Approximation using partial deriatives

Question

I am having trouble figuring out how to find the approximate value of :

$\frac{(2.03)^4}{(3.998)^2}$

I know its going to be done with partial derivatives and differentials but I just cant seem to get a grasp on how to do it. Any help would be much appreciated.

Answer 1

An immediate approximation is $$\frac{2^4}{4^2} = 1$$ A better one, given your comments, might be obtained by defining a function

$$f(x,y) = \frac{(2+x)^4}{(4+y)^2}$$ and then expand as Taylor series around $(0,0)$,

$$f(x,y) = f(0,0) + \frac{ \partial f}{\partial x}(0,0) \cdot x + \frac{ \partial f}{\partial y}(0,0) \cdot y + \dots$$

and stop at the order you wish, depending on the quality of the approximation desired. The trivial approximation is given by the leading term.

Answer 2

Take $$z = \frac{x^4}{y^2} \implies \ln z = 4\ln x - 2\ln y \implies \frac{dz}{z} = 4\frac{dx}{x}-2\frac{dy}{y} \implies \color{blue}{dz = z\left(4\frac{dx}{x}-2\frac{dy}{y}\right)}$$

• Then let $x = 2, dx = + 0.03 $ and $y = 4, dy = - 0.002$
• Evaluate $z$ at $x=2, y = 4$ and then find $dz$
• The approximate value will be $\color{blue}{z+dz}$