Misc Calc

Question 1

What is a trace?

Answer 1

The intersections of a sphere (or other shape) with the coordinate planes

Question 2

How do you do write equation of a sphere in set notation?

Answer 2

{(x,y,z): (insert equation here)}

Question 3

What does it mean if a set is compact?

Answer 3

It is both closed and bounded

(Closed = ALL boundary points of the set are in it)
(Bounded = can be contained in a finite disk, aka doesn’t go to infinity anywhere)

Question 4

When using polar paths, how do you know whether the limit exists or not?

Answer 4

If the limit is dependant upon theta, it does not exist (DNE)

If the limit is independent of theta, it exists and equals whatever number you got

Question 5

When drawing level curves if a function, what do you have to label each curve with?

Answer 5

It’s corresponding z value (the height)

Question 6

What is a level set?

Answer 6

A level set are the slices f(x,y,z) = constant in R^3

Question 7

What are the six fundamental quadric surfaces? (Graphs of 3d shapes)

Answer 7

Ellipsoid
Cone
Elliptic Paraboloid
Hyperboloid of One Sheet
Hyperbolic Paraboloid (saddle)
Hyperboloid of Two Sheets

Question 8

Which 3D graphs have equations that = 0? (Don’t have constants unless they’re multiplied by x, y, or z

Answer 8

Cone and paraboloids (elliptic paraboloid and hyperbolic paraboloid aka saddle)

Question 9

Which 3D graphs have equations that = 1?

Answer 9

Ellipsoid and hyperboloids (hyperboloid of one sheet, hyperboloid of two sheets)

Question 10

Is the component of B along A a vector or scalar?

Answer 10

CompA of B is a scalar and equals the length of the projection |ProjA of B|

Question 11

Given point P and Normal vector N, how do you know if a point Q is on the plane?

Answer 11

If PQ is perpendicular to N, or PQ•N =0