For the polynomial x$^(4)-2$x$^(3)+3$x$^(2)-10$x$+8,$

$(1)$ Use the Matlab plot function to determine how many real$-$valued zeros it has.

$(2)$ Solve for the zeros by using the Matlab "roots" function.

The function erf$($x$)=(2)/(\backslash$sqrt$(\backslash$pi $))\backslash$int$\_0^$x e$^(-$t$^(2))$dt is called the error function. It is used in the field of

probability and cannot be calculated exactly for finite values of x$.$ However, one can

expand the integrand as a Taylor polynomial and conduct integration.

$(1)$ Use the Matlab function "integral" to get its value and use it as "true value" $($not

really, but close enough$).$

$(2)$ For the integrand e$^(-$t$^(2)),$ what is the three$-$term Taylor truncation around $0?$ This

means that you use the first three terms in the Taylor expansion as its

approximation.

$(3)$ What is the analytic solution of the integral when the truncated expression is used?

Note: Since only polynomials are involved, you can derive it by hand. But you can

also use Matlab Symbolic Math to get it$.$ The result will still be a polynomial of x$.$

$(4)$ What is the approximate value of erf$(3\mathrm{.}0)$ using the first three terms of the Taylor

series around t$=0?$ Please use Matlab codes to get this value.

$(5)$ What is the true error?