Question $4)f(x,y,z)=x^3+y{z}^2,x=u^2+v,y=u+v^2,z=3uv$ with $(w,v)=(-2,2)$

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{u}}=3x^2*2u+z^2*1+2yz*3v$

$3(u^2+v{)}^2*2u+(3uv{)}^2+2(u+v^2)(3uv)*3(v)$

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{v}}=3x^2*1+z^2*2v+2yz*3u$

$3(u^2+v{)}^2+(3uv{)}^2*2v+2(u+v^2)(3uv)*3(u)$

$$ Is a graph a tree?

A tree is a graph $(V,E)$ with two special properties:

Every vertex has at most one incoming edge.

Either there are no vertices, or there is a vertex with no incoming edges, called the root, from which all other vertices are reachable.

If the second property does not hold, incidentally, the graph is called a forest.

Write an is$\_$tree property that has value True if the graph is a tree, and has value False otherwise.$[29]$ # Tests $10$ points: is$\_$tree

$$ Is a graph a tree?

A tree is a graph $(V,E)$ with two special properties:

Every vertex has at most one incoming edge.

Either there are no vertices, or there is a vertex with no incoming edges, called the root, from which all other vertices are reachable.

If the second property does not hold, incidentally, the graph is called a forest.

Write an is$\_$tree property that has value True if the graph is a tree, and has value False otherwise.$[29]$ # Tests $10$ points: is$\_$tree

$[1]$ class Graph$($object$)$:$[2]$ g$=$ Graph$($vertices$=\{\text{'}$a$\text{'},\text{'}$b$\text{'},\text{'}$c$\text{'},\text{'}$d$\text{'},\text{'}$e$\text{'},\text{'}$f$\text{'},\text{'}$g$\text{'}\},$ edges$=\{(\text{'}$a$\text{'},\text{'}$b$\text{'}),(\text{'}$a$\text{'},\text{'}$c$\text{'}),(\text{'}$a$\text{'},\text{'}$d$\text{'}),(\text{'}$b$\text{'},\text{'}$d$\text{'}),$

$(\text{'}c\text{'},\text{'}a\text{'}),(\text{'}c\text{'},\text{'}e\text{'}),(\text{'}d\text{'},\text{'}b\text{'}),(\text{'}d\text{'},\text{'}c\text{'}),$

$(\text{'}f\text{'},\text{'}g\text{'}),(\text{'}g\text{'},\text{'}f\text{'})$

Great, but, how do we display graphs? And what can we do with them?

Let's first of all add a method $.$show$()$ that will enable us to look at a graph; this uses the library networkx.

os

$[3]$ import networkx as nx # Library for displaying graphs.

class Graph$($object$)$:

$[1]$ class Graph$($object$)$:$[2]$ g$=$ Graph$($vertices$=\{\text{'}$a$\text{'},\text{'}$b$\text{'},\text{'}$c$\text{'},\text{'}$d$\text{'},\text{'}$e$\text{'},\text{'}$f$\text{'},\text{'}$g$\text{'}\},$ edges$=\{(\text{'}$a$\text{'},\text{'}$b$\text{'}),(\text{'}$a$\text{'},\text{'}$c$\text{'}),(\text{'}$a$\text{'},\text{'}$d$\text{'}),(\text{'}$b$\text{'},\text{'}$d$\text{'}),$

$(\text{'}c\text{'},\text{'}a\text{'}),(\text{'}c\text{'},\text{'}e\text{'}),(\text{'}d\text{'},\text{'}b\text{'}),(\text{'}d\text{'},\text{'}c\text{'}),$

$(\text{'}f\text{'},\text{'}g\text{'}),(\text{'}g\text{'},\text{'}f\text{'})$

Great, but, how do we display graphs? And what can we do with them?

Let's first of all add a method $.$show$()$ that will enable us to look at a graph; this uses the library networkx.

os

$[3]$ import networkx as nx # Library for displaying graphs.

class Graph$($object$)$:

Exercise: Complete the following implementation, in which an averagerator is used in order to replace values that are more than num$\_$stds away from the average, with the average itself.

The CleanData class is initialized by passing a discount factor for its averagerator.

Every piece $x$ of data is then filtered via a call to filter$(x,$ num$\_$stdevs$)$ ; this call returns:

$x$ if the value of $x$ is closer than num$\_$stdevs standard deviations from the running average,

the running average if the value of $x$ differs from the running average by more than num$\_$stdevs standard deviations.

Exercise: Implement the CleanData class

#@title Exercise: Implement the class

class CleanData$($object$)$: