📘 multivariable calculus

1. **Problem Statement:** Find the first partial derivatives of the functions: 8. $f(u,v) = e^{uv}$

1. The problem is to identify and solve the surface defined by the equation $$x^2 - y^2 + z^2 = 1$$. 2. This is a quadratic surface equation involving three variables $x$, $y$, and

1. **State the problem:** Find the value of $z$ at the point $(2,-1)$ for the function $z = 3y^2 - 2x^2 + x$. 2. **Recall the formula:** The function is given by

1. **State the problem:** Given the function $f(x,y) = 6x^2 \sin(x + y^2)$, find the mixed partial derivative $f_{xy}$ and evaluate it at $x=1$, $y=7$. 2. **Recall the formula and

1. **State the problem:** We want to find the volume of the solid $D$ bounded above by the plane $z=3x+6y+24$, below by the plane $z=-1$, and laterally by the triangular region $T$

1. Zadání problému: Vypočítáme Taylorův polynom 1. řádu funkce $$f(x,y) = \frac{e^{3x}}{x^2 + xy + y^2}$$ v bodě $$A = (0,1)$$. 2. Vzorec pro Taylorův polynom 1. řádu funkce dvou p

1. **Problem statement:** Find the stationary points and their nature for the functions: (a) $$z = 2x^2y^2 + 4xy^2 - 4y^3 + 16y + 5$$

1. The problem asks to evaluate the volume element in spherical coordinates at $r=2$, $\theta=30^\circ$, with $dr=d\theta=d\phi=1$. 2. The volume element in spherical coordinates i

1. **Problem statement:** Find and classify the critical points of the function $$f(x,y) = x^3 - 8y^3 - 12xy + 5.$$\n\n2. **Step 1: Find the partial derivatives.**\nWe calculate th

1. **Problem Statement:** Determine the signs of the partial derivatives $f_x(2,-2)$, $f_{xx}(2,-2)$, $f_y(2,-2)$, and $f_{yy}(2,-2)$ for the given saddle-shaped surface symmetric

1. The problem asks about the meaning of the Jacobian \(|J(u,v)|\) in the transformation of a double integral from Cartesian coordinates \((x,y)\) to new coordinates \((u,v)\). 2.

1. **State the problem:** We need to compute the surface integral of the function $$G(x,y,z) = y \sqrt{\frac{2}{x^2} + 4}$$ over the surface defined by the parabolic cylinder $$x^2

1. **State the problem:** Find all critical points of the function $$f(x,y) = xy + \frac{6}{x} + \frac{7}{y}.$$ 2. **Recall the definition of critical points:** Critical points occ

1. The problem asks to evaluate the expression $\left(\frac{\partial f}{\partial v} + \frac{\partial f}{\partial x} = \frac{\partial f}{\partial w} + 7 \frac{\partial f}{\partial v

1. **State the problem:** Find the vector equation of the tangent line to the curve formed by the intersection of the cylinders defined by the equations $$x^2 + y^2 = 25$$ and $$y^

1. **State the problem:** Find the minimum and maximum values of the function $$f(x,y) = x^2 - xy + y^2$$ on the quarter circle defined by $$x^2 + y^2 = 1$$ with $$x, y \geq 0$$. 2

1. **Problem Statement:** Investigate the existence of limits for various functions as $(x,y) \to (0,0)$ using path tests and polar coordinates, analyze continuity, and interpret g

1. We are given four functions and asked to find and sketch their domain, range, level curves, and region properties. 2. Let's analyze each function one by one.

1. The problem asks to find and sketch the domain of each function given. 2. The domain of a function $f(x,y)$ is the set of all points $(x,y)$ for which the function is defined.

1. **Problem Statement:** Analyze the functions for domain, range, level curves, boundary, and domain properties. ---

1. The problem is to understand and describe the surface defined by the equation $x=0$ in $\mathbb{R}^3$. 2. The equation $x=0$ represents all points in three-dimensional space whe