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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.

    Plot of points

  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.

    Plot of <span class="katex-math">x + y = 2</span>

  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.

    Left: <span class="katex-math">\mathbb{R}^3</span> plot of <span class="katex-math">x=4</span>, Right: <span class="katex-math">\mathbb{R}^2</span> plot of
<span class="katex-math">x=4</span>

  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

    Plot of elliptic paraboloid <span class="katex-math">x = y^2 + (2z)^2</span>.

    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}

    !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.

  1. y = z^2 - x^2

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

    !Plot of hyperbolic paraboloid y = 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.

    Plot of <span class="katex-math">f(x,y) = y</span>

  1. f(x,y) = \cos x

    See next page.

    Plot of <span class="katex-math">f(x,y) = \cos x</span>

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

    Plot of <span class="katex-math">f(x,y) = \sqrt{x^2 + y^2}</span>

  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?

    Provided with exercise 32

    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.

    Provided with exercise 34

    Contour map

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

    Provided with exercise 35

    Approximation of <span class="katex-math">f</span>


Page 4

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

    Contour map of <span class="katex-math">f(x,y) = y - \ln x</span>

  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)

    Isothermals of <span class="katex-math">T(x,y) = 100/(1 + x^2 + 2y^2)</span>

  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.

    Equipotential curves for <span class="katex-math">V(x,y) = c/\sqrt{r^2 - x^2 - y^2}, c=1, r=0</span>


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


© Emberlynn McKinney