Unit 2 MVC

What should candidate points look like f(x,y)

(X,y)

Steps to using Lagrange

1. Find gradient and 3 main equations (including constraint) 2.

How to phrase that limit exists explanation

Based on above results it appears that limit =0 since both pathways generated same result.

Now to find directional derivative

Duf(x,y) = gradient of f(x,y) . unit vector

Method of Lagrange multipliers

Finding max and min values of f on constraint g (Gradient of f)= lambda (gradient of g)

Fermat's theorem

If F has a local max or min at (a,b) and the first partial derivatives of f exist here, then Fx(a,b)=0 and Fy(a,b)=0

Clue that z=x or some kind of relationship might be a case

If systems has equations with those two variables and lambda

How to tell that contour map with ewually spaced circular level curves is a cone, and that contour map with steeper circles as radius increases is paraboloid

They increase at constant rate just like a cone does Paraboloid circles have steeper

Linear approximation /tangent plane approximation of f at a,b

f(x,y) approx = c + fx(a,b)(x-a) + fy(a,b)(y-b)

Law of Sines

sinA/a=sinB/b=sinC/c

Partial derivative of y with respect to x

0

Steps to find local Max min and saddle points

1. Find Fx and critical points of x and y 2. Use the critical points of 1. as different "cases" and find coordinates for every case 3 . Make a table of coordinates and find Fxx Fyy and Fxy and D to make conclusion.

Fxy vs Fyx

1. How does Fx change as y changes 2. How does Fy change as x changes

Even when finding partial derivatives don't forget ... apply this to z = tan(xy)

Chain rule. Zx = sec2(xy)y

How to notate domain and range for f(x) = x/sqrt(x-2)

D: { x| x>2 } R: { y| >= 2sqrt2 }

Gradient

Delta f (x,y) = Fx i + Fy j

How to find tangent plane to parametric surface when you are given parametrics and point

Find values of partial derivatives of both variables at given point. Cross these vectors to get n vector then solve for d using given point.

Ftt (40,15)

Ft(40,20)-Ft(40,10)/10 (time varying and velocity held constant)

Fx(x,y) vs Fy(x,y)

Fx, delf/delx, delz/delx, Dx Fy, delf/dely, delz/dely, Dy

What if y equals any value for case 1

If y is IR, assign it a variable like b

If f is a function of two variables with domain D, then the graph of f is the set of all points (x,y,z)...

In IR^3 such that z=f(x,y) and (x,y) is in D.

When finding limits don't forget about

Indeterminant form

How to find in WHAY DIRECTION function changes the fastest at a point

It's the directional derivative vector with the negative of the coefficient applied

How to find direction of rate of change of vector not given

Just the gradient vector's direction.

Point to always check for lagranges

Origin

For partial derivatives with a fraction use

Quotient rule and differentiate with respect to certain variable

When creating level curve graph never forget

To check for "holes" where z is undefined

When using vector don't forget to make it

Unit vector (maginigude of 1)

Law of Cosines

c²=a²+b²-2abcosC

Total Differential

dz = fx(x,y)dx + fy(x,y)dy = delz/delx (dx) + delz/dely (dy)

level curves

f(x,y) = k (contour map)

If z= f(x,y), dz/dt =

fx (dx/dt) + fy (dy/dt)

Clairaut's Theorem

fxy=fyx if they are both continuous on D

Equation of tangent plane to surface z=f(x,y) at point P (a,b,c)

z-c = fx(a,b)(x-a) + fy (a,b)(y-b)

Equation of tangent plane to surface z=f(x,y) at point a,b,c

z-c = fx(x-a) + fy(y-b) (if you bring over c then it becomes L(x,y) =)

How to notate domain and range for f(x,y) = xsquared + ysquared

D : IR^2 or { (x,y) | x exists on IR and y exists in IR } R: [0, infinity] or {z | z>= 0}

Formula for D

FxxFyy - (Fxy)^2

3 variables is how many dimensions

4 dimensions

How to find if limit of f(x,y) exists

Compare limits of two pathways and isolating one variable. Try to find failure. If those limits aren't equal, the limit of f(x,y) as (x,y) approaches point (a,b) DNE

If z= f(x,y) and x= g(s,t) and y=h(s,t), find two partial derivatives of Z.

Delz/dels = delz/delx(delx/dels) + delz/dely (dely/dels) Delz/delt = delz/delx(delx/delt) + delz/dely (dely/delt)

Meanings of partial derivative delh/delv or delh/delt in context

Describes how the wave heights change when the wind speed changes for a fixed duration. Describes how wave heights change when the duration changes and wind speed is fixed.

Largest possible error means find

Differwntial

In which direction does the function change fastest

Directionnnnn of gradient vector (neg from coefficient applied if applicable)

When finding magnitude of vector, what do you do with preexisitng coefficient

Don't distribute it until you've found square root of vector components squared

If square rooting both variables

Don't forget absolute value sign

When finding cases and you take a square root

Don't forget absolute value sign aro variable and plus or minus around numbers

When lines are evenLy spaced,

Double derivatives are 0

How to find fxy using level curve

Draw a constant y increasing with varying Fx (change in x) values (Looks like an E) Find Change in Z as y increases. If general distance between z values is increasing , Fxy is negative because it is getting negative faster.

How to find tangent plane equation when you can't isolate z

Either use Fx(x-a) + Fy(y-b) + Fz(z-c) =0 Or cross gradients

How to phrase lagranges conclusion

F(x) has a Max of ______ given the constraint g(x) at the points

Strategy for finding all second partial derivatives

Find Fx and Fy first. Fxx is derivative of Fx with respect to x, Fyy is derivative of Fy with respect to y, Fxy is derivative of Fx with respect to y. Fyx is derivative of Fy with respect to x.

