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

   (a)

   For $h \neq 0$, find $\dfrac{f(x+h) - f(x)}{h}$ in simplified form.

   [1 mark]

   • A$6x^{2} - 5$
   • B$6x^{2} + 6xh + 2h^{2} - 5h$
   • C$6x^{2} + 3xh + h^{2} - 5$
   • D$6x^{2} + 6xh + 2h^{2} - 5$

   (b)

   Find $f''(-2)$.

   [1 mark]

   • A$19$
   • B$-24$
   • C$24$
   • D$-48$

   (c)

   Use the definition of the derivative from first principles to find the gradient of the curve $y = f(x)$ at the point where $x = 1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the functions $p(x) = \dfrac{x^{2} - 9}{x - 3}$, $x \neq 3$, $q(x) = \dfrac{1}{(x-3)^{2}}$, $x \neq 3$, and $r(x) = |x - 3|$, $x \in \mathbb{R}$.

   (a)

   Find $\lim_{x \to 3} p(x)$.

   [1 mark]

   • A$0$
   • B$3$
   • C$6$
   • DThe limit does not exist because $p(3)$ is undefined.

   (b)

   Which statement is true?

   [1 mark]

   • AAs $x \to 3$, $q(x)$ increases without bound, so $\lim_{x \to 3} q(x)$ does not exist.
   • B$r$ is not continuous at $x = 3$.
   • C$r$ is differentiable at $x = 3$ and $r'(3) = 0$.
   • D$p$ is continuous at $x = 3$ because $\lim_{x \to 3} p(x)$ exists.

   (c)

   Explain why $r$ is continuous at $x = 3$ but is not differentiable at $x = 3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A ball is thrown vertically upwards. Its height above the ground, in metres, $t$ seconds after it is thrown is $s(t) = 20t - 5t^{2}$, for $0 \le t \le 4$.

   (a)

   Show from first principles that the velocity of the ball at time $t$ is $v(t) = 20 - 10t$.

   [3 marks]

   (b)

   Using your working from part (a), find the average velocity of the ball between $t = 2$ and $t = 2 + h$ when $h = 0.1$ and when $h = -0.1$. Hence write down the instantaneous velocity of the ball at $t = 2$ and interpret this value in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x) = x\,e^{2x}$, $x \in \mathbb{R}$. The $n$th derivative of $f$ is written $f^{(n)}(x)$.

   (a)

   (i) Find $f'(x)$, $f''(x)$ and $f'''(x)$, writing each in the form $(ax + b)e^{2x}$.
   (ii) Hence find the coordinates of the point on the curve $y = f(x)$ where $f''(x) = 0$, and justify that it is a point of inflexion.

   [6 marks]

   (b)

   Prove by mathematical induction that $f^{(n)}(x) = 2^{n-1}(2x + n)e^{2x}$ for all $n \in \mathbb{Z}^{+}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions