calc 3 ch13

f(x,y) is a multivariable function, and it can be written as

f(x,y) = z just like how f(x) = y

ellipse equation

Where:

• the ellipse is centered at (h,k)

• a represents the radius along the x-axis (e.g. if the ellipse is centered at 0,0 and a=10 then at x=-10 and x=10 you will have a point on the ellipse… it is 10 units away from the center along the x-axis!)

• b represents the radius along the y-axis

• note the =1

elliptic cylinder equation & explanation of why it works & visualize in your head

this is the same equation of an ellipse… the reason this works is because z can be anything and therefore the elliptic shape stretches across all z values, making a cylinder

where

• a and b are the radius along the x and y axis respectively

• h,k is the center of the elliptical cross section

ellipsoid equation & visualize in your head

same as ellipse but with the z and c

where:

• (h,k,l) is the center of the ellipsoid

• a is the radius along the x-axis

• b is the radius along the y-axis

• c is the radius along the z-axis

cylinder equation & visualization & why does it work

where:

• just like the circle equation

• (h,k) is the center

• r is the radius

this works because z is not in the equation because z can be anything so it stretches across the entire z axis (the circle just goes up towards positive z and down towards negative z for infinity making a cylinder)

circular double cone equation + explanation + visualization

explanation: it has cross sections of +-sqrt(x²) aka |x| and -|x| which is why you have 2 cones rather than one

where:

• (h,k,l) is the point where the cones meet.

• same as cylinder equation except the radius = z axis… Think about why this works: as you increase the z, the circle gets wider making a conical shape when you have a bunch of those circles (of varying sizes) stacked on top of each other

• note the +- sqrt as that is why there are 2 cones

the basic (non shift) equation is x² + y² = z² if that helps to remember at first

elliptical double cone equation + explanation + visualization

this is also the same as the circular double cone equation but since a≠b in an ellipse (unlike a circle), a² b² and c² are all present in this one

where:

• (h,k,l) is the point where the cones meet

• a and b control the parameters in the elliptical cross section (same as they normally do for ellipses)

• c controls how fast the cone flares outwards as the cone expands up the z-axis (e.g. large c value makes the cone narrower and steeper)

the basic (non shift) equation is [x² / a²] + [y² / b²] = [z² / c²] if that helps to remember at first

note that double cones could have restrictions that lead to only 1 of the cones being graphed

e.g. z>0

paraboloid equation

(z-l) = (x-h)² + (y-k)²

where (h,k,l) is the bottom of the paraboloid

cross sections of 2d cone equation

x² = y²

y = +-sqrt(x²)

recall that sqrt(x²) = |x| and then +- sqrt(x²) would include -|x|

level sets defined + what are they used for

{ (x,y) : f(x,y) = c } is the level set of f at height c
f(x,y) = z → z=c… You use a level set to find what a 3D function looks like at a certain z level… imagine taking a slice out of a 3d function at z=c

level curves vs level surfaces… what do they give information for?

level curves are 2D objects that give information on a 3D object (function of 2 variables like f(x,y) )

level surfaces are 3D objects that give information on a 4D object (function of 3 variables like f(x,y,z) )

what is a level curve?

it’s just a 2D image showing what the function looks like (birds eye view) at different z levels as you plug in different c values into the level set

level surfaces + what are they used for

Let w=f(x,y,z)

Let c equal some constant.

You set c=f(x,y,z) just like a level curve.

Use case: When you have a function with 4 variables, it can be hard to visualize. You use a level surface to represent how the function looks when you set w=c.

For each level surface, f(x,y,z) is a constant on that entire surface!

note that for f(x,y) when a variable (such as y) is missing then…

that means the variable (in this e.g. y) can be anything; the function will stretch along that variable’s axis (in this e.g. it will stretch along the y-axis).

any f(x,y) can be written as

any f(x,y) can be written as g(x,y,z) = 0 for g(x,y,z) = z - f(x,y)

*note on the for: f(x,y) is just z so it’d be g(x,y,z) = z-z = 0

f_y(x,y) notation meaning

the partial derivative of f with respect to y

*respect to y as y is the subscript
∂f/∂y

f_x(x,y) notation meaning

the partial derivative of f with respect to x

*respect to x as x is the subscript
∂f/∂x

f_y(x,y) = *limit defn and other definition

= (∂f / ∂y)(x,y) = same as picture

for f_x you’d keep y as a constant instead

f_yx(x,y) meaning

take partial derivative of f with respect to y first then with respect to x

*you just read it from left to right and take partial derivative from left to right

how to take a partial derivative with respect to some variable (e.g. x)

if you are taking partial derivative with respect to x, treat “y” as if it is a constant. you can replace “y” with c and then take derivative of that as if c were a constant. Then, at the end, put the “y”s back to where they would be

e.g. 3y² → 3c² and this is just a constant so the derivative is 0

but 3xy² → 3xc² → 3c² → 3y²

3x²y → 3x²c → 6xc → 6xy

f_yx = f_xy IFF

f_yx = f_xy IFF second derivative is continuous