Let z = f(x, y) be a function on the plane and let (x, y) = r(t) be a curve in the plane. The composition z = g(t) = f(r(t)) = f(x(t), y(t)) is a scalar-valued function of one variable. The function g is called the restriction of f to the curve r, since it just gives us the values of f along the curve r. In this context, the Chain Rule for Functions on Curves states that 1. 2. 3. 4. g' (t) = Vf(r(t)) r(t) = fx(x(t), y(t))x'(t) + f(x(t), y(t))y'(t). Chain Rule: Let z = f(x, y) = y sin x where (x, y) = r(t) = (cost, t). (a) Form the composition g(t) = f(x(t), y(t)) and then use the single variable chain rule to calculate g' (t). (b) Use the Chain Rule for Functions on Curves to calculate g' (t). Chain Rule: Suppose that z = f(x, y) = sin(2x+3y) where x = x(t) and y = y(t). If X(0) = /4, y(0) = /3, x'(0) = 1, and y'(0) = -2, find dat t = 0. Chain Rule: Let z= f(x, y) where x = r cos 0 and y = r sin 0. (a) Show that = fx cos 0 + fy sin 0 and 12 = -fx sin 0 + fy cose. (b) Solve the equations in part 3a to express fx and fy in terms of 37 and 3. (c) Hence show that (82) + (33) = () + (. Parametrized Surfaces: (a) Identify the surface with parametrization x = 3 cos sin y = 4 sine sin z = 5 cos p where 0 0 2 and 0 . Hint: Find an equation of the form F(x, y, z) = 0 for and from the equations above. this surface by eliminating (b) Calculate a parametrization for the tangent plane to the surface at (0.) = (/3, /4).