Differentiation from first principlesEdexcel A-Level Maths: Subtopic test

1. 1
   The point $P(3,9)$ lies on the curve $y=x^{2}$. The point $Q$ on the same curve has $x$-coordinate $3+h$, where $h\neq0$.

   (a)

   Find the gradient of the chord $PQ$.

   [1 mark]

   • A$6$
   • B$3+h$
   • C$6+h^{2}$
   • D$6+h$

   (b)

   As $h$ tends to $0$, the gradient of $PQ$ tends to the gradient of the tangent at $P$. What is this gradient?

   [1 mark]

   • A$0$
   • B$3$
   • C$6$
   • D$9$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x)=x^{3}$.

   (a)

   Which expression is equal to $\frac{f(x+h)-f(x)}{h}$ after simplifying?

   [1 mark]

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

   (b)

   Hence, from first principles, what is $f'(x)$?

   [1 mark]

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

   (c)

   Given that $f'(x)=3x^{2}$, find $f''(x)$, and hence find the rate of change of the gradient of the curve $y=f(x)$ when $x=2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve has equation $y=x^{2}$.

   (a)

   Prove, from first principles, that $\frac{\mathrm{d}y}{\mathrm{d}x}=2x$.

   [3 marks]

   (b)

   Find the coordinates of the point on the curve at which the gradient of the tangent is $-6$, and find the equation of the tangent at this point.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle moves in a straight line. Its displacement from a fixed point $O$ after $t$ seconds is $s=t^{3}-6t^{2}$ metres, for $t\geq0$. Its velocity is $v=\frac{\mathrm{d}s}{\mathrm{d}t}$ and its acceleration is $a=\frac{\mathrm{d}^{2}s}{\mathrm{d}t^{2}}$.

   (a)

   (i) Show, from first principles, that the derivative of $t^{3}$ with respect to $t$ is $3t^{2}$.
   (ii) Given that the derivative of $t^{2}$ is $2t$, find expressions for $v$ and $a$ in terms of $t$.

   [6 marks]

   (b)

   (i) Find the times at which the particle is instantaneously at rest.
   (ii) Find the acceleration when $t=4$ and interpret its meaning.
   (iii) Find the gradient of the curve of $s$ against $t$ when $t=1$ and interpret its meaning.

   [6 marks]

   Total for question 4: 12 marks

End of questions