MAT 267
Practice Problems for Final Review Chapters 10, 11, 12
1. Find the equation of the plane containing the line
x
=
4
−
t , y
=
2
3
t , z
=−
1
−
2
t
and the point
(1,1,1).
2. a)
Find the equation of a straight line through the point ( 3, 7, -2) and parallel to the plane
4
x
−
6
y
2
z
=
3.
b) Find the distance from the line to the plane.
3. Find the equation of a straight line through the point ( 3, 7, -2) and perpendicular to the plane
4
x
−
6
y
2
z
=
3.
4. Find the equation of the line through the two points A(1,2,3) and B(2,0,-1).
5.
f
x , y
=
e
−
x
2
y
2
,
find the equation of the level curve at 0.5 and sketch its graph.
6. Find the equation of the plane which passes through the points
A
3,2,
−
1
,
B
1,
−
1,3
and
C
3,2,4
.
7. Find the area of the triangle defined by the three points in problem 7 and find the measures of the
angles of the triangles.
8. The motion of a particle is given by the vector function
r
t
=⟨
2
t
3
/
2
,
cos
2
t
,
sin
2
t
⟩
,
0
≤
t
≤
1.
a) Find its velocity,
acceleration and speed at
t
=
1
/
2
b) Calculate the length of the trajectory traced by the particle from
t
=
0
to
t
=
1.
c) Find the equation of the tangent line to the curve
r
t
at the point P(0,1,0).
9.
f
x , y
=
x
3
x y
2
y
2
3
x
2
−
3
≤
x
≤
3,
−
3
≤
y
≤
3
Determine the coordinates of the relative and absolute
maxima, relative and absolute minima, and saddle points.
10. The airline's luggage restrictions state that the sum of length, width, and height of a piece of
luggage cannot exceed 147 cm. Find the dimensions of a rectangular piece of luggage that
maximizes the volume of the luggage.
11.
Find an equation for the tangent plane to
f
x , y
=
x
2
y
−
x y
3
2
at the point (3,2).
12. Find an equation for the tangent plane to
2
x
2
y
y
2
z
x z
3
1
=
0
at the point P( 1, 2, -1)
13.
If the temperature is given by
T
x , y , z
=
3
x
2
−
5
y
2
2
z
2
and you are located at
0.
3
,
0.2,0.5
and want
to get cool as soon as possible, in which direction should you head?
14. Consider the function
f
x , y ,z
=
x
3
y
3
−
z
.
a)
Find the direction of steepest ascent at the point (1, 1, 2).
b) Find the directional derivative of the function at the point (1,1,2) in the direction of the vector
v
= < 1, 1, 1>
c)
Find the rate of change of this function at the point (1,1,2) in the direction toward the point (-2,3,5).
d) At the point (1,1,2) find a direction (not a zero vector) in which the directional derivative is zero.
15. Find
∂
z
∂
x
if
x
2
sin
2
y
2
z x
=
x
1
−
z
2
.
16. A cylindrical piece of steel is initially 8 inches long and has a diameter of 8 inches. During heat treatment the
length and diameter each increase by 0.1 in. Use differentials to find the approximate increase in the volume.