Calculus with hyperbolic functionsAQA A-Level Further Maths: Subtopic test

1. 1
   The function $f$ is defined by $f(x)=\cosh 2x$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$2\cosh 2x$
   • B$\sinh 2x$
   • C$2\sinh 2x$
   • D$-2\sinh 2x$

   (b)

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

   [1 mark]

   • A$2\sinh 2x+c$
   • B$\sinh 2x+c$
   • C$\frac12\cosh 2x+c$
   • D$\frac12\sinh 2x+c$

   (c)

   Find the exact value of $\int_0^{\ln2}f(x)\,dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g$ is defined by $g(x)=\tanh x$ for all real $x$.

   (a)

   Find $g'(x)$.

   [1 mark]

   • A$\operatorname{sech}^2x$
   • B$-\operatorname{sech}^2x$
   • C$\operatorname{sech}x\tanh x$
   • D$\operatorname{cosech}^2x$

   (b)

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

   [1 mark]

   • A$\ln\sinh x+c$
   • B$\ln\cosh x+c$
   • C$-\ln\cosh x+c$
   • D$\frac12\tanh^2x+c$

   (c)

   Find the exact gradient of the curve $y=g(x)$ at the point where $x=\ln3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve has equation $y=2\cosh x-3\sinh x$.

   (a)

   Show that the curve has no stationary points.

   [3 marks]

   (b)

   Find the exact area of the region bounded by the curve, the coordinate axes and the line $x=\ln2$. The curve is above the $x$-axis for $0\leq x\leq\ln2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   You may use $\cosh^2u-\sinh^2u=1$, $e^u=\cosh u+\sinh u$, $\operatorname{arcosh}z=\ln\left(z+\sqrt{z^2-1}\right)$ and $\int\frac{1}{\sqrt{x^2-a^2}}\,dx=\operatorname{arcosh}\frac xa+c$ for $x>a$.

   (a)

   Use the substitution $x=3\sinh u$ to show that $\int_0^4\frac{1}{\sqrt{x^2+9}}\,dx=\ln3$.

   [6 marks]

   (b)

   Find the exact value of $\int_5^7\frac{1}{\sqrt{x^2-6x+5}}\,dx$.

   [6 marks]

   Total for question 4: 12 marks

End of questions