Further Mathematics 9231: Differentiation Topical Past Paper Questions

The curve C has parametric equations $$ x=2\cosh2t+3t,\quad y=\frac{3}{2}\cosh2t-4t,\quad for0\leqslant t\leqslant1. $$ (a) Find, in terms of e, the length of C. [5] (b) Show that $$ \frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=\frac{a\cosh2t}{\left(b\sinh2t+c\right)^{n}}, $$ where $a…

The curve C has parametric equations $$ x=8\ln(\tan{\textstyle\frac{1}{2}}t)-3\cot t-3t,\quad y=6\ln(\tan{\textstyle\frac{1}{2}}t)+4\cot t+4t,\quad\mathrm{f o r}\;{\textstyle\frac{1}{6}}\pi\leqslant t\leqslant{\textstyle\frac{1}{3}}\pi. $$ (a) (i) Show that $ \frac{dx}{dt} = 8\cs…

It is given that $ x = 1 + \frac{1}{t} $ and $ y = \cos^{-1} t $ for 0 < t < 1. (a) Show that $ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{t^{2}}{\sqrt{1-t^{2}}} $. [2] (b) Show that $ \frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}} = -t^{a}(1-t^{2})^{b}(2-t^{2}) $, where a and b are consta…