We need to verify that $F$ is a $C^1$ function (continuously differentiable) in a neighborhood of $(0,0)$. We calculate the gradient $\nabla F(x, y)$: $$ \frac\partial F\partial x = -\sin(y) $$ $$ \frac\partial F\partial y = 1 - x \cos(y) $$ Team R2r Root Certificater2r Updated Validation & Testing
If your exercise asks for a Taylor expansion (Sviluppo di Taylor), here is the general method: Umax Usc 5800 Scanner Driver For Windows 10 [TESTED]
The implicit function $y(x)$ exists locally. Its graph passes through the origin with a horizontal tangent (derivative is 0). Alternative Possibility: Taylor Series Expansion In some editions of the "Lite" version, Exercise 77 may refer to a Taylor expansion problem (e.g., "Write the Taylor series of the second order for a function...").
We want to solve for $y$ as a function of $x$ (i.e., $y = y(x)$). The theorem requires that the partial derivative with respect to the dependent variable ($y$) is non-zero at the point $(x_0, y_0)$.
$$ f(x_0, y_0) + f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0) + \frac12[ \dots ] $$