>with(plots): >blob:=z=x^2+3.7*y^2; >implicitplot3d(blob,x=-15..15,y=-10..10,z=0..200,axes=boxed, numpoints=5000);The sphere can be entered as one implicit expression.

>g:=x^2+y^2+z^2=1; >implicitplot3d(g,x=-1..1,y=-1..1,z=-1..1,axes=boxed);To look at the cross-section of the sphere you cut the sphere along a plane - i.e. you hold a variable constant. So the intersection of the sphere and the plane is:

>implicitplot(subs(z=0.5,g,x=-1..1,y=-1..1,labels=[x,y]);Notice that the plot is a two-dimensional circle. To intersect vertical planes hold the or constant.

>implicitplot({subs(y=0.6,g),subs(y=-0.8,g)},x=-1..1,z=-1..1,labels=[x,z]);Other three-dimensional shapes can be made from known conic sections. A few of these will be analyzed in the exercises.

- Given
- A)
- Plot the three-dimensional shape over the intervals , , and .
- B)
- Is the given equation a function?
- C)
- Plot the intersections of this shape and two planes perpendicular to the z-axis. What two-dimensional shapes are graphed?
- D)
- Plot the intersections of this shape and two planes perpendicular to the y-axis. What two-dimensional shapes are graphed?
- E)
- Plot the intersections of this shape and two planes perpendicular to the x-axis. What two-dimensional shapes are graphed?
- F)
- What three-dimensional shape is the equation (a sphere, a paraboloid, an ellipsoid, an hyperboloid, or an hyperbolic paraboloid (saddle))?

- Given
- A)
- Plot the three-dimensional shape over the intervals , , and .
- B)
- Is the given equation a function?
- C)
- Plot the intersections of this shape and two planes perpendicular to the z-axis. What two-dimensional shapes are graphed?
- D)
- Plot the intersections of this shape and two planes perpendicular to the y-axis. What two-dimensional shapes are graphed?
- E)
- Plot the intersections of this shape and two planes perpendicular to the x-axis. What two-dimensional shapes are graphed?
- F)
- What three-dimensional shape is the equation (a sphere, a paraboloid, an ellipsoid, an hyperboloid, or an hyperbolic paraboloid (saddle))?

2014-09-04