Question $1.(4$ Marks$)$ Find $\frac{dy}{dx}$ for each of the following:

a$.y=e^{cos(x^2)}$

b$.ln(x+y)=xy-y^3$

c$.y=x^{lnx}$

d$.y=\frac{e^{2x}}{(x^2+1{)}^3(1+sinx{)}^5}$

$($Hint Start by taking the logarithm for both sides$)$

Question $2.(2$ Marks$)$ Find the first $5$ terms of the Maclurin series of the function $f(x)=ln(x+2).$

Question $3.(3$ Marks$).$ Find the dimensions of the rectangle of largest area that has its base on the $x-$axis and its other two vertices above the $x-$axis and lying on the parabola

$y=12-x^2$

Question$6.(5$ Marks$)$ In an experiment to determine the relationship between force and momentum, a force $x,$ is applied to a mass, by placing the mass on an inclined plane, and the time, $Y,$ for the velocity to change from $u\frac{m}{s}$ to $v\frac{m}{s}$ is measured. The results obtained are as follows:

$\backslash$table$[[$Force $(N),$Time $(s)$

Question$4.(5$ Marks$).$ Evaluate the following integrals:

$1{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^2}{(x^3+5{)}^2}dx$

$2{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{ln5x}{x}dx$

$3{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{e^{\sqrt[\phantom{2}]{x}}}{\sqrt[\phantom{2}]{x}}dx$

$4{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^3{e}^{x}dx$

$5{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^{2}lnxdx$

$6{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{1}{(x+5{)}^2(x-1)}dx$

$7{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^2-x+6}{x^3+3x}dx$

$8{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^2-5x+16}{(2x+1)(x-2{)}^2}dx$

Question$5.(3$ Marks$)$ Determine the area of the region bounded by the functions $f(x)=2x^2+10$ and $g(x)=4x+16.$