$[4$ marks$]$ Find $f$ :

$($a$)f^{\text{'}\text{'}}(x)=5x^3+6x^2+2,f(0)=3,f(1)=-2{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{3t^4-t^3\sqrt[\phantom{2}]{t}+6t^2}{t^4}dt{\displaystyle \underset{0}{\overset{3}{}}}(|x-1|+4)dx{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^3(1-x^4{)}^{2}dxf{f}^{\text{'}}(x)=x^{3}x+y=0ff(x)=\sqrt[\phantom{2}]{x}-2,1x6n=50,1\underset{n}{lim}\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}\sqrt[\phantom{2}]{4+\frac{5i}{n}}\underset{n}{lim}{\displaystyle \underset{j=1}{\overset{n}{}}}\frac{j^7}{n^8}{f}^{\text{'}}(t)=sec(t)(sec(t)+tan(t)),-\frac{}{2}$

$(b)f^{\text{'}\text{'}}(x)=5x^3+6x^2+2,f(0)=3,f(1)=-2.$

$[6$ marks$]$ Compute the following integrals:

$(a){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{3t^4-t^3\sqrt[\phantom{2}]{t}+6t^2}{t^4}dt.$

$(b){\displaystyle \underset{0}{\overset{3}{}}}(|x-1|+4)dx.$

$(c){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^3(1-x^4{)}^{2}dx.$

$[3$ marks$]$ Find a function $f$ such that $f^{\text{'}}(x)=x^3$ and the line $x+y=0is$ tangent $to$ the graph $off.$

$[3$ marks$]Iff(x)=\sqrt[\phantom{2}]{x}-2,1x6,$ find the Riemann sum with $n=5$ correct $to3$ decimal places taking the sample points $tobe$ the midpoints.

$[4$ marks$]$ Express the following $limitsas$ a definite integral $on$ the interval $0,1.$

$(a)\underset{n}{lim}\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}\sqrt[\phantom{2}]{4+\frac{5i}{n}}.$

$(b)\underset{n}{lim}{\displaystyle \underset{j=1}{\overset{n}{}}}\frac{j^7}{n^8}.$