Solution to Midterm I of MAT17C
Professor Qinglan Xia
1. Consider the function
2
f (x, y) = e yx .
(a) Determine and sketch the domain of f .
Solution: The domain of f is
D = (x, y) : y x2 .
Sketch it yourself.
(b) Sketch the level curve f (x, y) = e.
Sol

Solution to Practice Midterm II of MAT17C
Instructions. No calculator is allowed.
1. Consider dierential equations of the form
d
3
1
= Ax (t) where A =
.
6 4
dt
(a) Find the general solution of the system.
Solution: Solving
det (A I) = det
3
1
= (3 ) (4 )

Solution to Practice Midterm I of MAT17C
1. Let
f (x, y) = ln 16 x2 y 2 ,
determine and sketch the domain of f . Also sketch the level curve
f (x, y) = ln 7.
Solution: The domain of f is a disk
D = (x, y) : x2 + y 2 < 16 .
The level curve f (x, y) = ln 7

Practice Midterm II of MAT17C
Instructions. No calculator is allowed.
1. Consider dierential equations of the form
dx
3
1
= Ax (t) where A =
.
6 4
dt
(a) Find the general solution of the system.
(b) Find the solution with the initial condition x (0) =
0
.