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
