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

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

$(0)$

$h=0\mathrm{.}4$

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

$($i$)$

$n=0\mathrm{.}2$

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

$($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 $y=9e^x,0x0,4,$ together with the Euler approximations using the step sizes in part $($a$).$ Use your sketches to decide whether your eatimates in part $($a$)$ are underestimates or everestimates.

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 Culer's method to estimate the true value of $y(0,4),$ namely, $9e^{0\mathrm{.}4}.($Round your answers to flour decimal places.$)$

$h=0\mathrm{.}4$ error exact value