Further Pure Mathematics WFM01: HYPORBOLIC FUNCTIONS Topical Past Paper Questions

$$ y=\operatorname{artanh}\left(\frac{x}{\sqrt{1+x^{2}}}\right) $$ (a) Show that $$ \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{\sqrt{1+x^{2}}} $$ (b) Hence, by integrating the result in part (a), prove that $$ \operatorname{artanh}\left(\frac{x}{\sqrt{1+x^{2}}}\right)=\operatorname…

Figure 1 Figure 1 shows a sketch of the curve with equation $$ y=\cos2x\quad0\leqslant x\leqslant\frac{\pi}{4} $$ The curve is rotated through $ 2\pi $ radians about the x-axis. (a) Show that the area of the curved surface generated is given by $$ S=2\pi\int_{0}^{\frac{\pi}{4}}\c…

Figure 1 Figure 1 shows the curve with equation $$ y=\ln\left(\tanh\frac{x}{2}\right)\qquad1\leqslant x\leqslant2 $$ (a) Show that the length, s, of the curve is given by $$ s=\int_{1}^{2}\coth x\mathrm{d}x $$ (b) Hence show that $$ s=\ln\left(\mathrm{e}+\frac{1}{\mathrm{e}}\righ…