📘 multivariable calculus

1. **Sketch the vector-valued function** $\mathbf{r}(t) = \langle 1 + 2t, -1 + 3t \rangle$. - This is a vector function in 2D where the $x$-component is $1 + 2t$ and the $y$-compon

1. **Problem 1:** Given the function $$f(x,y,z) = x^2 y - 10 y^2 z^3 + 43 x - 7 \tan(5 y),$$ find the partial derivatives $$\frac{\partial f}{\partial x}, \frac{\partial f}{\partia

1. **Problem (i):** Find a unit normal vector to the surface defined by $f(x,y) = x^3$ at the point $(2,-1,8)$. 2. Compute the partial derivatives:

1. **State the problem:** We are given the values of the partial derivatives of a function $f$ at the point $P(4,2,5)$ and the function value at that point. We want to write the li

1. **Problem statement:** Find the linear approximation (linearization) of the functions at the given points. ---

1. **State the problem:** We want to show that the function $f(x,y) = x \cdot e^{xy}$ is differentiable at the point $(2,0)$. 2. **Evaluate the function at the point:**

1. **State the problem:** We want to find the points $(x,y)$ where the function $$f(x,y) = \frac{x^2 y + y^3}{x^2 + y^2 + 1}$$ is continuous. 2. **Analyze the function:** The funct

1. **State the problem:** We want to show that the limit $$\lim_{(x,y) \to (0,0)} \frac{xy}{x^4 + y^2}$$ does not exist by approaching along different paths, including the path $$y

1. **Problem (a):** Evaluate $$\lim_{(x,y) \to (1,1)} \frac{x^2 y - xy^2}{x - y}$$ Step 1: Factor the numerator:

1. **Problem statement:** Find the domain of the functions: (a) $f(x,y) = \sqrt{25 - x^2 - 4y^2}$

1. Problem: Find all first and second-order partial derivatives of $f(x,y) = x^2 y^3 + 4xy^2$. 2. First-order partial derivatives:

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