1. 1
   Let $f(x)=3\sqrt{x}+\dfrac{4}{x^{2}}-2\ln x$, for $x>0$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$\dfrac{3}{2\sqrt{x}}-\dfrac{8}{x^{3}}-\dfrac{2}{x}$
   • B$\dfrac{3}{2\sqrt{x}}+\dfrac{8}{x^{3}}-\dfrac{2}{x}$
   • C$2x^{3/2}-\dfrac{8}{x^{3}}-\dfrac{2}{x}$
   • D$\dfrac{3}{2\sqrt{x}}-\dfrac{8}{x^{3}}+\dfrac{2}{x^{2}}$

   (b)

   Find the gradient of the graph of $f$ at the point where $x=1$.

   [1 mark]

   • A$-9.5$
   • B$7.5$
   • C$-8.5$
   • D$-6.5$

   (c)

   Find the exact value of $f'(4)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $g(x)=\cos(4x)$ and $h(x)=e^{1-2x}$, for $x\in\mathbb{R}$.

   (a)

   Find $g'(x)$.

   [1 mark]

   • A$4\sin(4x)$
   • B$-4\sin(4x)$
   • C$-\sin(4x)$
   • D$-\dfrac{1}{4}\sin(4x)$

   (b)

   Find $h'(0)$.

   [1 mark]

   • A$e$
   • B$2e$
   • C$-2$
   • D$-2e$

   (c)

   The function $p$ is defined by $p(x)=\ln\big(g(x)+2\big)$. Find $p'(x)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=\dfrac{\ln x}{x}$, for $x>0$.

   (a)

   Show that $f'(x)=\dfrac{1-\ln x}{x^{2}}$.

   [3 marks]

   (b)

   Find the exact coordinates of the point on the graph of $f$ where $f'(x)=0$, and determine whether $f$ is increasing or decreasing for values of $x$ greater than this. Justify your answer.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In a clinical trial, the concentration of drug A in a patient's blood $t$ hours after an injection is modelled by $C(t)=20te^{-0.5t}$ mg L$^{-1}$, $t\ge0$. The concentration of drug B in a second patient is modelled by $D(t)=\dfrac{40t}{t^{2}+4}$ mg L$^{-1}$, $t\ge0$.

   (a)

   (i) Show that $C'(t)=10e^{-0.5t}(2-t)$.
   (ii) Find the exact value of $C'(1)$ and interpret this value in context.

   [6 marks]

   (b)

   (i) Find $D'(t)$.
   (ii) Without using a calculator, determine which drug's concentration is increasing faster at $t=1$. Justify your answer. You may use the fact that $2<e<3$.

   [6 marks]

   Total for question 4: 12 marks

End of questions