use partial derivatives to approximate the value of $$f(6.1,1.9) $$ where $$f(x,y) = (x^{2}-y^{3}-1)^{\frac{ 1 }{ 3 }}$$ at $$(6,2)$$

Question

use partial derivatives to approximate the value of $$f(6.1,1.9) $$  where $$f(x,y) = (x^{2}-y^{3}-1)^{\frac{ 1 }{ 3 }}$$ at $$(6,2)$$

anonymous · Answer

partial derivative with respect to x or y

anonymous · Answer

or both?

ray10 · Answer

The question doesn't state :S so I think both :)

anonymous · Answer

I'll do with respect to x first

anonymous · Answer

what is the partial derivative of f with respect to x? do you know how to do this?

ray10 · Answer

sure thing :)

ray10 · Answer

yeah I think I know :)

anonymous · Answer

okay so compute it and when you have this you simply plug in x=6 and y=2 the same for the partial with respect to y

anonymous · Answer

remember to use the chain rule

ray10 · Answer

I'm not sure if I got the correct f'??

anonymous · Answer

okay so df/dx=1/2 (x^2-y^3-1) ^(1/2- 1) * (2x)

anonymous · Answer

simplifying yields
df/dx= x*(x^2-y^3-1)^(-1/2)

anonymous · Answer

now you plug in your values

ray10 · Answer

ohh right! Now I get it, let me just go through that again quickly :)

ray10 · Answer

so I plug in my values into the df/dx ?

anonymous · Answer

yep

anonymous · Answer

so you have 6 (36 -8 -1)^(-1/2)

anonymous · Answer

whoops that doesn't look right

ray10 · Answer

hmm I seem to get $$\frac{ 2\sqrt{3} }{ 3 }$$ what did you get?

anonymous · Answer

so when you approximate a value for your f(x,y), using partial derivatives should be comparable to your value f(6.1,1.9) see what you get with your original function when x=6.1 and y=1.9

ray10 · Answer

do you have to use $$fxx \times fyy - (fxy)^{2}$$ in the steps to working it out? ?

anonymous · Answer

so this usually works with functions of one variable, i haven't approximated functional evaluations using partial derivatives in a while and was hoping it would be this simple....hmmm maybe you do need another step b/c logically it doesn't make sense

anonymous · Answer

sorry this is all i could offer :-( i tried

ray10 · Answer

ohh actually I can see a few steps I need to do, I'll try them and get back to you :) thank you for your insight though :)

anonymous · Answer

You probably just need to find the equation of the tangent plane at some point close to (x,y) = (6.1,1.9) and then use it to approximate f(6.1,1.9)

ray10 · Answer

how do I go about that? :)

ray10 · Answer

Still confused :S