M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY Interphase Calculus III Worksheet Instructor: Samuel S. Watson 22 July 2015 Topics: multivariable chain rule, directional derivatives, gradient 1. Find the plane tangent to the graph of f ( x, y) = 2. Consider the function f ( x, y) = q 14 − x2 − y2 at the point (1, 2, 3). e xy . Use a tangent plane to approximate f (0.99, 0.98). e (1 + x 3 ) 3. Suppose that x (t) andy(t) are two functions of a real variable t. Express the derivative with q respect to t of f (t) = sin x (t) + x (t)y(t) in terms of x 0 (t) and y0 (t). d g( x (t), y(t)) by using the definition of a derivative and approxidt mating g with its tangent plane at ( x (t), y(t)). Bonus: check that your formula works to solve the previous exercise. 4. Write down a formula for 5. In addition to taking a derivative of a function f ( x, y) in a coordinate direction, we can take a derivative in an arbitrary direction specified by a unit vector u = (u1 , u2 ). We define ( Du f )( x0 , y0 ) = lim h →0 f (( x0 , y0 ) + uh) − f ( x0 , y0 ) . h Use a tangent plane approximation to express ( Du f )( x0 , y0 ) in terms of u1 , u2 , and the partial derivatives of f . 6. The gradient of a function f is a vector of its partial derivatives: ∇ f = q gradient of the function x2 + y2 . 7. Comment on the geometric relationship between the gradient of q of x2 + y2 . Can you explain this phenomenon? q ∂f ∂f , . Find the ∂x ∂y x2 + y2 and the level curves 8. Find the equation of the plane tangent to the surface x2 + y2 + z2 = 14 at the point (1, 2, 3) by finding the gradient of x2 + y2 + z2 . 9. Find the set of points ( x, y) at which the directional derivative of f ( x, y) = xy2 in the direction u = (3/5, −4/5) equals 1. 10. Level lines of a function f ( x, y) are shown below. Find a point ( x0 , y0 ) where f x is positive. Repeat with f xx , f y , and f xy . Find a point where f xx = 0. 1 3 3 2 2 1 0 5 4 0 1 2 3 11. (Challenge problem) Suppose you have a function f that you want to maximize, and suppose that you can also conveniently calculate its partial derivatives f x and f y . Describe an algorithm for beginning with a point ( x0 , y0 ) near a local minimum and obtaining the point ( xmin , ymin ) where the minimum is achieved. ( xmin , ymin ) ( x0 , y0 )