Consider the intial$-$value problem $y^{\text{'}}=y,y(0)=9.$

$($a$)$ Use Euler's method with each of the following step sizes to estimate the value of $y(0\mathrm{.}4).$

$h=0\mathrm{.}4$

$v(0\mathrm{.}4)=$

$($ii$)$

$h=0\mathrm{.}2$

$M(0\mathrm{.}4)=$

$($iii$)$

$h=0\mathrm{.}1$

$M(0\mathrm{.}4)=$

$($b$)$ We know that the exact solution of the initial$-$value problem in part $($a$)$ is $y=9e^x.$ Draw, as accurately as you can, the graph of $y=9e^x,0x0\mathrm{.}4,$ together with the Euler approximations using the step sizes in part $($a$).$ Use your sketches to decide whether your estimates in part $[$a$]$ are underestimates or overestimates.

The estimates are $,$

$($c$)$ The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part $($a$)$ in using Euler's method to estimate the true value of $y(0\mathrm{.}4),$ namely, $9e^{0\mathrm{.}4}.($Round your answers to four decimal places.$)$

$h=0\mathrm{.}4$ error $=|||($exact value$)-($approximate value$)|,||=$

$h=0\mathrm{.}2$ error exact value