Approximate using differentials when partial derivatives are given?

Question

I have ran into this problem on my online math assignment, this week we are covering partial derivatives and higher order partial derivatives, but I don't think I have learnt anything that can help me solve this problem.

So I am given $f_x(9,1)=1, f_y(9,1)=4, f(9,1)=5$

from that I'm supposed to figure out: $f(9,2), f(10,1),f(10,2)$

my question is how do I approach this and figure out an approximation given the partial derivatives? I kind of remember doing some approximations with reiman sums and stuff but I don't think that would apply here.

Thanks in advance!

Answer 1

Use approximation formula $$\Delta f \approx f_x \Delta x + f_y \Delta y$$

then $$f(x,y)\approx f(9,1) + \Delta f$$

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For example, to find $f(9,2)$ $$\begin{align}\Delta f &\approx f_x \Delta x + f_y \Delta y\\&\approx 1(0) + 4(1)\\&\approx 4 \end{align}$$

therefore $$\begin{align}f(9,2) &\approx f(9,1) + \Delta f\\&\approx 5+4\\&\approx 9\end{align}$$