Differentiability and Linear Approximation

The derivative of a multivariable function is a linear map

Linear approximation in practice

Partials existing is weaker than differentiability

Using the approximation with error awareness

For $f(x, y) = x^2 y$ at the point $(1, 2)$, what is the tangent plane to the graph of $f$?

Using the linear approximation of $f(x, y) = x^2 y$ at $(1, 2)$, what is the estimate of $f(1.1, 2.05)$?

A function has both partial derivatives at the origin, both equal to $0$. What can be concluded?

For $f(x, y) = x^2 y$, the linear approximation at $(1, 2)$ estimates $f(1.1, 2.05)$ as $2.45$. Give the true value minus the estimate, as a decimal like 0.0305.

For $f(x, y, z) = x^2 z + yz^2$ at the base point $(1, 1, 1)$, the linear approximation is $f(1,1,1) + f_x(1,1,1)(x-1) + f_y(1,1,1)(y-1) + f_z(1,1,1)(z-1)$. Give the coefficient of the term $(x - 1)$, as a plain number.