Gradient of a curve and differentiation from first principlesAQA A-Level Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=x^2$. The point $P(3,9)$ lies on $C$, and $Q$ is the point on $C$ with $x$-coordinate $3+h$, where $h\neq0$.

   (a)

   Find the gradient of the chord $PQ$.

   [1 mark]

   • A$6+h$
   • B$6$
   • C$6h+h^2$
   • D$h$

   (b)

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

   [1 mark]

   • A$3$
   • B$6$
   • C$9$
   • D$6+h$

   (c)

   Find the equation of the tangent to $C$ at $P$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=x^3$.

   (a)

   Simplify $\frac{\mathrm{f}(x+h)-\mathrm{f}(x)}{h}$.

   [1 mark]

   • A$3x^2$
   • B$3x^2h+3xh^2+h^3$
   • C$h^2$
   • D$3x^2+3xh+h^2$

   (b)

   Use differentiation from first principles to find $\mathrm{f}'(x)$.

   [1 mark]

   • A$3x^2+3xh+h^2$
   • B$x^2$
   • C$3x^2$
   • D$3x$

   (c)

   Find the coordinates of the points on the curve $y=\mathrm{f}(x)$ at which the gradient is 12.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=3x^2-5x$.

   (a)

   Prove from first principles that $\mathrm{f}'(x)=6x-5$.

   [3 marks]

   (b)

   The tangent to the curve $y=\mathrm{f}(x)$ at the point where $x=2$ meets the $x$-axis at $A$. Find the coordinates of $A$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A model rocket is launched vertically. Its height $h$ metres above the ground, $t$ seconds after launch, is modelled by $h=12t-t^3$ for $0\le t\le3$.

   (a)

   (i) Find $\frac{\mathrm{d}h}{\mathrm{d}t}$ and $\frac{\mathrm{d}^2h}{\mathrm{d}t^2}$.
   (ii) Find the time at which the rocket reaches its greatest height, and that height.
   (iii) Interpret the value of $\frac{\mathrm{d}^2h}{\mathrm{d}t^2}$ at this time.

   [6 marks]

   (b)

   Find the mean velocity of the rocket between $t=1$ and $t=1.1$. Compare this with the velocity at $t=1$, and explain why they are different and how the mean velocity could be made closer to the velocity at $t=1$.

   [6 marks]

   Total for question 4: 12 marks

End of questions