📘 multivariable calculus

1. **Problem statement:** We have the function $$F(x_1, x_2) = 8x_1^{0.76}x_2^{0.12}$$ and the point $$a = (8, 3)$$. We want to find the exact change in $$x_2$$ when $$x_1$$ decrea

1. **Stating the problem:** We need to find the conditional extrema (maximum or minimum) of the function $$f(x,y) = 2x^2 - xy$$ subject to the constraint $$u(x,y) = 2x + 2y - 2 = 0

1. **Problem statement:** Calculate the double integral $$\iint \frac{xy}{\sqrt{x^2 + y^2}} dA$$ over the region defined by $$x^2 + y^2 = 1$$ (the unit disk). 2. **Formula and appr

1. **Problem Statement:** (a) Convert points A(2,2,1) and B(5,5,4) from Cartesian $(x,y,z)$ to cylindrical coordinates $(r,\theta,z)$ where $r=\sqrt{x^2+y^2}$ and $\theta=\tan^{-1}

1. **Problem Statement:** Convert points A(2, 2, 1) and B(5, 5, 4) from Cartesian to cylindrical coordinates and analyze their components. 2. **Formula for Cylindrical Coordinates:

1. The problem is to identify the geometric objects represented by the given equations and describe their orientation relative to the coordinate axes. 2. (a) Equation: $z = -1$

1. **Problem 44:** Find the solid cylinder that lies on or below the plane $z=8$ and on or above the disk in the $xy$-plane with center the origin and radius 2. 2. **Step 1:** Stat

1. **State the problem:** Find and classify the extrema of the function $F(x,y) = 2x^2 + 10xy + 3y^2$ on the boundary of the circle $x^2 + y^2 = 5$. 2. **Method:** Use Lagrange mul

1. The problem is to analyze the parametric vector function $$\mathbf{r}(t) = (-\sin(t), -\cos(t), 2t)$$ for $$-2 \leq t \leq 2$$. 2. This function describes a curve in 3D space wh

1. Problem statement: Change the order of integration of the given double integral. 2. The integral to be converted is

1. Let's start by understanding double integrals in polar coordinates. 2. The problem: Convert a double integral from Cartesian coordinates $(x,y)$ to polar coordinates $(r,\theta)

1. The problem is to understand double integrals in polar coordinates and triple integrals in cylindrical coordinates. 2. Double integrals in polar coordinates are used to integrat

1. **State the problem:** We are given the surface equation $$z = -2y^2 - x^2 + 54$$ and boundary planes $$x = 3$$ and $$y = 3$$. We want to understand the shape and behavior of th

1. **State the problem:** We are given the surface defined by the equation $$z = -2y^2 - x^2 + 54$$ and two planes $$x = 3$$ and $$y = 3$$.

1. **State the problem:** We are given the function $$f(x,y) = x^4 + y^4 + 4xy$$ and we want to analyze or work with it. 2. **Understand the function:** This is a function of two v

1. Masalah: Tentukan luas permukaan dari permukaan $z = x^2 + 2y$ di atas segitiga $T$ dengan titik sudut $(0,0)$, $(1,0)$, dan $(1,1)$.\n\n2. Rumus luas permukaan permukaan $z = f

1. Statement: Problem 8: Evaluate the double integral $$\iint_R (4-3x^2-y^2)\,dx\,dy$$ where $R$ is bounded by $x=0$, $y=0$, and $x+y-2=0$. 2. Formula and plan: Use iterated integr

1. **State the problem:** We need to evaluate the line integral of the scalar function $f(x,y,z) = x - 3y^2 + z$ over the curve $C = C_1 \cup C_2$, where: - $C_1$ is the line segme

1. The problem is to graphically illustrate the equation $z=1.46$. 2. This equation represents a constant value for $z$, meaning it is a horizontal plane in three-dimensional space

1. **State the problem:** Find the area of the surface $S$ defined by $f(x,y) = 1 + x - 2y$ over the region $R$, which is a square with vertices $(0,0)$, $(3,0)$, $(0,3)$, and $(3,

1. **Problem Statement:** Identify the local maxima and minima points from the contour plot of the function $f(x,y)$, list their coordinates and function values, and determine if a