Using Taylor’s formula to find the quadratic and cubic approximations of the function 𝑓 (𝑥, 𝑦) = 𝑒 𝑥 cos 𝑦 at the origin.

Answer 1

Is that e^xcosy?

You can write out the Taylor (Maclaurin) series for both and multiply the terms that are low enough order:

e^x = 1 + x + x²/2! +x³/3!

cos(y) = 1 - y²/2! + y⁴/4!

There is no cubic term in cos(y) Taylor series: It's an even function and every odd derivative, +/- sin(0) = 0