How to solve all

$($a$)$ Consider the curve defined by the equation

$xln(y+1)+\frac{y}{x^2}=2$

Use implicit differentiation to find $\frac{dy}{(d)x}$ at all points on the curve.

$($b$)$ Consider the function $f(x)=e^{3x}-4x,$ with $0x3.$ Find the critical points of $f.$ Deduce the locations of the absolute maxima and minima of $f,$ and give the values of $f$ at these points.

$($c$)$ Use Newton's Method to find a solution of $tanx=1-x,$ giving your answers accurate to $3$ decimal places. Take $x_0=0\mathrm{.}5$ as your initial guess.

$($d$)$ Sketch the graphs of $y=2x^2-4x-16$ and $y=-x^2-17$ on the same diagram, showing their points of intersection with each other and with the coordinate axes. Find the area enclosed between these two graphs.

$($e$)$ Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}7r^{2}se{c}^2(2r^3-1)dr.$

$($f$)$ Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}tsin(7t)dt.$

$($g$)$ Solve the initial$-$value problem

$\frac{dx}{(d)t}=4t^3{e}^{-x},$ with $x(0)=ln8$

Express your answer in the form $x=f(t).$

$2.($a$)$ Use implicit differentiation and a trigonometric identity to prove that

$\frac{d}{(d)x}ta{n}^{-1}x=\frac{1}{1+x^2}.$

$($b$)$ A boat leaves a dock at $10$am and travels due west at a speed of $18k\frac{m}{h}.$ Another boat has been heading due north at a speed of $16k\frac{m}{h}$ and reaches the same dock at $1$ pm $.$ At what time were the two boats closest together?

$3.($a$)$ Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{7x+5}{(x-3)(x^2+x+1)}dx.$

$($b$)$ The speedometer reading $(v(km)h^{-1})$ in a car is recorded at one minute intervals $t$ over a $10$ minute period. The readings are:

$\backslash$table$[[t,0,1,2,3,4,5,6,7,8,9,10],[v,44,41,46,47,50,49,51,52,48,43,35]]$

Use the Trapezoidal Rule to estimate the distance travelled by the car during this time.