Differentiating exponential, logarithmic and trigonometric functionsEdexcel International A Level Maths: Subtopic test

1. 1
   The curve $C_1$ has equation $y=5e^{2x}-\ln3x$ for $x>0$.

   (a)

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

   [1 mark]

   • A$10e^{2x}-\dfrac{1}{3x}$
   • B$5e^{2x}-\dfrac1x$
   • C$10xe^{2x-1}-\dfrac1x$
   • D$10e^{2x}-\dfrac1x$

   (b)

   Find the gradient of $C_1$ at $x=1$, to 3 significant figures.

   [1 mark]

   • A$73.9$
   • B$35.9$
   • C$72.9$
   • D$74.9$

   (c)

   Find $\dfrac{d^{2}y}{dx^{2}}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C_2$ has equation $y=4\sin3x+\cos2x$, where $x$ is in radians.

   (a)

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

   [1 mark]

   • A$12\cos3x+2\sin2x$
   • B$12\cos3x-2\sin2x$
   • C$4\cos3x-\sin2x$
   • D$-12\cos3x-2\sin2x$

   (b)

   Find the exact gradient of $C_2$ at $x=\dfrac{\pi}{6}$.

   [1 mark]

   • A$-\sqrt3$
   • B$12-\sqrt3$
   • C$\sqrt3$
   • D$-2\sqrt3$

   (c)

   Find $\dfrac{d^{2}y}{dx^{2}}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C_3$ has equation $y=2^{x}$.

   (a)

   Find the gradient of $C_3$ at $x=3$, giving your answer in exact form and to 3 significant figures.

   [3 marks]

   (b)

   Show that the tangent to $C_3$ at $x=3$ meets the $x$-axis at the point where $x=3-\dfrac{1}{\ln2}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C_4$ has equation $y=3\sin2x-2\cos x$ for $0\le x\le\pi$, where $x$ is in radians.

   (a)

   (i) Find $\dfrac{dy}{dx}$.
   (ii) Find the gradient of $C_4$ at $x=\dfrac{\pi}{2}$.
   (iii) Show that the $x$-coordinates of the stationary points of $C_4$ satisfy $6\sin^{2}x-\sin x-3=0$.

   [6 marks]

   (b)

   Hence find the $x$-coordinates of the stationary points of $C_4$. Give your answers in radians to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions