📘 multivariable calculus

1. We are asked to compute the normal vector and the tangent plane at a given point $P$ on a surface. 2. The normal vector to a surface defined by a function $f(x,y,z) = 0$ at a po

1. **State the problem:** Find the part of the plane $z = x + 3$ that lies inside the cylinder defined by $x^2 + y^2 = 1$, and verify the point $P = (1, 0, 4)$ lies on this surface

1. **State the problem:** Find the gradient and the equation of the tangent plane to the surface given by $$z = 3x^2 - xy$$ at the point $$(1,2,1)$$. 2. **Recall the formula for th

1. **Problem statement:** We need to find the Jacobian determinants for two cases:

1. **State the problem:** We have the function $$f(x,y,z) = x^2 y + y^2 z + z^2 x$$. We need to find all critical points, classify them using the Hessian matrix, and compute the di

1. **Problem 1:** Examine the function $$z = 8x^3 + 2y - 3x^2 + y^2 + 1$$ for maximum, minimum, or saddle points. 2. **Step 1: Find the first partial derivatives.**

1. **State the problem:** We are given the function $$f(x,y,z) = x^2 y - 10 y^2 z^3 + 43 x - 7 \tan(5 y)$$ and need to find the partial derivatives $$\frac{\partial f}{\partial x}$

1. **State the problem:** We are given the differential form $$df = \frac{1}{x^2yz} \, dx + \frac{1}{xy^2z} \, dy + \frac{1}{xyz^2} \, dz$$ and we want to find the function $f(x,y,

1. **State the problem:** We are given the function $$f(x,y,z) = x^2 y - 10 y^2 z^3 + 43 x - 7 \tan(5 y)$$ and need to find the partial derivatives $$\frac{\partial f}{\partial x}$

1. **State the problem:** We are given the function $$f(x,y,z) = x^2 y - 10 y^2 z^3 + 43 x - 7 \tan(5 y)$$ and need to find the partial derivatives $$\frac{\partial f}{\partial x}$

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: