1. 1
   The function $f$ is defined by $f(x)=4x^{3}-5x^{2}+7$, for $x\in\mathbb{R}$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$12x^{2}-10x$
   • B$12x^{2}-10x+7$
   • C$12x^{3}-10x^{2}$
   • D$12x^{2}-5x$

   (b)

   Find the gradient of the curve $y=f(x)$ at the point where $x=-1$.

   [1 mark]

   • A$2$
   • B$22$
   • C$-22$
   • D$-2$

   (c)

   Find the $x$-coordinates of the points on the curve $y=f(x)$ where the gradient is zero.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g$ is defined by $g(x)=\frac{8}{x^{2}}-3x$, for $x\neq0$.

   (a)

   Find $g'(x)$.

   [1 mark]

   • A$-\frac{16}{x}-3$
   • B$\frac{16}{x^{3}}-3$
   • C$-\frac{16}{x^{3}}-3$
   • D$-\frac{8}{x^{3}}-3$

   (b)

   Find the gradient of the curve $y=g(x)$ at the point where $x=2$.

   [1 mark]

   • A$-4$
   • B$-11$
   • C$-1$
   • D$-5$

   (c)

   Find the $x$-coordinate of the point on the curve $y=g(x)$ where the gradient is $-1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=(x^{2}-3)(2x+1)$.

   (a)

   Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

   [3 marks]

   (b)

   Find the coordinates of the points on $C$ at which the gradient is 14.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A workshop makes $x$ bicycles per week, where $10\le x\le100$. The average cost of making each bicycle, in dollars, is modelled by $A(x)=0.5x+20+\frac{800}{x}$. The total weekly cost, in dollars, is $T(x)=x\,A(x)$.

   (a)

   (i) Find $A'(x)$.
   (ii) Find $A'(10)$ and interpret your answer in context.
   (iii) Show that the average cost per bicycle is increasing for $x>40$.

   [6 marks]

   (b)

   The marginal cost is $T'(x)$.
   (i) Show that $T'(x)=x+20$.
   (ii) Find the value of $x$ for which the marginal cost equals the average cost per bicycle.
   (iii) Comment on your answer to (b)(ii) with reference to part (a).

   [6 marks]

   Total for question 4: 12 marks

End of questions