Register to reply

Using the approximation, explain why the second derivative test works.

by jumboopizza
Tags: approximation, derivative, explain, test
Share this thread:
jumboopizza
#1
Jul18-11, 10:14 AM
P: 13
[b]1. The problem statement, all variables and given/known data[/
Using the approximation, explain why the second derivative test works

approximation=f(x0+delta x, y0+delta y)

delta x and delta y are small...


2. Relevant equations

f(x0+delta x,y0+delta y)

3. The attempt at a solution

ok so i know the first derivative of it is:

fx(x0+y0)*delta x+fy(x0+y0)*delta y

and second derivative is:

fxx(x0,y0)*delta x^2+fxy(x0,y0)*delta x*delta y+fyy(x0,y0)*delta y^2

it justs seems like since delta x and delta y are getting smaller,that everything will be going towards 0....so why does it work?
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution
Phys.Org News Partner Science news on Phys.org
What lit up the universe?
Sheepdogs use just two simple rules to round up large herds of sheep
Animals first flex their muscles
SammyS
#2
Jul18-11, 11:05 AM
Emeritus
Sci Advisor
HW Helper
PF Gold
P: 7,808
What is the problem as written?

f appears to be a function of two variables. What is the second derivative test for a function of two variables?
jumboopizza
#3
Jul18-11, 11:11 AM
P: 13
Using the approximation, explain why the second derivative works. Give three exam-
ples for each scenario of the second derivative test.

isnt that what the approximation is? the f(x+delta x,y +delta y)?

its asking about finding local mins,local max and saddle points....now i can show those examples,but how can i explain how it works?its like a proof or something

Ray Vickson
#4
Jul18-11, 11:14 AM
Sci Advisor
HW Helper
Thanks
P: 5,060
Using the approximation, explain why the second derivative test works.

Quote Quote by jumboopizza View Post
[b]1. The problem statement, all variables and given/known data[/
Using the approximation, explain why the second derivative test works

approximation=f(x0+delta x, y0+delta y)

delta x and delta y are small...


2. Relevant equations

f(x0+delta x,y0+delta y)

3. The attempt at a solution

ok so i know the first derivative of it is:

fx(x0+y0)*delta x+fy(x0+y0)*delta y

and second derivative is:

fxx(x0,y0)*delta x^2+fxy(x0,y0)*delta x*delta y+fyy(x0,y0)*delta y^2

it justs seems like since delta x and delta y are getting smaller,that everything will be going towards 0....so why does it work?
1. The problem statement, all variables and given/known data


2. Relevant equations



3. The attempt at a solution
You don't say what you are testing for. If you mean the second-order test for a maximum or minimum, you still need to be more specific: the necessary conditions and the (most common) sufficient conditions are a bit different. You need to tell us which ones you want.

RGV
SammyS
#5
Jul18-11, 11:56 AM
Emeritus
Sci Advisor
HW Helper
PF Gold
P: 7,808
Mathematical language is very exacting. You need to say what you mean & mean what you say.

f(x0+Δx, y0+Δy) is the (exact) value of the function, f, at the point (x0+Δx, y0+Δy).

If the first derivatives of f are zero at the point, (x0, y0), then the following is an approximation to f at the point (x0+Δx, y0+Δy).

f(x0+Δx, y0+Δy) ≈ f(x0, y0) + (1/2)[ fxx(x0, y0)(Δx)2 +2 fxy(x0, y0)(Δx)(Δy) + fyy(x0, y0)(Δy)2 ]


Register to reply

Related Discussions
Can someone please explain to me how this works? General Math 4
Can someone explain to me how a minor degree works? Career Guidance 2
Video inside: could someone explain how this works? General Discussion 3
How can you explain how a balance works vectorially? General Physics 5
Can someone explain how attraction works? High Energy, Nuclear, Particle Physics 13