1 Homework week 3
Consider the function
f x y x2
x
y2
1. Find:
x
x2 y2
a) Domain: we do
b) level curves:
1
c
x
x2 y2
1
0
so
D x y x2 y2
0
so
y cx x2
2. Plot the surface which it represents:
a)
3. Find if the function is continuous at 0 0 by using limits: we want to calculate
x
∞
x ylim
0 0 x2 y2
so the function is not continuous at the point 0 0 , see Domain and figure.
4. Calculate the partial derivatives of this same function: Take f x y x x 2 y2 ∂f
∂x
2x2
x2 y2 and
∂f
∂y
2
1
x2 y2 2yx
y2 x2
1
2
5. Find the slope of the tangent lines to the surface at the points 0 0 2 and 0 2 0 . Draw them
in your Plot. Solution: We evaluate the previous formulas
∂f
∂x
∂f
∂y
∂f
∂x
∂f
∂y
2 02 02 0 2 0 0 2
0 0 2
0 2 0
0 2 0
2 2
2 0 2 0
02 0 2 2 2 0 2 2
0 2 2 02 2 0 0 2
0 2 2 02 1
02
2
2
2
1
0 2
2
25
0
0
0 2
1
2
02 25
1 Homework week 3
y= +
3
c<0
cx−x2
c>0
y= +
y
cx−x2
c=1/6
c=1/5
c=1/4
c=1/3
c=1/2
c=1
2
1
1
2
3
4
5
x
6
−1
−2
c<0
y= −
cx−x2
c>0
−3
y= −
Figure 1.1: Level curves
2
cx−x2
1 Homework week 3
z
z=1
z=1/2
z=1/4
y
x
Figure 1.2: Surface
3
1 Homework week 3
m=−25
z
m=0
(0.2,0)
y
x
Figure 1.3: Tangent lines at (0.2,0) the slopes corresponding to the derivatives of the function with
respect to x and y are -25 and 0 respectively.
4