Final

Question 1

Find the equation of a line given: 2 points

Question 2

Find the equation of a line given: 1 point and 1 vector

Question 3

Find the equation of a line given: 1 point and 1 perpendicular plane

Question 4

Find the equation of a line given: 1 point and 1 perpendicular line

Question 5

Find the equation of a line given: 2 intersecting planes

Question 6

Find the equation of a plane given: 1 point and 1 normal vector

Question 7

Find the equation of a plane given: 1 point and 1 line

Question 8

Find the equation of a plane given: 3 points

Question 9

Find the equation of a plane given: 2 lines

Question 10

Find the equation of a plane given: a tangent plane

Question 11

Find the equation of a plane given: 1 line and 1 perpendicular plane

Question 12

(Line Testing) Parallel Line

Question 13

(Line Testing) Intersecting Line and Perpendicular Line

Question 14

(Line Testing) Skew Line

Answer 14

not parallel or intersecting

Question 15

Distance Equation

Question 16

Directional Derivative

Question 17

Direction of the Maximum Rate of Change

Question 18

Maximum Rate of Change

Question 19

Tangent Plane Equation/Linear Approximation

Question 20

Prove a limit does NOT exist

Question 21

Prove a limit DOES exist

Answer 21

• can be continuous and have a constant denominator
• can simplify to get something to get a non-undetermined limit
• can apply the squeeze theorem

Question 22

Find and Classify local extrema

Question 23

Find global extrema

Question 24

Lagrange Multipliers

Question 25

What do double integrals find?

Answer 25

Volume

Question 26

Polar Coordinates (2D)

Answer 26

*Add an extra r

Question 27

What do triple integrals find?

Answer 27

Mass or Density

Question 28

Cylindrical Coordinates (3D)

Answer 28

*Add an extra r

Question 29

Spherical Coordinates (3D)

Answer 29

*Add an extra

Question 30

Scalar Line Integrals

Question 31

Change of Coordinates

Answer 31

(1) graph the given bounds and find the equation for each line segment
(2) put these equations equal to u and v
(3) solve for x and y to find the Jacobian

(4) solve using

Question 32

Change Order of Integration

Answer 32

(1) sketch the given integrals
(2) reverse the order in which the figure is being cut
(3) keep what is inside of the integrand the same

Question 33

Vector Line Integrals

Question 34

Fundamental Theory of Line Integrals Criteria (Line Integrals)

Answer 34

F is conservative, C is not closed, and the Potential Function can be found

Question 35

Potential Function

Question 36

Fundamental Theory of Line Integrals Formula (Line Integrals)

Question 37

What is the line integral if F is conservative and C is closed

Answer 37

equals 0

Question 38

Green’s Theorem Criteria (Line Integrals)

Answer 38

F is not conservative, C is closed, has continuous partials, and positively oriented (if not add a - in front)

Question 39

Green’s Theorem Formula (Line Integrals)

Question 40

curl F

Answer 40

Tells how rotational something is

Question 41

div F

Answer 41

Tells how much an object expands/contracts under the velocity field

Question 42

Parametric Surfaces

Answer 42

(1) draw the function given
(2) put in terms of r(u,v) and assign u and v

Question 43

Tangent Plane to S (Parametric Surfaces)

Question 44

Surface Integrals

Question 45

area(S)

Question 46

Flux

Answer 46

How much coffee is made per second

Question 47

Stokes Theorem Criteria (Surface Integrals)

Answer 47

F is a curl field, S is closed, and you can find G such that

Question 48

Stokes Theorem Formula (Surface Integrals)

Question 49

What is the surface integral if F is a curl field and S is closed?

Answer 49

equals 0

Question 50

Divergence Theorem Criteria (Surface Integrals)

Question 51

Divergence Theorem Formula (Surface Integrals)