C Roettger, Fall 14
Math 265 Quiz 6A – 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) =
x2 y + y 2
.
5
Find an equation for the tangent plane to the graph of f at P0 = (1, −2).
Solution. First, the partial derivatives
∂f
2xy
=
∂x
5
∂f
x2 + 2y
=
.
∂y
5
At P0 , these have the values −4/5 and −3/5, respectively. Finally, we need
the value f (P0 ) = 2/5. So the tangent plane to the graph of f at P0 is
described by the equation
z=
2 4
3
4
3
− (x − 1) − (y + 2) = − x − y.
5 5
5
5
5