Sec2Sess24

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Function of 1 variable
f(x) = sin(x)
Function of 2 variables
given (x,y) -> get a number f(x,y)

Example
<math>f(x,y)\ =\ x^2\ +\ y^2</math>

functions can be defined only in limited domains
f(x,y) = sqrt(y) is only defined when y >= 0
f(x,y) = 1 / x+y only if x+y <> 0

Example
f(x,y) = temperature at point (x,y)

or...3 or more parameters!

For simplicity, focus mostly on 2 (or 3) variables

How to visualize function of 2 variables?
-> graph
z = f(x,y)

Example: f(x,y)=-y
graph: z=-y

<math>\mathrm{Example\ 2:\ f(x,y)\ =\ 1-x^2-y^2}</math>

intersections:
-> in yz-plane
<math>x=0,\ z=1-y^2</math>

-> in xz-plane
<math>y=0,\ z=1-x^2</math>

-> in xy-plane
<math>\mathrm{z=0,\ 1-x^2-y^2=0,\ x^2+y^2=1,\ a\ unit\ circle}</math>

Graphing:
The yz-traces are those curves found by setting x = a constant.

Traces where z = constant are generally called "level curves".

variable that are not a part of the graph equation can change and don't change the shape of the graph. They can shift the position of the graph
shape along the axis of the variable by changing the variable's value

shape terminology list:
https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-a-functions-of-two-variables-tangent-approximation-and-optimization/session-24-functions-of-two-variables-graphs/MIT18_02SC_notes_14.pdf