Page 1

Sketch the points (0, 5, 2), (4, 0, 1), (2, 4, 6), and (1, 1, 2) on a single set of axes.

Which of the points P(6, 2, 3), Q(5, 1, 4), R(0, 3, 8) is closest to the xzplane? Which point lies on the yzplane?
Q is the closest to the xzplane since \min{2, 1, 3} = 1
R lies on the yzplane since R_x = 0.

Describe and sketch the surface in \mathbb{R}^3 represented by the equation x + y = 2.
The surface is a plane parallel to the zaxis whose intersection with the xyplane is a line in the xyplane with slope 1 and yintercept 2.

a) What does the equation x = 4 represent in \mathbb{R}^2? What does it represent in \mathbb{R}^3? Illustrate with sketches.
In \mathbb{R}^2, x=4 represents a line parallel to the yaxis with an xintercept at 4.
In \mathbb{R}^3, x=4 represents a plane parallel to the yzplane whose intersection with the xyplane is the abovedescribed line in the xyplane.

Find an equation of the sphere that passes through the point (4, 3, 1) and has center (3, 8, 1).
r = \sqrt{(43)^2 + (38)^2 + (11)^2}
r = \sqrt{1 + 25 + 4} = \sqrt{30}(x3)^2 + (y8)^2 + (z1)^2 = 30

Find an equation of the sphere that passes through the origin and whose center is (1, 2, 3).
r = \sqrt{(01)^2 + (02)^2 + (03)^2}
r = \sqrt{1 + 4 + 9} = \sqrt{14}(x1)^2 + (y2)^2 + (z3)^2 = 14
 2332.

Describe in words the region of \mathbb{R}^3 represented by the equation or inequality.

y = 4
A plane parallel to the xzplane that intersects the yaxis at 4.

x = 10
A plane parallel to the yzplane that intersects the xaxis at 10.

x > 3
A partial space expanding in the positive x direction out from but not including a plane parallel to the yzplane that intersects the xaxis at 3.

y \ge 0
A partial space expanding in the positive y direction out from and including the xzplane.

0 \le z \le 6
A partial space between and including the xyplane and a plane parallel to the xyplane that intercepts the zaxis at 6.

x^2 = 1
A surface composed of the parabola x^2 = 1 in every plane parallel to the yzplane.

x^2 + y^2 + z^2 \le 3
A volume inside and including the sphere centered on the origin with radius \sqrt 3.

x = z
A plane composed of the line x = z in every plane parallel to the xzplane.

x^2 + z^2 \le 9
A partial space expanding inwards to the yaxis bounded by and including the surface composed of the circle centered at x = 0, z = 0 with radius 3 in every plane parallel to the xzplane.

x^2 + y^2 + z^2 > 2z
A partial space expanding outwards from and not including the sphere centered on (0, 0, 1) with radius 1.
Page 2
 1120.

Use traces to sketch and identify the surface.

x = y^2 + 4z^2
x = f(y,z) = y^2 + (2z)^2
Note that the axes in Figures 4 and  are rotated from the usual “view,” but the coordinate system is unchanged.

x^2 = y^2 + 4z^2
x = f(y,z) = \pm\sqrt{y^2 + (2z)^2}
!Plot of elliptic cone x^2 = y^2 + 4z^2
Note that the axes in Figures 4 and  are rotated from the usual “view,” but the coordinate system is unchanged.

y = z^2  x^2
y = f(x,z) = z^2  x^2

Find an equation for the surface obtained by rotating the parabola y=x^2 about the yaxis.
y = x^2 + z^2

Traditionally, the earth’s surface has been modeled as a sphere, but the World Geodetic System of 1984 (WGS84) uses an ellipsoid as a more accurate model. It places the center of the earth at the origin and the north pole on the positive zaxis. The distance from the center to the poles is 6356.523 km and the distance to a point on the equator is 6378.137 km.
a) Find an equation of the earth’s surface as used by WGS84.
({x \over 6378.137\text{km}})^2 + ({y \over 6378.137\text{km}})^2 + ({z \over 6356.523\text{km}})^2 = 1
b) Curves of equal latitude are traces in the planes z=k. What is the shape of these curves?
Circles
c) Meridians (curves of equal longitude) are traces in planes of the form y = mx. What is the shape of these meridians?
Ellipses
Page 3
 2129.

Sketch the graph of the function.

f(x,y) = y
See next page.

f(x,y) = \cos x
See next page.

f(x,y) = \sqrt{x^2 + y^2}

Two contour maps are shown. One is for a function f whose graph is a cone. The other is for a function g whose graph is a paraboloid. Which is which, and why?
I is function g and II is function f since its slope (indicated by the distances between the concentric circles) is constant.

Make a rough sketch of a contour map for the function whose graph is shown.

A contour map of a function is shown. Use it to make a rough sketch of the graph of f.
Page 4

Draw a contour map of the function showing several level curves.
f(x,y) = y  \ln x

A thin metal plate, located in the xyplane, has temperature T(x,y) at the point (x,y). The level curves of T are called isothermals because at all points on an isothermal the temperature is the same. Sketch some isothermals if the temperature function is given by T(x,y) = 100/(1 + x^2 + 2y^2)

If V(x,y) is the electric potential at a point (x,y) in the xyplane, the the level curves of V are called equipotential curves because at all points on such a curve the electric potential is the same. Sketch some equipotential curves if V(x,y) = c/\sqrt{r^2  x^2  y^2}, where c is a positive constant.
If r = \sqrt{x^2 + y^2} then V(x,y) = c/\sqrt{x^2 + y^2  x^2  y^2} = c/0, so it is not true that r = \sqrt{x^2 + y^2}. Thus assuming r = 0.
Page 5

Express each of the following loci [surfaces] in spherical coordinates:
a) the sphere x^2 + y^2 + z^2 = 9
r = 3
b) the cone z^2 = 3(x^2 + y^2)
\theta\in{\arctan{1\over 3},\pi\arctan{1\over 3}}
c) the paraboloid z = x^2 + y^2
r = \sqrt{x^2 + y^2 + (x^2 + y^2)^2}
\theta = \arccos{\left({x^2+y^2\over\sqrt{x^2+y^2+z^2}}\right)}
d) the plane z = 0
\theta = {\pi \over 2}
e) the plane y = x
\phi \in {{\pi \over 4},{3\pi \over 4}}

If \rho,\phi,z are cylindrical coordinates, describe each of the following loci and write the equation of each locus in rectangular coordinates:
a) \rho = 4, z = 0
x^2 + y^2 = 16, z = 0
b) \rho = 4
x^2 + y^2 = 16
c) \phi = \pi/2
y = 0, x > 0
d) \phi = \pi/3, z = 1
y = \sqrt 3 x, z = 1, x > 0
© Emberlynn McKinney