Math
21D each of the following equations in two-dimensional space.
1.) Graph
Vogler
c.) y = x d . ) y = 3
a.) y =Sheet
3
= —2
Discussion
1x
e.) y = x
3
1
g.) y l n x
h.) y = i . )
= y
Y= —
1.) Graph each of the following equations in two-dimensional space.
2
x
2.) Sketch
each of the
using
c.) following
y = x dequations
. ) y =(surfaces)
3
e.)the
y =following
x
a.) y =the
3 level xcurves =for—2
values of z : -3, -2, -1, 0, 1, 2, 3
3
1
x
i . )
= y
a.) z y
b .g.)
) yz l=n1 ——
y h.) y c=.
Y= —
2
)
x
z (i.e., xequations
3.) Sketch the
all three
coordinate
plane
0 , y 0 (surfaces)
, and z 0using
) forthe
each
of the
2.)
level curves
for each
of traces
the following
following
2
values
of zequations
: -3, -2, -1,
0, 1, 2, 3
following
(surfaces).
f
.
y
e
x
f
.
y
e
x
x
2
. ——
) z y c . c+. )
z =
a.) zx y+ 2y +
b 3z
. )= 6z b
=1
)y
z2 (i.e.,
4.)
space each
the following
Use traces
3.) Sketch in
allthree-dimensional
three coordinate plane
tracesof
x 0 , y equations
0 , and z(surfaces).
0 ) for each
of the
2(
and/or level
curves (surfaces).
i f necessary.
following
equations
xI
2
e
y +
= 2y
3 +b 3z. =) 6 x b=. —2
c .c ) .y+. =) x z =y ( = 3
e.) y = x
a.)
x
)
z
d.) hx-K1
m (s i n x
e.) lx-*()
i m -x
f.) l7i1 -m
)
*
00
n
3
yx
1
4.) eSketch
the following1equations (surfaces). Use traces
22=1/x
2 in three-dimensional
g.) y = ln x space
h -. each
) y of
+
Y= —
x _
(+
and/or
level curves i f necessary. g.)
l
i
m
(1
+
2
x
)
x
+
0
x
—
)
I
_
1 ,
y
x
la.) y = 3 b . ) x = —2 c 1. ) y.2 = x
5
y
=
3
e.)
y = x
1
) 20 0
/
)+
-+ )
/ xc nxE- LM
h
.
4
3
)
+w
xz
1
x
l ) +n)i l i mn0
g.) y = ln x h . ) y 22=
n
Y= —
2
I+ + +n+ + +the
+ + graph
+ + + + o+f+y+ =+ +ln(x
+ +—
+ +=+1)+ i+n +the
+ +xy-plane.
+ + + + + +Fi
+ n+d+ an
+ + equation
+ + + + + for the
5.) (ax.)+ Consider
y9
x
x l2 created
)
surface
by revolving this graph about
the
2
i c+
+
The following
problem is for recreational purposes
only.
z
i.)
x-axis
i
i
.
)
y-axis
y
. )
2
)ax2
=the
5.)
.)
Consider
the
graph
o
f
y
=
ln(x
—
1) ixz-plane.
n the xy-plane.
Fi equation
n d an equation
for the
b.)
Consider
graph
of
z
=
sin
x
in
Find an
for the surface
10.) T w
+ o bicyclists are twelve miles apart. 9Th e y begin riding toward each other, one
l 2
f
.
y
=
e
x
O
.
)
zf
2.
y
==
xe
x
2
+
y
2
p
.
)
z
2
=
x
2
+
y
O
2
.+
)Z
z=
2y
=2
xx
22
+
y
2
p
.
)
z
2
=
x
2
+
y
2
+
Z
=
y
2
x
2
=
2
+
y
2
y
2
x
2
1
1
)
Z
2
=
X
2
+
y
2
x
=
2
+
y
2
y
2
x
2
1
1
)
Z
2
=
X
2
+
y
2
x
surface
created
byand
revolving
thisabout
about
createdat
by4revolving
this
the
pedaling
mph
the graph
other
atgraph
2 mph.
A t the
the same time a bumblebee begins flying
z
i
+
back and forth between the riders at a constant speed of 10 mph. What is the total distance
i the
. ) time
y-axis
z-axis
my2 i.) x-axis
the bumblebee
travelsi by
the riders meet ?
=
( 2
b.)
the graph
of function.
z = sin x in the xz-plane. Find an equation for the surface
6.) xSketch
the domain
of each
4 Consider
+
created
by revolving this graph about the
) a.)
z1 ( x , y) = l n (x
. +i.) x-axis
t 2
2
yy) = i i1 . ) z-axis
c.)
(
x
,
d
.
)
( x , y) = V ( x
)
a2
=
—
4
—
V25
—
x
2x — the4domain
) ( y of each function.
6.) nSketch
4 )
4
2 — 1y )
2+
— following
7.) Evaluate
the
limits or determine that the limit does not exist.
x a.)
(1 ( x ,xy)
= l2n (x
2
,. + y
2
c.)
( x , y) =
1
d
.
) 1 ( x , y) = V ( x
y)y
2
—
2x+ — 4 ) 4( —
y V25 — x
)
4
)
2
— 1y )
2+
— following
3
7.) Evaluate
the
limits or determine that the limit does not exist.
=
(
x
2
z
n
,l2
1
(yy=
1
6
+
)+
m
Thanks to Dr. Kouba
3
+
=
.yz
l
n