Write using R code. Refer to the image. Problem $2.$ Assume you have a function check.prime that can be used to check whether

a given natural number is a prime. $[$Modify your own code or one of mine posted in the

solution of homework $3].$

a$.$ Based on the function check.prime, create a new function that can count the number

of primes smaller than or equal to the given natural number. Name this new function

count.prime.

b$.$ Consider a sequence $x=100**seq(1$:$500).$ Let $y$ be a vector with the same length as $x$

and the $i$ th element of $y($i$.$c$.,y_i)$ is the count of prime numbers up to the $i$ th clement

of $x($i$.$e$.,x_i).$ Define $z=\frac{x}{log(x)}.$ Display the last $20$ values of $x,y,$ and $z.$ For

example, you can use the command tail $($cbind $(x,y,z).$

c$.$ Plot $y$ versus $x.$ On the same figure, add a curve $z$ versus $x.$ Distinguish the two

curves with different line types and colors. Add legend too. In a separate figure, plot

$\frac{y}{z}$ versus $x.$ Make comments on the relationship between $y$ and $z$ when $x$ is large.

Search "Prime number theorem" for reference to help make your comments.