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# Page 1

1. Sketch the points (0, 5, 2), (4, 0, -1), (2, 4, 6), and (1, -1, 2) on a single set of axes. 2. Which of the points P(6, 2, 3), Q(-5, -1, 4), R(0, 3, 8) is closest to the xz-plane? Which point lies on the yz-plane?

Q is the closest to the xz-plane since \min{2, -1, 3} = -1

R lies on the yz-plane since R_x = 0.

1. Describe and sketch the surface in \mathbb{R}^3 represented by the equation x + y = 2.

The surface is a plane parallel to the z-axis whose intersection with the xy-plane is a line in the xy-plane with slope -1 and y-intercept 2. 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 y-axis with an x-intercept at 4.

In \mathbb{R}^3, x=4 represents a plane parallel to the yz-plane whose intersection with the xy-plane is the above-described line in the xy-plane. 1. Find an equation of the sphere that passes through the point (4, 3, -1) and has center (3, 8, 1).

r = \sqrt{(4-3)^2 + (3-8)^2 + (-1-1)^2}
r = \sqrt{1 + 25 + 4} = \sqrt{30}

(x-3)^2 + (y-8)^2 + (z-1)^2 = 30

1. Find an equation of the sphere that passes through the origin and whose center is (1, 2, 3).

r = \sqrt{(0-1)^2 + (0-2)^2 + (0-3)^2}
r = \sqrt{1 + 4 + 9} = \sqrt{14}

(x-1)^2 + (y-2)^2 + (z-3)^2 = 14

23-32.

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

1. y = -4

A plane parallel to the xz-plane that intersects the y-axis at -4.

2. x = 10

A plane parallel to the yz-plane that intersects the x-axis at 10.

3. x > 3

A partial space expanding in the positive x direction out from but not including a plane parallel to the yz-plane that intersects the x-axis at 3.

4. y \ge 0

A partial space expanding in the positive y direction out from and including the xz-plane.

5. 0 \le z \le 6

A partial space between and including the xy-plane and a plane parallel to the xy-plane that intercepts the z-axis at 6.

6. x^2 = 1

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

7. x^2 + y^2 + z^2 \le 3

A volume inside and including the sphere centered on the origin with radius \sqrt 3.

8. x = z

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

9. x^2 + z^2 \le 9

A partial space expanding inwards to the y-axis 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 xz-plane.

10. 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

11-20.

Use traces to sketch and identify the surface.

1. 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.

2. x^2 = y^2 + 4z^2

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

Note that the axes in Figures 4 and - are rotated from the usual “view,” but the coordinate system is unchanged.

1. y = z^2 - x^2

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

2. Find an equation for the surface obtained by rotating the parabola y=x^2 about the y-axis.

y = x^2 + z^2

1. Traditionally, the earth’s surface has been modeled as a sphere, but the World Geodetic System of 1984 (WGS-84) 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 z-axis. 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 WGS-84.

({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

21-29.

Sketch the graph of the function.

1. f(x,y) = y

See next page. 1. f(x,y) = \cos x

See next page. 2. f(x,y) = \sqrt{x^2 + y^2} 1. 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.

2. Make a rough sketch of a contour map for the function whose graph is shown.  3. A contour map of a function is shown. Use it to make a rough sketch of the graph of f.  # Page 4

1. Draw a contour map of the function showing several level curves.
f(x,y) = y - \ln x 1. A thin metal plate, located in the xy-plane, 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) 2. If V(x,y) is the electric potential at a point (x,y) in the xy-plane, 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

1. 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}}

1. 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