C Roettger, Fall 14
Math 265 Quiz 6B – Solutions
Graph of f (x, y) in blue, tangent plane red. Intersection curves of the graph
with planes x = x0 and y = y0 green and yellow, respectively.
Problem 1 Consider the function
f (x, y) =
x3 + xy
.
2
Find an equation for the tangent plane to the graph of f at P0 = (−1, 2).
Solution. First, the partial derivatives
∂f
3x2 + y
=
∂x
2
∂f
x
=
.
∂y
2
At P0 , these have the values 5/2 and −1/2, respectively. Finally, we need
the value f (P0 ) = −3/2. So the tangent plane to the graph of f at P0 is
described by the equation
1
5
1
3 5
z = − + (x + 1) − (y − 2) = 2 + x − y.
2 2
2
2
2