Differentiating hyperbolic functionsEdexcel International A Level Further Maths: Subtopic test

1. 1
   The hyperbolic functions are defined by $\sinh x=\frac{e^x-e^{-x}}{2}$, $\cosh x=\frac{e^x+e^{-x}}{2}$ and $\tanh x=\frac{\sinh x}{\cosh x}$, and $\operatorname{sech}x=\frac{1}{\cosh x}$.

   (a)

   Find $\frac{d}{dx}\left(\tanh3x\right)$.

   [1 mark]

   • A$\operatorname{sech}^23x$
   • B$3\operatorname{sech}^23x$
   • C$3\operatorname{sech}^2x$
   • D$-3\operatorname{sech}^23x$

   (b)

   The curve $y=\cosh x-2x$ has a stationary point. Find its $x$-coordinate.

   [1 mark]

   • A$\ln\left(2+\sqrt3\right)$
   • B$\ln2$
   • C$\ln\left(\sqrt5-2\right)$
   • D$\ln\left(2+\sqrt5\right)$

   (c)

   Use the quotient rule to show that $\frac{d}{dx}\left(\tanh x\right)=\operatorname{sech}^2x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=x\sinh^2x$.

   (a)

   Find $\frac{dy}{dx}$.

   [1 mark]

   • A$\sinh^2x+2x\sinh x\cosh x$
   • B$2x\sinh x\cosh x$
   • C$\sinh^2x+x\cosh^2x$
   • D$\sinh^2x+2x\cosh x$

   (b)

   Use your calculator to find the gradient of $C$ at $x=1$, to 3 significant figures.

   [1 mark]

   • A$3.63$
   • B$4.47$
   • C$5.01$
   • D$3.76$

   (c)

   Show that $C$ has exactly one stationary point.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f$ is defined by $f(x)=\frac{\cosh2x}{x+1}$ for $x>-1$.

   (a)

   Find $f'(x)$.

   [3 marks]

   (b)

   Find an equation of the normal to the curve $y=f(x)$ at the point where $x=0$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=4\cosh x-3\sinh x$.

   (a)

   Find the exact coordinates of the stationary point of $C$ and determine its nature.

   [6 marks]

   (b)

   Find the exact coordinates of the point on $C$ at which the gradient is $3$.

   [6 marks]

   Total for question 4: 12 marks

End of questions