Final: 13.9 to 14.6, 15.7-15.8

Question 1

How do you find maximum volume/profit?

Answer 1

Volume = Construct a function for volume based on x, y and z as a function of x and y. Use partials and second partials test (set = 0) to find maximum.

Profit - Same, but function is provided

Question 2

How do you use Lagrange multipliers?

Answer 2

Set g(x,y) and h(x,y) = to constraint. Equate gradient of f = (lambda)*gradient of g + (lambda) * gradient of h. Solve for x and y, making sure to consider 0,0.

Question 3

How do you find Center of Mass (plane)?

Answer 3

Mx = intintR xrho(x,y) dA, My = intintR yrho(x,y) dA, m = intintR rho(x,y) dA. COM = (Mx/m, My/m)

Question 4

What do the different differential abbrevations mean?

Answer 4

ds = ||r’(t)|| dt, dS = sqrt(1+fx(x,y)^2+fy(x,y)^2) dA, dr = r’(t) dt

Question 5

How do you find the surface area given a closed region R in the xy-plane (shadow)?

Answer 5

intintR ds = intintR sqrt(1+fx(x,y)^2+fy(x,y)^2) dA

Question 6

How do you find the center of Mass (space)?

Answer 6

Mx = intintintQ xrho(x,y,z) dV, My = intintintQ yrho(x,y,z) dV, Mz = intintintQ z*rho(x,y,z) dV. m = intintintQ rho(x,y,z) dv. COM = (Mx/m, My/m, Mz/m).

Question 7

What is the divergence Theorem?

Answer 7

intintS F dot N dS = intintintQ div F dV

Question 8

What is Stokes’ Theorem?

Answer 8

intC F dot dr = intintS curl F dot N dS