Taylor approximation with 2 variables, not sure how to solve this

Question

This is the problem I'm trying to solve:

Let $f(x,y)=ay+sin(bx)+c$. Evaluate the Taylor polynomial at $P(0,0)$ and find the values for $a$, $b$ and $c$ if $P(x,y)=-1+2x-y$

I do know the linearization formula, but it requires me to use the value of $f$ at the given point. However, since there are 3 unknown constants, I don't know how to work this out... Thanks for your help.

Answer 1

The linear Taylor-Polynomial of $f$ with respect to $(0,0)$ is given by:

$$f(x,y) \approx P(x,y;(0,0))= f(0,0) + f'_x(0,0)(x-0) + f'_y(0,0)(y-0)$$

Can you take it from here? The spoiler contains the next step - try to do it yourself, first.

$$P(x,y) = [c] + [b\cos(b0)x] + [ay] = c + bx + ay\overset{!}{=}-1+2x-y$$

Answer 2

The second order taylor polynomial of $f(x, y)$ at $(0, 0)$ is $$f_{approx}(x, y) = c + b\cos(bx_0)x+ay+0xy = c+bx+ay.$$

But we are given that $f_{approx}(x, y) = -1+2x-y$, so we set the expressions equal to solve for $a, b,$ and $c$. This yields:

$$-1+2x-y=c+bx+ay\\\implies a = -1, b = 2, c = -1.$$