Differentiating powers of xAQA A-Level Maths: Subtopic test

1. 1
   A curve has equation $y=4x^3-5x^2+7x-9$.

   (a)

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

   [1 mark]

   • A$12x^2-10x+7-9$
   • B$12x^2-10x+7$
   • C$4x^2-5x+7$
   • D$x^4-\frac53x^3+\frac72x^2-9x$

   (b)

   Find the gradient of the curve at the point where $x=-1$.

   [1 mark]

   • A$29$
   • B$9$
   • C$5$
   • D$-25$

   (c)

   Find the values of $x$ at which the gradient of the curve is 7.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve has equation $y=5\sqrt{x}-\frac{12}{x^3}$, where $x>0$.

   (a)

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

   [1 mark]

   • A$\frac52x^{-\frac12}-36x^{-4}$
   • B$5x^{-\frac12}+36x^{-4}$
   • C$\frac52x^{-\frac12}+36x^{-2}$
   • D$\frac52x^{-\frac12}+36x^{-4}$

   (b)

   Find the gradient of the curve at the point where $x=1$.

   [1 mark]

   • A$-\frac{67}{2}$
   • B$41$
   • C$\frac{77}{2}$
   • D$\frac52$

   (c)

   Find the gradient of the curve at the point where $x=4$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\frac{(x+2)^2}{\sqrt{x}}$, where $x>0$.

   (a)

   Show that $\mathrm{f}(x)=x^{\frac32}+4x^{\frac12}+4x^{-\frac12}$.

   [3 marks]

   (b)

   Hence find $\mathrm{f}'(x)$, and find the gradient of the curve $y=\mathrm{f}(x)$ at the point where $x=4$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=x^2+\frac{16}{x}+1$, where $x>0$.

   (a)

   (i) Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.
   (ii) Find the coordinates of the point on $C$ at which the gradient is zero.
   (iii) Find the gradient of $C$ at the point where $x=1$.

   [6 marks]

   (b)

   The tangent to $C$ at the point where $x=4$ meets the $y$-axis at the point $P$. Find the equation of the tangent and the coordinates of $P$.

   [6 marks]

   Total for question 4: 12 marks

End of questions