Math 2720: Multiviariable Calculus
Homework #3
Due Tuesday, January 20
In class, we talked about the graphs of degree one functions like
p(x, y) = 3x + 2y + 1,
and found that the graphs are planes.
Problem 1 Consider the function f (x, y) = x2 y + y 2 x.
(a) Determine the slope of f in the x-direction at (1, 1).
(b) Determine the slope of f in the y-direction at (1, 1).
(c) Find a degree one function p(x, y) with the same x-direction slope
and y-direction slope as f at the point (1, 1).
(d) If necessary, modify your answer to (c) so that it is also true that
p(1, 1) = f (1, 1).
You have found the tangent plane to the graph of f at (1, 1).
Problem 2 Consider the following 2-variable trajectory problem:
You launch a ball in R2 from the origin at angle θ and velocity v;
where does it cross the x-axis?
Solving this problem gives you a function
S(θ, v) = the solution to the problem .
(a) Find a formula for the function S.
(b) Find the tangent plane to the graph of X over the point θ =
v = 12 .
π
,
6
(c) Find the tangent plane to the graph of X over the point θ =
v = 32 .
π
,
4