Approximating a function of several variables for small absolute values of the inputs

Question

Context: I am working through the book Differential and Integral Calculus by N. Piskunov. The exercise in the image is from Chapter VIII which has the following subsections:

What I have tried:

• Using the first order differential approximation: $f(x, y) \approx f(x_0, y_0) + f_{x}(x_0, y_0)\Delta x + f_{y}(x_0, y_0)\Delta y$, which I realized leads to more square roots.
• Intuitively, I thought it's something related to: $\sqrt{1+x} \approx 1 + \frac{1}{2}x$ and $\frac{1}{\sqrt{1+x}} \approx 1 - \frac{1}{2}x$ (for small values of $x$), but I don't know how arrive at the given answer formally.

A hint or a full answer would be greatly appreciated.

Answer 1

Your last approximations $(1+x)^\alpha\approx 1+\alpha x$ point in the correct direction. After that use that for small numbers $x,y$ also $(1+x)(1+y)\approx 1+x+y$.

If you want to make it a real challenge then try to collect the next-order terms, that is, everything quadratic in $x,y,z$, to get an idea of how large the truncation error is.

Answer 2

This is a problem of multivariable Taylor series.

An easy way to do it for $$f(x,y,z) \qquad \text{around}\qquad (a,b,c)$$ is to introduce a dummy variable $t$ and only expand it around $t=0$. This would give $$f( (x-a)t+ (y-b)t+ (z-c)t)=$$ $$f(0)+\Big((x-a)+(y-b)+(z-c)\Big) t\, f'(0)+$$ $$\frac 12 \Big((x-a)+(y-b)+(z-c)\Big)^2t^2 f''(0)+\cdots$$ and, later, make $t=1$.