📘 multivariable calculus

1. **State the problem:** We want to evaluate the improper integral $$\iiint_{\mathbb{R}^3} \frac{e^{-(x^2 + 4y^2 + 9z^2)}}{\sqrt{x^2 + y^2 + z^2}} \, dx \, dy \, dz.$$

1. **Problem statement:** We have four vectors normal to surfaces at point P and four tangent plane equations at P. We need to match each vector and each plane equation to one of f

1. **Problem statement:** Find all first and second order partial derivatives for each function. ---

1. **Problem 1:** Given $$z = x^2 \tan^{-1}\left(\frac{y}{x}\right) - y^2 \tan^{-1}\left(\frac{x}{y}\right),$$ show that $$\frac{\partial^2 z}{\partial x \partial y} = \frac{x^2 -

1. **State the problem:** We need to evaluate the triple integral $$\iiint_V xy^2 z^3 \, dx \, dz \, dy$$ over the region $$V = \{(x,y,z) : z^2 \leq x \leq y, 0 \leq y \leq 2, \sqr

1. Find the domain and range of each function. 2. For graphing functions 1 to 12:

1. **State the problem:** We need to analyze the function $$f(x,y) = \frac{2x^2y}{x^4 + y^2}$$ and understand its properties. 2. **Examine the domain:** The denominator is $$x^4 +

1. Problem: Sketch the level curves of the function $$f(x,y) = \sqrt{9 - x^2 - y^2}$$ for $$k = 0,1,2,3$$. Step 1: Set $$f(x,y) = k$$, so $$\sqrt{9 - x^2 - y^2} = k$$.

1. **Problem statement:** Find the limit $$\lim_{(x,y)\to(0,0)} \frac{xy^2}{x^2 + y^2}$$ and analyze it along the curve $$y = mx$$ where $$m$$ is a constant slope. 2. **Substitute

1. **State the problem:** Find all critical points (minima and maxima) of the function $$f(n,y) = n^4 + y^4 - 4ny + 1$$ by solving where the partial derivatives are zero. 2. **Find

1. **Problem statement:** Analyze the function $$f(a,b) = (2a + 2b)^2 + (2a - 2b)^2$$ over variables $a$ and $b$. 2. **Simplify the function:**

1. Problem 17: Find the maximum and minimum values of the function $f(x,y)=x^3+3xy^2-15x^2-15y^2+72x$. 2. Compute partial derivatives and critical points.

1. Problem 17: Find and classify critical points of $f(x,y)=x^3+3xy^2-15x^2-15y^2+72x$. 2. Compute partial derivatives and set gradient to zero.

**Problem:** Find the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for the function $f(x,y) = e^{xy} \sin(4y^2)$. 1. **Write down the fun

1. **Given:** $f(x,y) = x^2 + xy^3$ a. $f(0,0) = 0^2 + 0\cdot0^3 = 0$

1. Find the function values for \( f(x,y) = x^2 + xy^3 \). 1.a. Calculate \( f(0,0) = 0^2 + 0 \times 0^3 = 0 \).

1. **Problem 1:** Given $z=x^3 + y^3 - 3x^2y$, verify $x^2 \frac{\partial^2 z}{\partial x^2} + 2xy \frac{\partial^2 z}{\partial x \partial y} + y^2 \frac{\partial^2 z}{\partial y^2

1. **State the problem:** We have a function of two variables: $$u(x_1, x_2) = 2 x_2$$.