Differentiating powers of x and stationary pointsEdexcel A-Level Maths: Subtopic test

1. 1
   A curve has equation $y=(2x+5)(x-1)$.

   (a)

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

   [1 mark]

   • A$4x+3$
   • B$2$
   • C$4x+5$
   • D$4x-3$

   (b)

   Find the gradient of the curve at the point where $x=2$.

   [1 mark]

   • A$8$
   • B$11$
   • C$9$
   • D$7$

   (c)

   Find the $x$-coordinate of the stationary point of the curve.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f$ is defined by $f(x)=\frac{x^{2}+3x-5}{4x^{1/2}}$ for $x>0$.

   (a)

   Which expression is equal to $f(x)$?

   [1 mark]

   • A$\frac14x^{3/2}+\frac34x^{1/2}-\frac54$
   • B$\frac14x^{5/2}+\frac34x^{3/2}-\frac54x^{1/2}$
   • C$\frac14x^{3/2}+\frac34x^{1/2}-\frac54x^{-1/2}$
   • D$\frac14x^{3/2}+\frac34x^{1/2}-\frac54x^{1/2}$

   (b)

   Which expression is $f'(x)$?

   [1 mark]

   • A$\frac38x^{1/2}+\frac38x^{-1/2}-\frac58x^{-3/2}$
   • B$\frac38x^{1/2}+\frac38x^{-1/2}+\frac58x^{-1/2}$
   • C$\frac14x^{1/2}+\frac34x^{-1/2}+\frac54x^{-3/2}$
   • D$\frac38x^{1/2}+\frac38x^{-1/2}+\frac58x^{-3/2}$

   (c)

   Find the exact value of $f'(4)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve has equation $y=x^{3}-6x^{2}+9x+2$.

   (a)

   Find the coordinates of the stationary points of the curve.

   [3 marks]

   (b)

   Determine the nature of each stationary point, and state the values of $x$ for which $y$ is decreasing.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A manufacturer makes an open-topped box from $1200$ cm$^{2}$ of thin card, with no wastage. The box has a square base of side $x$ cm and height $h$ cm, and its volume is $V$ cm$^{3}$.

   (a)

   (i) Show that $V=300x-\frac{x^{3}}{4}$.
   (ii) Find the value of $x$ for which $V$ is stationary.

   [6 marks]

   (b)

   (i) Justify that this value of $x$ gives a maximum volume.
   (ii) Find the maximum volume.
   (iii) A designer claims that using $2400$ cm$^{2}$ of card, double the amount, would double the maximum volume. Evaluate this claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions