The derivative as a gradient and rate of changeEdexcel International A Level Maths: Subtopic test

1. 1
   A curve has equation $y=x^2+3x$. The point $P$ has coordinates $(2,10)$ and the point $Q$ on the curve has $x$-coordinate $2+h$, where $h\neq0$.

   (a)

   Find the gradient of the chord $PQ$ in terms of $h$.

   [1 mark]

   • A$7h+h^2$
   • B$7+h$
   • C$4+h$
   • D$7+h^2$

   (b)

   What is the gradient of the tangent to the curve at $P$?

   [1 mark]

   • A$10$
   • B$8$
   • C$4$
   • D$7$

   (c)

   Explain why the gradient of the chord $PQ$ is not equal to the gradient of the tangent at $P$, and how the gradient of the tangent can be found from it.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The volume $V$ cm$^3$ of water in a tank at time $t$ minutes is modelled by $V=40+6t-0.5t^2$ for $0\le t\le10$.

   (a)

   What are the units of $\frac{\mathrm{d}V}{\mathrm{d}t}$?

   [1 mark]

   • Acm$^3$ min$^{-1}$
   • Bcm$^3$
   • Cmin cm$^{-3}$
   • Dcm$^3$ min

   (b)

   Find the value of $\frac{\mathrm{d}V}{\mathrm{d}t}$ when $t=4$.

   [1 mark]

   • A$56$
   • B$6$
   • C$2$
   • D$-2$

   (c)

   Find the time at which the volume of water momentarily stops changing, and explain what the sign of $\frac{\mathrm{d}V}{\mathrm{d}t}$ tells you about the volume just before this time.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $y=f(x)$ has equation $f(x)=x^3-6x^2+9x+2$.

   (a)

   Find $f'(x)$ and hence solve $f'(x)=0$.

   [3 marks]

   (b)

   Find $f''(x)$. Explain what $f''(x)$ represents, and find the values of $x$ for which the gradient of the curve is increasing.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=f(x)$, where $f(x)=2x^2-5x$.

   (a)

   (i) Use differentiation from first principles to show that $f'(x)=4x-5$.
   (ii) Hence find the coordinates of the point on $C$ where the gradient is $3$.

   [6 marks]

   (b)

   (i) Show that the gradient of the chord joining the points on $C$ where $x=1$ and $x=1+h$ is $2h-1$.
   (ii) Using $f'(x)=4x-5$, find the value of $h$ for which this chord has the same gradient as the tangent to $C$ at $x=2$.
   (iii) State the gradient of the tangent at $x=1$ and explain how it relates to the chord gradient $2h-1$.

   [6 marks]

   Total for question 4: 12 marks

End of questions