How to find absolute extrema of 3d function over a region

Find Interior points (critical values) Find corner points (x and y values as coordinates for domain and range) Boundary points (find boundaries of region as equations, plug them into original function, and find where derivative =0) Find function value of all these points and make decision.

How to find equation of tangent plane when given parametric and a specific point (x=..... y=..... z=......)

Find Ru and Rv vectors (partial derivative VECTORS with respect to each variable). Then plug point in and find cross product of Ru and Rv to find n vector. Then find plane

How to find parametric equations for tangent line of plane/cylinder intersection .

Find both gradients at point of intersection (vector perpendicular to plane and cylinder respectively) Then cross these gradients to get a vector that will be tangent.

Boundary points

Find boundary ad equations and find the derivative and find when it equals 0

How to find tangent plane if given two vectors at point

Find derivative of each vector at point and cross to get j vector

If u have a surface described by two curves r(t) and need to find tangent plane

Find derivatives if both curves and cross them

Partial derivatives of f(x,y)

Find fx and fy, meaning only find the derivative of that one variable.

How to find in which direction the function changes the fastest at

Find gradient vector and you can factor number out but make sure negative is in. Note sign

How to estimate largest possible error of V when you have length width and height correct to within x cm

Find partial derivatives of V using V=lwh. The change in V will be approximately Vl(dL) + Vh(dh) + Vw(dw) And dl=dw=dh=x cm

What to do if you aren't given vector but you're given a point and another point it's in the direction of

Find vector between twk points starting at thr orginial one

How to find n vector when you're given parametrics of multivaribows

Find vectors that are partial derivatives with respect to one variable. Cross these to find n vector

Ftv(40, 15)

Ft(50,15)-Ft(30,15)/20 (Second variable varies first, first variable is seen in second step:

What does it mean for Fxx if level curves are decreasing but at a slower rate

Fx is negative but Fxx is positive, since level curves are getting less negative.

Define Fx vs Fy

Fx: differentiate with respect to x varying and y constant Fy: differentiate with respect to y varying and x constant

Lagranges with multiple constraints

Gradient f = gradient g (lambda) + gradient h (meu)

When you see constraint like sphere equation

If 1 variable can be 0, try with all. Then try with 2, then with none and all equal.

Extreme value theorem for functions of 2 variables

If f is continuous on a closed, bounded set D in R^2, f attains an absolute maximum and minimum value.

How to find if error is a lot or not (volume)

Largest error/actual volume as a percentage

How to approach Fxy problems

Look at how z changes as y increases. (Is it going in negative direction or positive direction. Look at how z changes as Fx changes (does it become more or less steep, look at distance between lines.) If less steep and negative direction, Fxy positive If less steep and positive direction, Fxy negative

How to find limit as t approaches infinity of delh/delt when you are given a table

Look at t values changing and wind speed constant. See what happens to CHANGESZ in heights as t approaches infinity. NOT THE HEIGHTD THEMSELVES

Maximum rate of change of f

Magnitude of the gradient

How to find maximum rate of increase

Magnitude of the gradient, don't forget to multiply coefficient top

What to do if variables on both sides

Make function all on one side (F(x,y,z))

What if all the sets of x, y, or z produce Same extrema of f

Mention this extrema of f is found at (a,b,c)

What pathways should you usually try if limit approaching 0

Pathway x=0 or y=0 or x= some relationship of y

gradient gives

Perpendicular vector to point

If finding partial derivative chain at specific point

Plug in points at the very end

What to do if you get variables equal to each other with absolute value in Lagrange

Plug them into constraint (they are all technically equal)

Directional derivative in words

Rate of change of a function at a given point along a particular direction.

Where to plug in different case points

RelTionship between x and y established by Fx and Fy

What to do if all 3 variables approaching 0

Set two variables equal to 0 or each other as pathways

For partial derivative chain rule never forget

Showing line by line work

How to find boundary points

Take boundary and substitute into original function. Then take the derivative and find when =0. Plug this into boundary.

partial derivative

Taking a derivative of a function with respect to a specific variable while treating the other variables as if they were constants.

What if you get y=2x from Fx=0 equation ANS it works in fy

That is your candidate point (y=2x)

If function looks factorable and you're finding limit

Try to see if you can factor before canceling

Classifying relative extrema

Use Fxx fyy fxy

Second derivative test (when to use and outcomes)

Use after finding critical points of f using fermat's theorem If D>0 and Fxx(a,b) > 0, f(a,b) is a local MINIMUM. If D>0 and Fxx(a,b) < 0, then f(a,b) is a local MAXIMUM. If D<0 then f(a,b) is neither a local max or min (saddle point) If d=0 inconclusive

How to find points where distance is closest to origin on the surface

Use distance formula (x2+y2+z2) as function and surface as constraint

Corner points

Use domain and range to find coordinates of corners . These are all pointsto consider

Steps to lagranges multiplyer with 3 variables

Use gradients to make systems (including constraint) Solve for lambda, find if 0,0 is possibility Find if two variables have a relationship with each other (remember cases can be things like z=x)

Interior points

Use partials to find critical points inside the region

How to know when to use partial vs total derivative

Use total (d and not del) if it is with respect to 1 variable

Crossing two gradients at the same point makes

Vector tangent to curve because the gradients are both perpendicular

When to use partial derivatives of all three variables in finding equation of plane

When equation isn't z= (z is distributed throughout)

When lines are vertical / horizontal how does this affect partial derivatives

When vertical, Fy is 0 When horizontal, Fx is 0

If f(x) is the same thing as y, what is f(x,y)

Z

saddle point

both a peak and a pit

Differential means

dz = fx(dx) + fy(dy) where u usually don't do anything to solve for dx and dy