Calculus of hyperbolic functionsEdexcel A-Level Further Maths: Subtopic test

1. 1
   Let $f(x)=\cosh3x$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$\sinh3x$
   • B$3\cosh3x$
   • C$-3\sinh3x$
   • D$3\sinh3x$

   (b)

   Find $\int f(x)\,dx$.

   [1 mark]

   • A$3\sinh3x+c$
   • B$\frac13\cosh3x+c$
   • C$\frac13\sinh3x+c$
   • D$\sinh3x+c$

   (c)

   Find the exact gradient of the curve $y=f(x)$ at the point where $x=\frac13\ln2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=\tanh3x$.

   (a)

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

   [1 mark]

   • A$\frac{3}{\cosh^23x}$
   • B$\frac{1}{\cosh^23x}$
   • C$\frac13\cdot\frac{1}{\cosh^23x}$
   • D$3\cosh^23x$

   (b)

   Find the gradient of $C$ at the point where $x=\frac13\ln2$.

   [1 mark]

   • A$\frac{12}{5}$
   • B$\frac{48}{25}$
   • C$\frac{75}{16}$
   • D$\frac{16}{25}$

   (c)

   Find the equation of the tangent to $C$ at the origin.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x\sinh^2x$.

   (a)

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

   [3 marks]

   (b)

   Find the exact value of the gradient of $C$ at the point where $x=\ln3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=2\cosh x-3\sinh x$.

   (a)

   (i) Find $\frac{dy}{dx}$.
   (ii) Show that $C$ has no stationary points.

   [6 marks]

   (b)

   (i) Show that $C$ crosses the $x$-axis at $x=\frac12\ln5$.
   (ii) The curve lies above the $x$-axis for $0\le x\le\ln2$. Find the exact area of the region bounded by $C$, the coordinate axes and the line $x=\ln2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions