1. a: . The surface 2 = a: Let f(:r:,y) = '/1+'.ry2. (a) Find the gradient of f at the point P = [2. 2). [1)] Find the maximal and minimal rates of change in f at the point [2. 2). (c) Find an equation for the plane tangent to the graph z = f(;r. y) at the point (Ev2.f[2,2)l [d] Find an equation for the line tangent to the level curve f(;r. y) = H2, 2). 5'5 2 2 3 4 . Let 1? = <. i and ti>. and suppose that f(s:. y) is a function with the '2 2 directional derivatives D(1,2) = 3v/2 and Dul. 2) = 5 at the point (1. 2). Find the gradient vector Vf(1. 2). . '2 . Find an equation for the tangent plane to the surface given by ye": sin[2] 2) = U at the point (0,1. E). 2 y2 and the surface xyz = 3{) intersect along a curve in R3. Find a vector tangent to this curve at the point P = (3. 2.5). Hint: such a vector is parallel to tangent planes at P to both surfaces. . Atmospheric temperature depends on position and time. If we denote position by three spatial coordinates :c, y. and 2 (measured in kilometres) and time by t (measured in hours), then the temperature (in degree Celsius) is a function of four variables. T(:I:, y, z, t). Suppose t hat my 1+2 and a thermometer is attached to a weather balloon moving along the curve :r=t. y=2t. z=tt2. T(x:yazat) = (1+t) (a) What physical rates of change are represented by the expressions % and (ff1:? dT 6T 3T 3T 6T (b) Give a formula for E in terms of t. a, ay, a. and E. (c) Find the rate of change in the temperature being recorded by the weather balloon at time t = 1