Differentiating exponentials, logarithms and trigonometric functionsAQA A-Level Maths: Subtopic test

1. 1
   A curve has equation $y=e^{3x}-4x$.

   (a)

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

   [1 mark]

   • A$e^{3x}-4$
   • B$3e^{3x}-4x$
   • C$\frac{e^{3x}}{3}-4$
   • D$3e^{3x}-4$

   (b)

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

   [1 mark]

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

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve has equation $y=\tan2x$ for $-\frac{\pi}{4}<x<\frac{\pi}{4}$, where $x$ is in radians.

   (a)

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

   [1 mark]

   • A$\sec^22x$
   • B$2\sec^22x$
   • C$2\sec2x$
   • D$\frac12\sec^22x$

   (b)

   Find the gradient of the curve where $x=\frac{\pi}{8}$.

   [1 mark]

   • A$2$
   • B$8$
   • C$4$
   • D$1$

   (c)

   Find the equation of the tangent to the curve at $x=\frac{\pi}{8}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The population $P$ of a colony of bacteria, in thousands, is modelled by $P=5\times2^{0.4t}$, where $t$ is the time in hours after the colony is first observed.

   (a)

   Find $\frac{dP}{dt}$ in terms of $t$.

   [3 marks]

   (b)

   Find the rate of growth of the population when $t=5$, and the time at which the rate of growth first reaches 20 thousand per hour. Give both answers to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question $x$ is measured in radians. You may use the standard limits $\lim_{h\to0}\frac{\sin h}{h}=1$ and $\lim_{h\to0}\frac{\cos h-1}{h}=0$.

   (a)

   Prove from first principles that $\frac{d}{dx}(\sin x)=\cos x$.

   [6 marks]

   (b)

   (i) Prove from first principles that $\frac{d}{dx}(\cos x)=-\sin x$. (ii) Hence find the gradient of $y=3\cos2x$ at $x=\frac{\pi}{4}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions