MATHEMATICS 2A: Additional Problems 1
Week 2-2023
1. Find an equation of the plane that passes through the point
P0 with a normal vector n.
(a) P0 (1, 0, −3); n =< 1, −1, 2 >
(b) P0 (1, 2, −1); n =< −1, 4, −2 >
2. Find an equation of the plane that passes through the given
points
(a) (−1, 1, 1), (0, 4, 2) and (1,1,1)
(b) (2, −1, 4), (−4, 1, 1) and (1, 1, −1)
3. Find an equation of the plane parallel to the plane Q passing
through the point P0 .
(a) Q : 2x + y − z = 1; P0 (0, 2, −2)
(b) Q : 4x + 3y − 2z = 12; P0 (1, −1, 3)
4. The plane through (1, 2, −1) that is perpendicular to the line
of intersection of the planes 2x+y+z = 2 and x+2y+z = 3.
5. The plane through the points P1 (−2, 1, 4), P2 (1, 0, 3) that is
perpendicular to the plane 4x − y + 3z = 2.
6. The plane through (−1, 2, −5) that is perpendicular to the
planes 2x − y + z = 1 and x + y − 2z = 3.
7. Show that the distance D between a point (x0 , y0 , z0 ) and a
plane ax + by + cz = d is given by
D=
|ax0 + by0 + cz0 − d|
√
a 2 + b2 + c 2
8. Show that the point in the plane ax+by +cz = d nearest the
origin is P (ad/D2 , bd/D 2 , cd/D 2 ), where D2 = a2 + b2 + c2 .
Conclude that the least distance from the plane to the origin
is |d|/D.
9. Find the domain of the following functions.
a. f (x, y) = 2xy − 3x + 5y 2
b. f (x, y) = cos(x2 − y 2 )
3
c. f (x, y) = ln(x2 − y)
d. f (x, y) = 2
y − x2
10. Find and sketch the domain of the function.
ln(y − x)
st
b. g(x, y) = √
a. f (s, t) = 2
2
s −t
x−y+1
√
√
√
c. f (x, y) = xy − 1
d. f (x, y) = y − x
11. Sketch a graph of the following functions. In each case identify the surface, and state the domain and range of the function.
√
a. f (x, y) = 3x − 6y + 12
b. f (x, y) = 1 − x2 − y 2
√
c. f (x, y) = x2 − y 2
d. f (x, y) = x2 + y 2 − 1
12. Sketch the level curves f (x, y) = c of the function f for the
indicated values of c.
√
(a) f (x, y) = 16 − x2 − 4y 2 ; c = 0, 2, 3, 4
(b) f (x, y) = ln(x + y);
c = −1, 0, 1, 2
(c) f (x, y) = y − x2 − 1;
c = −2, −1, 0, 1, 2