Please solve in python!!!

Recall that if $n0$ is an integer, then we write $n!$ to denote the product of all positive integers less than or equal to $n.$ Specifically,

$01$ and $nn*(n-1)cdots3*2*1$ for $n1.$

$($a$)$ Write a Python function that accepts a nonnegative integer $n$ as input and returns its factorial as output. Print your function's output for $n=0,1,2,3,4,5$ and use plt$.$plot$()$ to graph these values as isolated points using the option $\text{'}$bo$\text{'}.$

$($b$)$ Use Python to compute the antiderivative $G_n(t)$ of $g_n(t)=t^n{e}^{-t}$ for $n=0,1,2,3,4,5$ with constants of integration $C=0.$ What integration technique$($s$)$ could you have used to find these results by hand?

$($c$)$ For each of the antiderivatives $G_n(t)$ you found in part $($b$),$ evaluate $G_n(0).$ Use this information to propose a formula for $G_n(0)$ when $n0$ is arbitrary.

Recall that if $n0$ is an integer, then we write $n!$ to denote the product of all positive integers less than or equal to $n.$ Specifically,

$01$ and $nn*(n-1)cdots3*2*1$ for $n1.$

$($a$)$ Write a Python function that accepts a nonnegative integer n as input and returns its factorial as output. Print your function's output for $n=0,1,2,3,4,5$ and use plt$.$plot$()$ to graph these values as isolated points using the option $\text{'}$bo$\text{'}.$

$($b$)$ Use Python to compute the antiderivative $G_n(t)$ of $g_n(t)=t^n{e}^{-t}$ for $n=0,1,2,3,4,5$ with constants of integration $C=0.$ What integration technique$($s$)$ could you have used to find these results by hand?

$($c$)$ For each of the antiderivatives $G_n(t)$ you found in part $($b$),$ evaluate $G_n(0).$ Use this information to propose a formula for $G_n(0)$ when $n0$ is arbitrary.

$2$

$($d$)$ Use Python to compute $\underset{t}{lim}{G}_n(t)$ for $n=0,1,2,3,4,5$ and propose a formula for $\underset{t}{lim}{G}_n(t)$ when $n0$ is an arbitrary integer. What technique could you use to find this limit by hand? Print a statement explaining why this method guarantees that your proposed formula holds.

$($e$)$ Use your results from parts $($a$)$ and $($b$)$ and the fundamental theorem of definite improper integrals to propose a formula for $n!.$

$($f$)$ The equation that you derived in part $($e$)$ is only valid for integers $n0.$ However, there is nothing technically preventing us from plugging non$-$integer values into the formula.

Write a Python function that accepts a nonnegative number x as input and returns the result of plugging $x$ into your formula from part $($e$).$ Use plt$.$plot$()$ and np$.$ linspace$()$ to graph your function on the $x-$range $0,5.$ In the same window, also plot the points that you graphed in part $($a$).$

d$)$ Use Python to compute $\underset{t}{lim}{G}_n(t)$ for $n=0,1,2,3,4,5$ and propose a formula for $\underset{t}{lim}{G}_n(t)$ when $n0$ is an arbitrary integer. What technique could you use to find this limit by hand? Print a statement explaining why this method guarantees that your proposed formula holds.

$($e$)$ Use your results from parts $($a$)$ and $($b$)$ and the fundamental theorem of definite improper integrals to propose a formula for $n!.$

$($f$)$ The equation that you derived in part $($e$)$ is only valid for integers $n0.$ However, there is nothing technically preventing us from plugging non$-$integer values into the formula.

Write a Python function that accepts a nonnegative number $x$ as input and returns the result of plugging $x$ into your formula from part $($e$).$ Use plt$.$ plot$()$ and np $.$ linspace$()$ to graph your function on the $x-$range $0,5.$ In the same window, also plot the points that you graphed in part $($a$).$