Equation Root$-$Finding Unsurprisingly, python functions that find approxi$-$

mate roots to intractible equations exit. In fact, there are at least three: fsolve,

brentq, and root. The second $($brentq$)$ can be used only on scalar functions$-$

functions whose independent variable is one$-$dimensional. The last one $($root$)$

is for vector functions $($functions whose independent variable is multiple dimen$-$

sions$).$ The first is the most general, and the one to be introduced here.

Recall, that the roots of a function, $f(x)=0,$ with all the terms of the

equation moved to one side of an equal sign $($where the other side then equals

zero$),$ are the values $x$ that make that function equal zero, and so are also called

the zeros of a function. Therefore, plotting the function for a range of $x$ values

allows you to estimate $x$ values that give zero, which can then then be provided

the python root$-$finding function as a starting point $($guess$)$ to come up with

more precise values for the equation's roots.

The arguments of fsolve are the function name $($with no arguments$)$ and a

starting point:

root $=$ fsolve$($function$,$ guess$)$

It is available in scipy, not pylab, so you have to add an additional import

statement:

from pylab import $*$

from scipy.optimze import fsolve

Use the projectile motion function from the previous assignment. From the

plot of this function, esimate a solution and input it to the program:

guess $=$ float$($input$("$Root estimate: $"))$

Print the value of fsolve$($launchcurve$,$ guess$)($substitute the function

name in your program for launchcurve$).$

Submit all roots of the function under each of the following initial conditions:

The original problem: $R=2\mathrm{.}5m,h=1\mathrm{.}2m,$ and $v_i=4\mathrm{.}8\frac{m}{s}$

$R=1m,$ and all other initial conditions the same

$R=5m,$ and all other parameters the same

$R=5m,h=3\mathrm{.}2m,v_i=6\mathrm{.}8\frac{m}{s}$

Finally, find the root$($s$)$ of the equation $sin(x)=$exp$(-x).$