1. 1
   Let $f(x) = x^{3} - 6x^{2} + 9x + 2$, for $x \in \mathbb{R}$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$3x^{2}-12x+9$
   • B$3x^{2}-6x+9$
   • C$x^{2}-12x+9$
   • D$3x^{2}-12x$

   (b)

   Find the interval on which $f$ is decreasing.

   [1 mark]

   • A$x<1$ or $x>3$
   • B$1<x<3$
   • C$x<1$
   • D$x>3$

   (c)

   Find the equation of the tangent to the curve $y = f(x)$ at the point where $x = 0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A function $h$ is defined by $h(x) = x^{2} + 3x$, for $x \in \mathbb{R}$. The table shows values of $\dfrac{h(2+d)-h(2)}{d}$ for small positive $d$: $d=0.1$ gives $7.1$; $d=0.01$ gives $7.01$; $d=0.001$ gives $7.001$.

   (a)

   Using the table, estimate $h'(2)$.

   [1 mark]

   • A$7.1$
   • B$14$
   • C$7$
   • D$0$

   (b)

   Find the equation of the normal to the curve $y=h(x)$ at the point where $x=2$.

   [1 mark]

   • A$y-10=7(x-2)$
   • B$y-10=\frac{1}{7}(x-2)$
   • C$y-10=-\frac{1}{7}(x+2)$
   • D$y-10=-\frac{1}{7}(x-2)$

   (c)

   Find $h''(x)$, and state whether $h$ is concave-up or concave-down for all $x \in \mathbb{R}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $g(x) = x^{2}e^{x}$, for $x \in \mathbb{R}$.

   (a)

   Find $g'(x)$.

   [3 marks]

   (b)

   Hence find the coordinates of the stationary points of the curve $y = g(x)$, and use the second derivative to determine their nature.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A ball is thrown vertically upwards from the top of a building. Its height above the ground, $h$ metres, $t$ seconds after it is thrown, is modelled by $h(t) = -5t^{2} + 20t + 25$, for $0 \le t \le T$, where $T$ is the time at which the ball hits the ground.

   (a)

   Find the maximum height reached by the ball, and the time at which it occurs. Justify that this is a maximum.

   [6 marks]

   (b)

   Find the time $T$ at which the ball hits the ground, and find the speed of the ball at this instant.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The rate of growth of a plant's height is modelled by $\dfrac{dh}{dt} = 3t^{2} - 2t$, where $h$ is the height in cm and $t$ is the time in weeks after planting, for $t \ge 0$. When $t=1$, the height is $h=6$ cm.

   (a)

   Find $h(t)$.

   [1 mark]

   • A$t^{3}-t^{2}+6$
   • B$3t^{3}-t^{2}+6$
   • C$t^{3}+t^{2}+6$
   • D$t^{3}-t^{2}$

   (b)

   Find the height of the plant after 3 weeks.

   [1 mark]

   • A$18$ cm
   • B$24$ cm
   • C$42$ cm
   • D$78$ cm

   (c)

   Find $\displaystyle\int_{1}^{3} \dfrac{dh}{dt}\,dt$, and interpret this value in context.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $f'(x) = 4x(x^{2}+3)^{3}$, for $x \in \mathbb{R}$.

   (a)

   Find $f(x)$, given that $f(0) = 5$.

   [1 mark]

   • A$(x^{2}+3)^{4}-76$
   • B$\dfrac{(x^{2}+3)^{4}}{2}+5$
   • C$\dfrac{(x^{2}+3)^{4}-71}{2}$
   • D$2(x^{2}+3)^{4}-157$

   (b)

   Find $\displaystyle\int \sin(3x) \, dx$.

   [1 mark]

   • A$\frac{1}{3}\cos(3x)+C$
   • B$-3\cos(3x)+C$
   • C$-\cos(3x)+C$
   • D$-\frac{1}{3}\cos(3x)+C$

   (c)

   Find $\displaystyle\int 6x^{2}\sqrt{2x^{3}+1}\,dx$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The curve $C$ has equation $y = 4 - x^{2}$, for $-3 \le x \le 3$.

   (a)

   Find the area of the region enclosed by $C$ and the $x$-axis between $x=-2$ and $x=2$.

   [3 marks]

   (b)

   Find the total area of the regions enclosed by $C$, the $x$-axis, and the lines $x=-3$ and $x=3$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A cylindrical can with a closed top and base is designed to hold $500\pi$ cm$^{3}$ of liquid. The radius of the base is $r$ cm and the height is $h$ cm.

   (a)

   Show that the surface area of the can is given by $A(r) = 2\pi r^{2} + \dfrac{1000\pi}{r}$, and find the value of $r$ that minimises $A$.

   [6 marks]

   (b)

   Use the second derivative to show that this value of $r$ gives a minimum surface area, and find the minimum surface area correct to 3 significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions