1. 1
   A ball is thrown into the air. Its height, $h$ metres, after $t$ seconds is given by $h = 20t - 5t^2$.

   (a)

   A ball's height, $h$ metres, after $t$ seconds is given by $h = 20t - 5t^2$. Find $\frac{dh}{dt}$.

   [1 mark]

   • A$20t - 10t$
   • B$20 - 10t$
   • C$20 - 5t$
   • D$10 - 10t$

   (b)

   Using $\frac{dh}{dt} = 20-10t$, find the value of $t$ at which the ball reaches its maximum height.

   [1 mark]

   • A$t = 20$
   • B$t = 4$
   • C$t = 2$
   • D$t = 10$

   (c)

   At the maximum height ($t=2$), what type of stationary point does the graph of $h$ against $t$ have?

   [1 mark]

   • ANo stationary point
   • BMinimum point
   • CPoint of inflection
   • DMaximum point

   Total for question 1: 3 marks

2. 2
   A manufacturer's cost, $C$ dollars, of producing $x$ items is modelled by $C = x^3 - 6x^2 + 20x$, for $x > 0$.

   (a)

   Find $\frac{dC}{dx}$.

   [2 marks]

   (b)

   Find the value of $\frac{dC}{dx}$ when $x = 4$, and explain what this value represents in context.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An engineer models the area, $A$ cm$^2$, enclosed by a length of wire as $A = 12x - 2x^2$, where $x$ cm is a design parameter, for $0 < x < 6$.

   (a)

   The area, $A$ cm$^2$, enclosed by a length of wire bent into a shape is modelled by $A = 12x - 2x^2$, where $x$ cm is a design parameter. Find $\frac{dA}{dx}$ and hence find the value of $x$ at the stationary point.

   [3 marks]

   (b)

   Determine, by inspecting the gradient just before and just after $x=3$, whether this stationary point is a maximum or a minimum, and state the maximum or minimum area.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A drone's height, $h$ metres, above a launch pad $t$ seconds after take-off is modelled by $h = -\frac{1}{3}t^3 + 4t^2$, for $0 \leq t \leq 12$.

   (a)

   Find $\frac{dh}{dt}$ and hence find the two values of $t$ at which the drone's height is momentarily stationary.

   [4 marks]

   (b)

   By inspecting the gradient either side of $t=8$, determine whether this is a maximum or minimum point, and find the drone's height at this point.

   [4 marks]

   (c)

   Explain what the stationary point at $t=0$ represents physically, and describe (using the sign of $\frac{dh}{dt}$) how the drone's height behaves for $8 < t < 12$.

   [5 marks]

   Total for question 4: 13 marks

End of questions