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 )