Calc III Midterm 3

Flat tangent plane

If F has a local max or min at (a,b) then fx(a,b) = 0 and fy(a,b)=0

Critical point

A point at (a,b) where fx(a,b) and fy(a,b) both are either 0 or do not exist.

Not all of them

Do all critical points give maxes or mins?

the local minimum

If D > 0 and Fxx(a,b) > 0 (a,b) are:

the local maximum

If D > 0 and fxx(a,b) < 0, (a,b) are:

Saddle point

If D < 0, it is a:

Not enough information

If D = 0, then there is:

D = fxx(a,b) * fyy(a,b) - (fyx(a,b))²

Second Derivative Test

What does the second derivative test show?

It is the determinant that shows the critical points of a function z=f(x,y) as the local minima, local maxima, or saddle points

Extreme Value Theorem

If y = f(x) is continuous on a closed interval, then f(x) has a global max and a global min. They occur at either the critical points or at end points.

global max and min

If f(x,y) is continuous on a closed and bounded set R in a 2D space. The, f attains a _________ on R and it occurs either at critical points or boundary points.

Set

A ___ is bounded if it can be enclosed in a desk.

How do you find the dimensions of a box w/o the top? Given that you have the volume.

Get the surface area equation (ex: S = (LW)(2HW)(2LH). Then, get the volume equation (ex., V = LWH). Find the expression for each variable to replace H. Then, take the partial derivatives with respect to L and W. With that, you will be able to plug them into each other to get a dimension for one of them. Keep plugging until you find the dimensions for all of them.

Surface

z=f(x,y)

Curve

g(x,y) = k

same

At max or min gradient of f points in the ___ direction of gradient of g.

Methods of Lagrange Multiplers

To find the max/min → 1) find all x, y, lambda values where ∇f(x,y) = λ∇g(x,y) 2) evaluate in f(x,y) the largest is the max and the smallest is the min.

What do three variable langrage multipliers look like?

fx = λgx + μλhx, fy = λgy + μλhy, fz = λgz + μλhz