$Given that \frac{d}{dx} ({\sin }^{-1} x+{\cos }^{-1}x)=0 , the unique value of {\sin }^{-1}x+ {\cos }^{-1} x is?$

$b) Find \frac{d}{dx}({\tan }^{-1} 2x^2)$

$f) Find an expression for \sin ({\tan }^{-1 }x)$

$$\begin{gathered} A function is defined as f\left(x\right) =1-\cos \frac{x}{2} where 0\le x\le a \\ ¡) find the largest value of a for which the inverse function f^{-1}(x) exists. \\ ii. show that f^{-1}(x)=2{\cos }^{-1}(1-x) \\ iii. sketch the graph of y= f^{-1}(x) \\ iv. find the area enclosed between the curve y=f^{-1}(x), the x axis and x=2. \end{gathered}$$

$$\begin{gathered} Show that {\tan }^{-1}\left(4\right)-{\tan }^{-1}\left(\frac{3}{5}\right)=\frac{\mathrm{\pi }}{4} \\ If 2{\sin }^{-1}\mathcal{x}={\cos }^{-1}\mathcal{x} find \mathcal{x} when 0\le \mathcal{x}\le 1 \end{gathered}$$

$$\begin{gathered} A projectile is fired from O, at the bottom of the inclined road, with a speed of V m/s \\ at an angle of elevation \theta to the horizontal as shown below. U\sin g the axis above you \\ may assume that the position of the projectile is given by \\ x=Vt \cos \theta and y=Vt \sin \theta -\frac{1}{2}g{t}^2 \\ Where t is the time in second after firing and g is the acceleration due \\ to gravity \\ For simplicity assuming that \frac{2V^2}{g}=1 \\ i) Show that the path of the trajectory of the projectile is \\ y=x \tan \theta -x^2 se{c}^2\theta . \\ ii) Show that the range of the projectile r=OT metres \\ inclined road is given by \\ r=\frac{\sin (\theta -\alpha ) \cos \theta }{{\cos }^2 \alpha } \end{gathered}$$

$$\begin{gathered} Consider the functions f\left(\mathcal{x}\right)={\cos }^{-1}\left(2\mathcal{x}\right) and g\left(\mathcal{x}\right)={\sin }^{-1}\mathcal{x} \\ \left(i\right) Sketch f\left(\mathcal{x}\right)={\cos }^{-1}\left(2\mathcal{x}\right) \\ \left(ii\right) Prove that the \mathcal{x}-oordinates of the point of intersection of \\ f\left(\mathcal{x}\right) and g\left(\mathcal{x}\right) is \frac{1}{\sqrt{5}} \\ \left(iii\right) Show that the gradients of the \tan gents f\left(\mathcal{x}\right) and g\left(\mathcal{x}\right) at their point of intersection \\ -2\sqrt{5} and \frac{\sqrt{5}}{2}respectively. \\ \left(iv\right) Write the equation of \tan (\beta -\alpha ) \\ \left(v\right) Hence or otherwisw find the acute angke between f\left(\mathcal{x}\right) and g\left(\mathcal{x}\right) at their point \\ of intersection (to the naerest degree) \end{gathered}$$

$$\begin{gathered} The curve on the right has an asymptote at \mathcal{x}=0 \\ The shaded area "goes forever" but its value \\ has a limit. Steffi Graph was trying to find this limit but was unable to \\ integrate f\left(\mathcal{x}\right). \\ Her friend monica suggested she use the inverse function to find the \\ shaded area. \end{gathered}$$

$$\begin{gathered} State the domain and range of the inverse function f^{-1}\left(\mathcal{x}\right) \\ Roughly sketch the inverse function f^{-1}\left(\mathcal{x}\right) \\ Show that f^{-1}\left(\mathcal{x}\right)=\frac{2}{1+{\mathcal{x}}^2} \\ Find the limit that the value of the shaded area appraoches. \end{gathered}$$

$$\begin{gathered} A function is defined as f\left(x\right) =1-\cos \frac{x}{2} where 0\le x\le a \\ \left(i\right) Find the largest value of a for which the inverse function f^{-1}\left(x\right) exists. \\ \left(ii\right) Find f^{-1}\left(x\right) \\ \left(iii\right) Sketch the graph ofy= f^{-1}\left(x\right) \\ \left(iv\right) Find the area enclosed between the curve y = f^{-1}\left(x\right), the x axis and x = 2. \end{gathered}$$

$$\begin{gathered} A cricketer hits the ball from ground level with a speed of 20 m/s and an angle of elevation of \alpha . \\ It flies towards a high wall 20 metres away. Take g=10m/s^2 \\ i) Given that the horizontal and vertical displacements at time are, respectively (do not need to derive these): \\ x=20t\cos \alpha \\ y=-5t^2+20t\sin \alpha \\ Show that the value of h , the height up the wall at which the ball will collide, is given by \\ h=-5se{c}^2\alpha +20\sin \alpha \\ ii) Show that the maximum value of h is obtained when \tan \alpha =2 \\ iii) Assuming \tan \alpha =2 , find the speed at which the ball will hit the wall. \end{gathered}$$