a jar contains 6 jellybeans, 4 green jellybeans, and 4 blue jelly beans. if we choose a jellybean, then another without putting
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If 
represent a family of surfaces for different values of the constant 
. The gradient of the function 
defined as 
is a vector normal to the surface .
Given <span>the paraboloid
.
We can rewrite it as a scalar value function f as follows:
The normal to the </span><span>paraboloid at any point is given by:
Also, the normal to the given plane
is given by:
Equating the two normal vectors, we have:
</span>
Since, -1 = 2 is not possible, therefore there exist no such point <span>on the paraboloid such that the tangent plane is parallel to the plane 3x + 2y + 7z = 2</span>.
Given <span>the paraboloid
We can rewrite it as a scalar value function f as follows:
The normal to the </span><span>paraboloid at any point is given by:
Also, the normal to the given plane
Equating the two normal vectors, we have:
</span>
Since, -1 = 2 is not possible, therefore there exist no such point <span>on the paraboloid
Consider the top half of a sphere centered at the origin with radius 
, which can be described by the equation
and consider a plane
with
. Call the region between the two surfaces 
. The volume of is given by the triple integral
Converting to polar coordinates will help make this computation easier. Set
Now, the volume can be computed with the integral
You should get
and consider a plane
with
Converting to polar coordinates will help make this computation easier. Set
Now, the volume can be computed with the integral
You should get