Consider a function f(x,y) = sqrt(20-x2-7y2). Using linear approximation of f near (2,1), find how large the approximated value of f(1.95, 1.08) is from the exact value a) 0.44% b) 0.28% c) 0.33% d) 0.55% e) None of the above

using linear approximation eh .... gonna have to define a tangent plane maybe?

unless you have some other methods in mind

\[f(x,y,z)=(20-x^2-7y^2)^{1/2}-z\] \[f_x=-x(20-x^2-7y^2)^{-1/2}\] \[f_x=-7y(20-x^2-7y^2)^{-1/2}\] \[f_z=-1\] these partials give us the equations for the normal to the plane defined by using the given x,y point values \[f_x=-2(20-2^2-7)^{-1/2}=-2/3\]\[f_x=-7/3\]\[f_z=-1\] N=<2,7,3> and using the point (2,1,3) as the anchor gives us the tangent plane as: 2(x-2)+7(y-1)+3(z-3)=0

now inputting the (1.95, 1.08) and solving for z gives us a value to compare with, so z=2.84667 by comparison is there a way that we can evaluate

Thanks very much for your assistance but please how did you get the 2,7,3

the gradient of an equation is a set of its partial derivatives. Those partial derivates define equations that relate to a parametric setup for the normal of a surface. By using the point given (2,1) i was able to establish the normal of the tangent plane at the given point

since the normal is a vector, and vectors can be scaled, i scaled the partial results by a factor of -3 -3(-2/3) = 2 -3(-7/3) = 7 -3(-1) = 3

ok

im not sure how to determine the % accuracy without using a calculator to determine the value of f(1.95,1.08) to begin with, and im not sure if you knew of a method i might be forgetting

I don't know of any method but am still working on it. Thanks very much for the assistance

http://www.wolframalpha.com/input/?i=%28sqrt%2820-7%281.08%29%5E2-%281.95%29%5E2%29-2.84667%29%2Fsqrt%2820-7%281.08%29%5E2-%281.95%29%5E2%29 the difference is about 0.44 %

great....why did you divide it??

suppose you have 2 number, p and q, and you want to know how much of a percentage difference there is between them. first get the value of the difference: p-q and divide that by the number you are wanting to find the percentage by p-q is what percentage of p? (p-q)/p

prolly miss worded that one all over the place :)

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Great, the answer showed negative, is it allowed to take it positive???

positive is fine

ok. Thanks very much! am very greatful for this help. God bless you

youre welcome, i hope it made sense :)

Find the maximum and minimum values of the function f(x,y) = xy subject to the constraint x2/2 + y2/2 = 1. Give the maximum and minimum values and the points at which they are attained. please can you help with this one??

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