 5x3 y3 + 22. Find the maximum value M for the directional derivative at the point (1, 2, 1). (Use symbolic notation and fractions where needed.) mm Find an equation of the tangent plane to the surface at the given point. 5x2 + 2y2 + 42,2 = 26, P = (2,1,1) (Express numbers in exact form. Use symbolic notation and fractions where needed. Let f (x, y, z) and give the equation in terms of x, y, and z.) Find an equation of the tangent plane at P = (2, 3, i) to the given surface. E 5x2 + zze'" = 29 (Express numbers in exact form. Use symbolic notation and fractions where needed. Let f (x, y, z) and give the equation in terms of x, y, and z.) equation: Find a function f (x, y, 2) such that V f is the constant vector (7, 4, 5). (Use symbolic notation and fractions where needed. Use C for the constant of integration.) mz) = Find a function f (x, y, 2) such that V f = (82,4y3, 8x). (Use symbolic notation and fractions where needed. Use C for the constant of integration.) f(x, y, Z) = Suppose that a duck is swimming in the circle x = cos(8t), y = sin(8t) and that the water temperature is given by the formula dT T = 2x2 ey 3x y4. Find ?' the rate of change in temperature the duck might feel. (Use symbolic notation and fractions where needed.) Word of caution: this problem is lengthy because of the number of terms. Be careful when entering your answer. dT