1. 1
   Let $f(x) = 5x^4 - 8x^3 + 2x$, for $x \in \mathbb{R}$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$20x^3-24x^2+2$
   • B$20x^3-24x^2$
   • C$5x^3-8x^2+2$
   • D$20x^3-24x^2+2x$

   (b)

   Find $f''(x)$.

   [1 mark]

   • A$60x^2-48x+2$
   • B$60x^2-48x$
   • C$60x^3-48x^2$
   • D$20x^2-24x$

   (c)

   Find the equation of the tangent to the curve $y=f(x)$ at the point where $x=1$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   Find $g'(x)$.

   [1 mark]

   • A$2xe^{4x}+x^2e^{4x}$
   • B$8xe^{4x}$
   • C$2xe^{4x}+4x^2e^{4x}$
   • D$2xe^{4x}$

   (b)

   Find the gradient of the curve $y=g(x)$ at the point where $x=0$.

   [1 mark]

   • A$1$
   • B$4$
   • C$2$
   • D$0$

   (c)

   Show that $g$ is an increasing function for all $x>0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle moves in a straight line so that its velocity, in m s$^{-1}$, at time $t$ seconds is $v(t)=3t^2-12t+9$, for $t\ge0$. When $t=0$ the particle's displacement from a fixed point $O$ is $s=4$ m.

   (a)

   Find the times at which the particle is instantaneously at rest.

   [3 marks]

   (b)

   Find the displacement of the particle from $O$ when $t=3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An open-topped rectangular tank is to be made from thin sheet metal. The base is a square of side $x$ metres and the height is $h$ metres. The tank must hold $32$ m$^3$ of water, and the total area of metal used, in m$^2$, is $A(x)=x^2+\dfrac{128}{x}$, for $x>0$.

   (a)

   Show that $A'(x)=2x-\dfrac{128}{x^2}$, and hence find the value of $x$ that minimises $A$, justifying that it gives a minimum.

   [6 marks]

   (b)

   (i) Using $x=4$, find the minimum area of metal required and the height $h$ of the tank. (ii) The tank is then filled with water at a constant rate of $2$ m$^3$ per minute. Let $y$ metres be the depth of water at time $t$ minutes, so that the volume of water is $V=16y$. Find the rate at which the depth of the water is increasing.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Let $f'(x)=6x^2-\dfrac{2}{x^3}$, for $x\neq0$, and $f(1)=5$.

   (a)

   Find the indefinite integral of $f'(x)$.

   [1 mark]

   • A$2x^3-\dfrac{1}{x^2}+C$
   • B$2x^3+\dfrac{1}{x^2}+C$
   • C$2x^3+\dfrac{2}{3x^2}+C$
   • D$12x+\dfrac{6}{x^4}+C$

   (b)

   Given that $f(1)=5$, find $f(x)$.

   [1 mark]

   • A$2x^3+\dfrac{1}{x^2}+5$
   • B$2x^3+\dfrac{1}{x^2}-2$
   • C$2x^3+\dfrac{1}{x^2}+2$
   • D$2x^3+\dfrac{1}{x^2}+3$

   (c)

   Find $\displaystyle\int_1^2 f'(x)\,dx$, and interpret this value in terms of $f$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $f(x)=x^2-5x$, for $x\in\mathbb{R}$.

   (a)

   Using the definition of the derivative from first principles, find the value of $\displaystyle\lim_{h\to0}\dfrac{f(3+h)-f(3)}{h}$.

   [1 mark]

   • A$6$
   • B$-1$
   • C$-2$
   • D$1$

   (b)

   Find $\displaystyle\lim_{x\to5}\dfrac{f(x)}{x-5}$.

   [1 mark]

   • A$5$
   • B$0$
   • CThe limit does not exist
   • D$\infty$

   (c)

   State, with a reason, whether $f$ is differentiable at $x=5$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Let $h(x)=\arctan(3x)$, for $x\in\mathbb{R}$.

   (a)

   Find $h'(x)$.

   [3 marks]

   (b)

   Hence, using integration by parts, find $\displaystyle\int \arctan(3x)\,dx$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A biologist models the population $N$ (in thousands) of a bacterial colony by the differential equation $\dfrac{dN}{dt}=0.02N(100-N)$, for $t\ge0$ hours, where $N=10$ when $t=0$.

   (a)

   Solve the differential equation to show that $N=\dfrac{100e^{2t}}{9+e^{2t}}$.

   [6 marks]

   (b)

   Using the differential equation directly (without using the result of part (a)), find the Maclaurin series for $N(t)$ up to and including the term in $t^2$.

   [6 marks]

   Total for question 8: 12 marks

End of questions