Product, quotient and chain rulesAQA A-Level Maths: Subtopic test

1. 1
   A curve has equation $y=x^2e^{3x}$.

   (a)

   Find $\frac{dy}{dx}$.

   [1 mark]

   • A$2xe^{3x}$
   • B$3x^2e^{3x}$
   • C$(2x+3x^2)e^{3x}$
   • D$6xe^{3x}$

   (b)

   Which values of $x$ give the stationary points of the curve?

   [1 mark]

   • A$x=0$ and $x=-\frac23$
   • B$x=0$ only
   • C$x=0$ and $x=-\frac32$
   • D$x=-\frac23$ only

   (c)

   Show that the stationary point at $x=0$ is a minimum.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve has equation $y=\frac{3x+1}{x^2+1}$.

   (a)

   Find $\frac{dy}{dx}$.

   [1 mark]

   • A$\frac{3}{2x}$
   • B$\frac{3x^2+2x-3}{(x^2+1)^2}$
   • C$\frac{9x^2+2x+3}{(x^2+1)^2}$
   • D$\frac{-3x^2-2x+3}{(x^2+1)^2}$

   (b)

   Find the gradient of the curve where $x=0$.

   [1 mark]

   • A$3$
   • B$0$
   • C$1$
   • D$-3$

   (c)

   Find the $x$-coordinates of the stationary points of the curve, correct to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve has equation $y=\sqrt{4x^2+9}$.

   (a)

   Find $\frac{dy}{dx}$, simplifying your answer.

   [3 marks]

   (b)

   Find the equation of the normal to the curve at the point where $x=2$. Give your answer in the form $ax+by=c$ with integer constants.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A spherical balloon is inflated so that air enters at a constant rate of $50$ cm$^3$ s$^{-1}$. At time $t$ seconds the balloon has radius $r$ cm and volume $V$ cm$^3$, where $V=\frac43\pi r^3$. The balloon is empty when $t=0$.

   (a)

   (i) Find the rate of increase of the radius when $r=5$. (ii) Find the rate of increase of the surface area $S=4\pi r^2$ at this instant.

   [6 marks]

   (b)

   (i) Show that $t=\frac{2\pi r^3}{75}$. (ii) Find $\frac{dt}{dr}$ and hence use the inverse relationship to write $\frac{dr}{dt}$ in terms of $r$. (iii) Find the time at which the radius is increasing at $0.1$ cm s$^{-1}$, correct to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions