Linearization of f(x,y) = sqrt(29 - 4x^2 - 4y^2) at (2,1)

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Homework Help Overview

The discussion revolves around finding the linearization of the function ƒ(x,y) = sqrt(29 - 4x² - 4y²) at the point (2,1). Participants are exploring the calculations involved in determining the linear approximation of this multivariable function.

Discussion Character

  • Mathematical reasoning, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to calculate the linearization using the defined formula and provides their computed values for ƒ(2,1), ƒx(2,1), and ƒy(2,1). Some participants question the validity of the simplification steps taken, particularly regarding the division by 3. Others suggest evaluating the function at nearby points to verify the linear approximation.

Discussion Status

Participants are actively engaging with the calculations and questioning the steps taken by the original poster. There is a mix of attempts to clarify the mathematical reasoning and to check assumptions about the operations performed. Guidance has been offered regarding the importance of maintaining the integrity of the expressions during simplification.

Contextual Notes

There appears to be some confusion regarding the simplification of the linearization expression, particularly related to the division by 3. The original poster expresses a need for assistance, indicating that they are encountering challenges with the problem.

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Homework Statement



Find the linearization of the function ƒ(x,y) = sqrt(29 - 4x2 - 4y2) at the point (2,1)

Homework Equations


[/B]
Point (a,b)
L(x,y) = Linearization
L(x,y) = ƒ(a,b) + ƒx(a,b)(x-a) + ƒy(a,b)(y-b)

The Attempt at a Solution


[/B]
ƒ(2,1) = 3

ƒx(x,y) = [1/2⋅-8x]/[sqrt(29-4x2-4y2]
ƒx(2,1) = -8/3

ƒy(x,y) = [1/2⋅-8y]/[sqrt(29-4x2-4y2]
ƒy(2,1) = -4/3

L(x,y) = 3 + [-8/3(x-2)] + [-4/3(y-1)]
= 3 + (-8/3x + 16/3) + (-4/3y+4/3)
= 3 -8/3x +16/3 - 4/3y + 4/3
Simplifying gives me
= -8x - 4y + 29

Which is wrong according to my Webwork

please any help would be amazing.
 
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3 -8/3x +16/3 - 4/3y + 4/3 gives (-8x - 4y + 29)/3 You can't just strike the /3 ...
 
i multiplied everything by 3, can i not do that?
 
Nope: f = x + 2 is something quite different from f = 3x + 6 !

Lean back a little, relax, take a breath or a break and it'll be obvious...

:) If you don't believe me you can always evaluate f(2.1,1) and f(2,1.1) on a calculator or a spreadsheet...
 
Thank you BvU appreciate the help, more questions may be on the way, its been a rough day.
 

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