5 Let z = f(x, y) be given by a differentiable function f, which implies x f(x,

Question

5 Let z = f(x, y) be given by a differentiable function f, which implies xf(x, y), yf(x, y), exist for all (x, y) in the Domain of f. Suppose f(x0, y0) = c and let C = {z = c} be the level curve at height c to the graph of z = f(x, y). Do the following problems,

• Suppose that y is an implicit function of x, y(x), in the equation for the level curve C. Assume also that yf(x0, y0) is not equal to 0. Find a parametric equation for the tangent line to the level curve C at the point (x0, y0) .

• Verify that the direction vector for the tangent line that you found above is perpendicular to f(x0, y0) = < xf(x0, y0), yf(x0, y0) > , i.e. the gradient vector at the point (x0, y0).

• Suppose that the level curve C is given parametrically by r(t) =< x(t), y(t) >, where x(t), y(t) are differentiable functions of t. Show that the tangent vector r’(t) =< x'(t), y'(t) > is equal to the direction vector (in the parametric equation) of the tangent line you found above.

• Do these calculations explicitly for z = f(x, y) given in problem 2 where the x and y coordinates are described by x = cost, y = sin t where t is time, at the point (x0, y0) = (1/ 2 , 1/ 2 ) . Note the level curve C = {z = 3 }.

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