Question

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

Answer 1

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

Answer 2

unless you have some other methods in mind

Answer 3

\[f(x,y,z)=(20-x^2-7y^2)^{1/2}-z\]
\[f_x=-x(20-x^2-7y^2)^{-1/2}\]
\[f_x=-7y(20-x^2-7y^2)^{-1/2}\]
\[f_z=-1\]
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

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

ok

Answer 9

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

Answer 10

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

Answer 11

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 %

Answer 12

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

Answer 13

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

Answer 14

prolly miss worded that one all over the place :)

Answer 15

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

Answer 16

positive is fine

Answer 17

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

Answer 18

youre welcome, i hope it made sense :)

Answer 19

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??