Problem $2(1$ point$)$

Run Euler's Method with an appropriate number of steps to obtain an approximation for $\backslash (\backslash$boldsymbol$\{$R$\}(\backslash$boldsymbol$\{$A$\})\backslash ),$ where your value of $\backslash (\backslash$boldsymbol$\{$A$\}\backslash )$ is given in the statement of Problem $2$ in Childsmath. Enter your answer correct to three decimal places. $($Recall that "correct to three decimal places" does not mean to round off the third decimal place but instead to quote the first three correct decimals; for example, if Python returns $0\mathrm{.}238776125,$ then your answer should be $0\mathrm{.}238$ and not $0\mathrm{.}239).$

How do you know what's an appropriate number of steps? Run the code with an increasing number of steps $($each time adjusting the step size$)$ until the approximation stabilizes, i$.$e$.,$ the first three decimal places do not change. Problem $\backslash$#$2$: For this question you will use $\backslash ($ A$=4\backslash ).$

Enter the value of $\backslash ($ R$($A$)\backslash ),$ correct to three decimals, into the answer box below.

Problem $\backslash$#$2$:

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Submit Problem $\backslash$#$2$ for Grading $```$

## code for problem $2$

# define variables; manually input initial values

t $=[$e$]$

R $=[0]$

# define parameters

n $=40$ ## number of steps $($adjust$)$

deltat $=0\mathrm{.}1$ #step size $($adjust$)$

tau $=1\mathrm{.}21$

k $=2\mathrm{.}08$

# copy, paste, and modify code from above to answer problem $2$

for i in range$(0,$n$)$:

tnew $=$ t$[$i$]+$ deltat

Rnew $=$ R$[$i$]+($t$[$i$]+$ R$[$i$])*$deltat ## make sure you understand this Line!

t$.$ append$($tnew$)$

R$.$ append$($Rnew$)$

# output statement

print$("$The population infected after", t$[$n$],$ "days is approximately", R$[$n$],$ "individuals."$)$

# plot solution

plt$.$plot$($t$,$R$)$

plt$.$scatter$($t$[$n$],$ R$[$n$],$ color$=\text{'}$r$\text{'})$

plt$.$title$("$Approximate Solution to a Modified Logistic Model"$)$

plt$.$xlabel$("$time$")$

plt$.$ylabel$("$population infected"$)$

plt$.$grid$()$

plt$.$show$()$

$```$