Unit 3 test: Partial Derivatives

What is the domain of $f(x, y) = \ln(x + y - 1)$?

The level curve $f(x, y) = k$ of $f(x, y) = x^2 - y$ passes through $(3, 4)$. Find $k$.

Evaluate $\displaystyle\lim_{(x, y) \to (0, 0)} \frac{x^2 + y^2}{\sqrt{x^2 + y^2 + 4} - 2}$.

What is $\displaystyle\lim_{(x, y) \to (0, 0)} \frac{xy}{x^2 + 2y^2}$?

Let $f(x, y) = x^2e^{3y} - y\sin x$. Find $f_x$.

Let $f(x, y) = x^3y^4 + y\ln x$. Find $f_{yx}$, the result of differentiating first with respect to $y$ and then with respect to $x$.

Find the tangent plane to $z = x^2 + xy$ at the point $(1, 2, 3)$. Enter it in the form $z = \ldots$.

Use the linearization of $f(x, y) = x^2y$ at $(2, 3)$ to estimate $f(1.98, 3.01)$.

Let $z = x^2 + y^2$ with $x = 2t$ and $y = t^2 + 1$. Find $\dfrac{dz}{dt}$ at $t = 1$.

The point $(1, 1)$ lies on the curve $x^2 + 3xy + y^3 = 5$. Find $\dfrac{dy}{dx}$ at that point.

Find $\nabla f(1, 2, -1)$ for $f(x, y, z) = xy^2 + xz^2$.

Find the directional derivative of $f(x, y) = x^2 - xy + 3y$ at $(1, 1)$ in the direction of $\mathbf{v} = \langle 1, 1\rangle$.

Find all critical points of $f(x, y) = x^3 - 3x + y^2 + 2y$.

Classify the critical points of $f(x, y) = x^3 - 3x + y^2 + 2y$.

Use Lagrange multipliers to find the maximum value of $f(x, y) = 3x + 4y$ on the circle $x^2 + y^2 = 25$.