Differentiation (A2)AQA A-Level Maths: Topic test

1. 1
   Let $f(x)=4e^{-2x}+\ln x$ for $x>0$.

   (a)

   What is $f'(x)$?

   [1 mark]

   • A$8e^{-2x}+\frac1x$
   • B$-8e^{-2x}+\frac1x$
   • C$-2e^{-2x}+\frac1x$
   • D$-8e^{-2x}-\frac{1}{x^2}$

   (b)

   What is $f''(x)$?

   [1 mark]

   • A$-16e^{-2x}-\frac{1}{x^2}$
   • B$16e^{-2x}+\frac{1}{x^2}$
   • C$16e^{-2x}-\frac1x$
   • D$16e^{-2x}-\frac{1}{x^2}$

   (c)

   Show that the stationary points of $f$ satisfy $e^{2x}=8x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $y=\dfrac{e^{2x}}{x+1}$ for $x>-1$.

   (a)

   What is $\dfrac{dy}{dx}$?

   [1 mark]

   • A$\dfrac{e^{2x}(2x+1)}{(x+1)^2}$
   • B$\dfrac{2e^{2x}}{(x+1)^2}$
   • C$\dfrac{e^{2x}(2x+3)}{(x+1)^2}$
   • D$\dfrac{e^{2x}(2x+1)}{x+1}$

   (b)

   What is the $x$-coordinate of the stationary point of the curve?

   [1 mark]

   • A$-1$
   • B$0$
   • C$-\frac12$
   • D$\frac12$

   (c)

   Show that the stationary point is a minimum.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve has equation $y^3+2xy=x^2+1$ and passes through the point $P(2,1)$.

   (a)

   Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$.

   [3 marks]

   (b)

   Find the equation of the normal to the curve at $P$, in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A spherical raindrop of radius $r$ mm evaporates so that its volume $V$ mm$^3$ decreases at a rate proportional to its surface area, with constant of proportionality $k>0$. Time $t$ is measured in seconds. For a sphere, $V=\frac43\pi r^3$ and the surface area is $4\pi r^2$.

   (a)

   Show that $\dfrac{dr}{dt}=-k$. Hence find, when $k=0.02$, the time taken for a drop of initial radius $3$ mm to evaporate completely.

   [6 marks]

   (b)

   Another drop evaporates in the same way, with a different value of $k$. At the instant when its radius is $2$ mm, its radius is decreasing at $0.05$ mm s$^{-1}$. Find the value of $k$, and the rate at which the drop's volume and its surface area are changing at this instant.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A curve has parametric equations $x=2t+\dfrac1t$ and $y=t^2-3$, for $t>0$.

   (a)

   What is the gradient of the curve at the point where $t=2$?

   [1 mark]

   • A$\frac{7}{16}$
   • B$\frac{16}{9}$
   • C$\frac{16}{7}$
   • D$4$

   (b)

   At what value of $t$ is the tangent to the curve parallel to the $y$-axis?

   [1 mark]

   • A$\frac{1}{\sqrt2}$
   • B$\frac12$
   • C$\sqrt2$
   • D$-\frac{1}{\sqrt2}$

   (c)

   Show that $\dfrac{dy}{dx}=\dfrac{2t^3}{2t^2-1}$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The function $h$ is defined by $h(t)=3^{2t}-\sin\dfrac t2$, where $t$ is in radians.

   (a)

   What is $h'(t)$?

   [1 mark]

   • A$2t\cdot3^{2t-1}-\frac12\cos\frac t2$
   • B$2\ln3\cdot3^{2t}+\frac12\cos\frac t2$
   • C$\ln3\cdot3^{2t}-\frac12\cos\frac t2$
   • D$2\ln3\cdot3^{2t}-\frac12\cos\frac t2$

   (b)

   What is the value of $h'(0)$?

   [1 mark]

   • A$\ln3-\frac12$
   • B$2\ln3-\frac12$
   • C$2\ln3+\frac12$
   • D$\frac32$

   (c)

   Find the exact value of $h''(0)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A population of $P$ rabbits on an island increases at a rate proportional to $P$, but $40$ rabbits per month are removed. Time $t$ is in months. At $t=0$, $P=500$ and the population is increasing at $60$ rabbits per month.

   (a)

   Construct a differential equation for $P$ and find the constant of proportionality.

   [3 marks]

   (b)

   Find the population for which the number of rabbits stays constant. Find also the value of $\dfrac{d^2P}{dt^2}$ at $t=0$ and interpret it in context.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The curve $C$ has equation $y=\dfrac{\ln x}{x^2}$ for $x>0$.

   (a)

   Find the exact coordinates of the stationary point of $C$ and determine its nature.

   [6 marks]

   (b)

   Find the equation of the tangent to $C$ at the point where $x=1$, and show that the $x$-coordinate of the point of inflection of $C$ is $e^{5/6}$.

   [6 marks]

   Total for question 8: 12 marks

End of questions