# calculus

Worksheet

1

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Example Graphs in space may be represented in words, by equations, or visually. Consider the
equation

x2z − y2z − 16z − 3×2 − 3y2 = −48.

To find out what this graph looks like, let’s try to rewrite the equation:

x2z − y2z − 16z − 3×2 − 3y2 + 48 = 0
(z − 3)(x2 + y2 − 16) = 0

z − 3 = 0 or x2 + y2 − 16 = 0
z = 3 or z2 + y2 = 16

Thus we can describe this graph in words as the set of all points that either have z-coordinate 3
(these points form a plane) or are on the cylinder of radius 4 centered on the z-axis.

Write an equation for the graph described in words, and sketch the graph.

1. The sphere of radius 6 centered at (3, 0, −1)

2. The set of all points whose x and z coordinates are equal.

3. The cylinder of radius 2 centered on the line through the point (1, 2, 3) parallel to the y-axis.

After rewriting the equation as needed, describe the graph in words, then sketch its graph.

4. x2y − 4×2 − 5xy + 20x = 0

5. y2 − 2y + z2 + 6z = 15
Hint: Complete the square.

6. x2y + y3 + yz2 − 9y + 2×2 + 2y2 + 2z2 = 18

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