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- Calculus 3
- Partial Derivatives

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- Intro Lesson: d2:56
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Functions with two variables are written in 2 forms:

$z = f(x,y)$

$F(x,y,z) = 0$

The domain and range of a function $f(x,y)$ is as follows:

$D = \{ (x,y) \in \mathbb{R}^2 : f(x,y) \}$

$R = \{ z \in \mathbb{R} : f(x,y) = z \}$

A trace of a surface is a set of intersecting points which is created by a surface intersecting one of the following planes: $xy$-plane, $xz$-plane, $yz$-plane.

Trace of the $xy$-plane happens at $z=0$.Trace of the $xz$-plane happens at $y=0$.

Trace of the $yz$-plane happens at $x=0$.

A level curve is a curve that is on the $xy$-plane with a specific value $z=z_0$.

- Introduction
**Functions of Several Variables Overview:**a)__2 Variable Functions__- How to represent them
- What they look like Visually

b)__Domain & Range__- Domain $\to$ looks at $x$ and $y$
- Range $\to$ looks at $z$
- Examples of finding Domain & Range

c)__Traces__- Traces $\to$ intersection of surface and planes
- What it looks like visually
- Example of finding the Trace

d)__Level Curves__- Curves on the $xy$-plane with a specific $z$ value
- Examples of finding level curves

- 1.
**Finding Domain & Range**

Find the domain and range of:$f(x,y) = \frac{1}{\sqrt{x^2 + y^2 - 4}}$

- 2.Find the domain and range of:
$f(x,y) = \sqrt{x} \sqrt{x^2+y^2} e^{1-y^2}$

- 3.Find the domain and range of:
$f(x,y) = ln(x^2 - 1)$

- 4.
**Finding the Traces**

Find and graph the trace of the $xz$ plane of$f(x,y) = \sqrt{x}\sqrt{x+y^2}$

- 5.Find and graph the trace of the $yz$ plane of
$5x + 2y + z = 3$

- 6.Find the level curve of $f(x,y) = \sqrt{x^2 + y^2}$ at $z_0 = 1$ and $z_1 = 2$
- 7.Find and graph the level curve of $2z + 4y^2 - x = 0$ at $z_0 = 0$.