FP3: DifferentiationEdexcel International A Level Further Maths: Topic test

1. 1
   The function $f$ is defined by $f(x)=\ln(\cosh x)$ for $x\in\mathbb{R}$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$\frac{1}{\cosh x}$
   • B$\coth x$
   • C$\tanh x$
   • D$\sinh x$

   (b)

   Find $f''(x)$.

   [1 mark]

   • A$\frac{1}{\cosh^2x}$
   • B$-\frac{1}{\cosh^2x}$
   • C$\frac{1}{\cosh x}$
   • D$\tanh^2x$

   (c)

   Find the exact value of $f'(\ln3)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=\arctan(2x)$.

   (a)

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

   [1 mark]

   • A$\frac{1}{1+4x^2}$
   • B$\frac{2}{1+2x^2}$
   • C$\frac{2}{\sqrt{1-4x^2}}$
   • D$\frac{2}{1+4x^2}$

   (b)

   Find the gradient of $C$ at the point where $x=\frac12$.

   [1 mark]

   • A$\frac12$
   • B$1$
   • C$2$
   • D$\frac43$

   (c)

   Find an equation of the tangent to $C$ at the point where $x=\frac12$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x\cosh3x-\frac13\sinh3x$.

   (a)

   Show that $\frac{dy}{dx}=3x\sinh3x$.

   [3 marks]

   (b)

   Show that $C$ has exactly one stationary point, and determine its nature.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $f$ is defined by $f(x)=x\operatorname{artanh}x+\frac12\ln\left(1-x^2\right)$ for $-1<x<1$.

   (a)

   Show that $f'(x)=\operatorname{artanh}x$, and hence find the exact value of $f'\left(\frac12\right)$.

   [6 marks]

   (b)

   Use the result of part (a) to find $f''(x)$. Hence show that $f$ has a single stationary point, which is a minimum, and deduce that $x\operatorname{artanh}x\ge-\frac12\ln\left(1-x^2\right)$ for $-1<x<1$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The curve $C$ has equation $y=\ln(\sinh2x)$ for $x>0$.

   (a)

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

   [1 mark]

   • A$2\tanh2x$
   • B$2\coth2x$
   • C$\coth2x$
   • D$\frac{2}{\sinh2x}$

   (b)

   Find the gradient of $C$ at the point where $x=\frac12\ln3$.

   [1 mark]

   • A$\frac85$
   • B$\frac54$
   • C$\frac43$
   • D$\frac52$

   (c)

   Find the exact value of $x$ at which the gradient of $C$ is $3$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The curve $C$ has equation $y=\arcsin(3x)$ for $-\frac13<x<\frac13$.

   (a)

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

   [1 mark]

   • A$\frac{3}{\sqrt{1-9x^2}}$
   • B$\frac{1}{\sqrt{1-9x^2}}$
   • C$\frac{3}{\sqrt{1-3x^2}}$
   • D$\frac{3}{1+9x^2}$

   (b)

   Find the gradient of $C$ at the point where $x=\frac16$.

   [1 mark]

   • A$3$
   • B$\frac{\sqrt3}{2}$
   • C$2\sqrt3$
   • D$\frac{3\sqrt3}{2}$

   (c)

   Show that the gradient of $C$ is at least $3$ at every point of $C$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The curve $C$ has equation $y=\operatorname{arsinh}(\tan x)$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$.

   (a)

   Show that $\frac{dy}{dx}=\sec x$.

   [3 marks]

   (b)

   Find an equation of the tangent to $C$ at the point where $x=\frac{\pi}{3}$, giving $y$ in terms of $x$ and exact constants.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The curve $C$ has equation $y=\arcsin(\tanh x)$ for $x\in\mathbb{R}$.

   (a)

   Show that $\frac{dy}{dx}=\operatorname{sech}x$. Hence state the maximum gradient of $C$ and the value of $x$ at which it occurs.

   [6 marks]

   (b)

   The point $P$ on $C$ has $x$-coordinate $\ln2$. Find an equation of the normal to $C$ at $P$, giving your answer in the form $y=mx+c$ with $m$ and $c$ to 3 significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions