📘 multivariable calculus

1. **State the problem:** Find the maximum and minimum values of the function $$f(x,y) = x^2 + y$$ subject to the constraint $$g(x,y) = x^2 + y^3 = 1$$ using Lagrange multipliers.

1. The problem asks for the geometric interpretation of the partial derivative $f_x(a,b)$ of a function $f(x,y)$ at the point $(a,b)$. 2. Recall that the partial derivative $f_x(a,

1. We are asked to find the point on the surface $z = xy + 1$ that is nearest to the origin $(0,0,0)$. 2. The distance $D$ from any point $(x,y,z)$ to the origin is given by the fo

1. **Problem Statement:** Correct and solve the first problem: Find the domain, global maximum, and global minimum of the function $$f(x,y) = \sqrt{64 - x^2 - y^2}$$.

1. **Problem statement:** Calculate the second partial derivatives $\frac{\partial^2 f}{\partial x^2}$, $\frac{\partial^2 f}{\partial \theta^2}$, and $\frac{\partial^2 f}{\partial

1. Diberikan fungsi dua peubah dengan persamaan $$z = f(x,y) := \sqrt{x^2 + y^2}$$. Kita diminta untuk menggambar grafik fungsi ini. 2. Fungsi ini merupakan fungsi jarak dari titik

1. **Problem Statement:** Given the function $$u = \frac{1}{\sqrt{x^2 + y^2}}$$, find the second partial derivatives $$\frac{\partial^2 u}{\partial x^2}$$ and $$\frac{\partial^2 u}

1. **Problem statement:** Find the volume of the solid in the first octant bounded by the coordinate planes ($x=0$, $y=0$, $z=0$), the cylinder $x^2 + y^2 = 4$, and the plane $z +

1. The problem asks to find the partial derivatives $f_u$, $f_v$, and $f_w$ of the function $$f(u,v,w) = e^{uv} \ln w.$$ 2. Recall the rules for partial derivatives: when different

1. **Problem Statement:** Given a function $u = f(x^2 + 2yz, y^2 + 2xz)$, find the value of $$z \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial z} - x \frac{\partial u

1. **State the problem:** Find all local maxima, minima, and saddle points of the function $$f(x,y) = \sqrt{56x^2 - 8y^2 - 16x - 31} + 1 - 8x$$. 2. **Domain consideration:** The ex

1. The first problem is to find the partial derivatives $\frac{\partial f}{\partial r}$ and $\frac{\partial f}{\partial \theta}$ for the function $f(x,y) = \frac{y}{x^2 + y^2}$ whe

1. **Problem statement:** Find the points where the function $f(x,y) = x + y$ attains its maximum and minimum values subject to the constraint $x^2 + y^2 = 1$. 2. **Method:** Use t

1. **Stating the problem:** We want to find the result of the second iteration using Newton's method for the function

1. **Problem Statement:** Calculate the double integral $$\iint_R f(x,y) \, dA$$ where $$R = \{(x,y): 0 \leq x \leq 4, 0 \leq y \leq 2\}$$ and the function $$f(x,y)$$ is defined as

1. Given $u = f(x - y, y - z, z - x)$, prove that $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0.$$ 2. Use chain ru

1. The problem is to find the third mixed partial derivative $f_{rst}$ of the function $f(r,s,t) = e^r \sin(st)$. 2. First, recall the function: $$f(r,s,t) = e^r \sin(st)$$

1. **Problem:** Compute the Jacobian \( \frac{\partial(u,v)}{\partial(x,y)} \) where \( u = x + y \) and \( v = xy + 1 \). 2. **Formula:** The Jacobian determinant for functions \(

1. The problem asks to find a unit vector in the direction where the function $f(x,y) = 6x^6 y^7$ increases most rapidly at the point $P(-1,1)$, and to find the rate of change of $

1. **State the problem:** We need to find the directional derivative of the temperature function $$T = x^3 y + y^3 z + z^3 x$$ at the point $P = (2, -1, 0)$ in the direction toward

1. The problem is to understand and compute surface and volume integrals. 2. Surface integrals calculate the integral of a function over a surface $S$. The formula is $$\iint_S f(x