📘 multivariable calculus

1. **State the problem:** Find the average value of the function $$F(x,y) = \frac{x^2 + 2xy + y^2}{x^2 + y^2}$$ over the region in the first quadrant bounded by the coordinate axes

1. **State the problem:** Find the volume of the solid $Q$ bounded above by the surface $$y = 4 - x^2 - z^2$$ and below by the $xz$-plane (where $y=0$). 2. **Set up the integral:**

1. **State the problem:** Evaluate the triple integral $$\int_{-1}^1 \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{x^2+y^2}^{2 - x^2 - y^2} (x^2 + y^2)^{3/2} \, dz \, dy \, dx.$$\n\n2.

1. The problem asks to determine and graph the domain of the function $$f(x,y) = \ln(x + y)$$. 2. The natural logarithm function $$\ln(z)$$ is defined only for $$z > 0$$.

1. **State the problem:** We have the function $$f(x,y) = (x^2 + 2xy)e^y$$ defined on $$\mathbb{R}^2$$. We need to find the partial derivatives $$\frac{\partial f}{\partial x}$$ an

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