Calculus III Advanced (Course) (11.6) (Homework)
Contents
Section 11.6 Homework
From Calculus 10e by Larson and Edwards, p. 802. Exercises 1, 2, 3, 4, 5, 6, 10, 16, 32, 44.
Exercises 11.6.1-6 Match the Equation with its Graph
Solution
1. c, 2. e, 3. f, 4. b, 5. d, 6. e.
Exercise 11.6.10 Sketching a Surface in Three Dimensions
- \(y^{2}+z=6\)
Solution
Exercise 11.6.16 Sketching a Quadric Surface
Describe and sketch
- \( -8x^{2}+18y^{2}+18z^{2}=2 \)
Solution One-sheet Hyperboloid
Exercise 11.6.32 Find the Equation for a Rotated Surface
Curve Equation | \(z=3y \) |
Coordinate Plane | \(yz\)-plane |
Axis of Revolution | \(y\)-axis |
Solution
To find an equation for the surface form by revolving the graph for \(z=3y\) about the \(y\)-axis, solve for \(z\) in \(y\) terms.
- \(z=3y=r(y)\)
The equation for the surface is
- \( 0x^{2}+z^{2}=[r(y)]^{2} \) Revolved about the \(y\)-axis
- \( z^{2}=(3y)^{2} \) Substitute \(3y\) for \(r(y)\)
- \( z^{2}=9y^{2} \) Equation for the surface.
Exercise 11.6.44 Find the Equation for a Surface
The set for all points equidistant from the point (0,0,4) and the \(xy\)-plane.
Solution
Find the distance from the point \(x=y\) and \(z=0\) to the point (0,0,4).
- \( d=\sqrt{x^{2}+y^{2}+(z-4)^{2}} \)
The length is the line from \((x,y,z)\) to the \(xy\)-plane is \( |z| \) so
\( |z| \) | \(= \sqrt{x^{2}+y^{2}+(z-4)^{2}} \) |
\(= x^{2}+y^{2}+z^{2}-8z+16 \) | |
\( 0 \) | \(= x^{2}+y^{2}-8z+16 \) |
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Parent Article: Calculus III Advanced (Course)