Hyperbolic functions and their graphsAQA A-Level Further Maths: Subtopic test

1. 1
   Let $x=\ln3$.

   (a)

   Find the value of $\sinh x$.

   [1 mark]

   • A$\frac43$
   • B$\frac53$
   • C$\frac83$
   • D$\frac34$

   (b)

   Find the value of $\cosh x$.

   [1 mark]

   • A$\frac43$
   • B$\frac{10}{3}$
   • C$\frac53$
   • D$\frac35$

   (c)

   Find the exact value of $\tanh x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the graphs of $y=\cosh x$ and $y=\tanh x$ for all real $x$.

   (a)

   What are the coordinates of the minimum point of $y=\cosh x$?

   [1 mark]

   • A$(0,0)$
   • B$(1,0)$
   • C$(0,-1)$
   • D$(0,1)$

   (b)

   Which statement describes the asymptotes of $y=\tanh x$?

   [1 mark]

   • AVertical asymptotes $x=1$ and $x=-1$
   • BHorizontal asymptotes $y=1$ and $y=-1$
   • CHorizontal asymptotes $y=0$ and $y=1$
   • DNo asymptotes, because it is defined for every real $x$

   (c)

   Describe the symmetry of each of the two graphs.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f$ is defined for all real $x$ by $f(x)=2\cosh x+3\sinh x$.

   (a)

   Express $f(x)$ in the form $Ae^x+Be^{-x}$, where $A$ and $B$ are constants.

   [3 marks]

   (b)

   Hence solve $f(x)=3$, giving your answer as an exact natural logarithm.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The functions $\sinh x$, $\cosh x$ and $\tanh x$ are defined for all real $x$ in terms of $e^x$.

   (a)

   (i) Show that $\tanh x=\frac{e^{2x}-1}{e^{2x}+1}$.
   (ii) Hence explain why $\tanh x<1$ for all real $x$, and state the limiting value of $\tanh x$ as $x\to\infty$ and as $x\to-\infty$.

   [6 marks]

   (b)

   (i) Show that $\cosh x-\sinh x=e^{-x}$.
   (ii) Hence explain why the graphs of $y=\cosh x$ and $y=\sinh x$ never meet, and describe how the gap between them behaves as $x\to\infty$.
   (iii) Find the value of $x$ for which $\cosh x=2\sinh x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions