1. 1
   Let $f(x)=e^{2x}$ and $g(x)=\sqrt{1+x}$, for $|x|<1$.

   (a)

   Find the coefficient of $x^{3}$ in the Maclaurin series for $f(x)$.

   [1 mark]

   • A$8$
   • B$\dfrac13$
   • C$\dfrac43$
   • D$\dfrac16$

   (b)

   Find the first three terms of the Maclaurin series for $g(x)$.

   [1 mark]

   • A$1+\dfrac{x}{2}-\dfrac{x^{2}}{8}$
   • B$1+\dfrac{x}{2}+\dfrac{x^{2}}{8}$
   • C$1+\dfrac{x}{2}-\dfrac{x^{2}}{4}$
   • D$1-\dfrac{x}{2}+\dfrac{3x^{2}}{8}$

   (c)

   Use the Maclaurin series for $f(x)$ to find $\displaystyle\lim_{x\to0}\frac{e^{2x}-1-2x}{x^{2}}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $h(x)=\ln(1+3x)$, for $-\dfrac13<x\le\dfrac13$.

   (a)

   Find the first three terms of the Maclaurin series for $h(x)$.

   [1 mark]

   • A$3x-\dfrac32x^{2}+x^{3}$
   • B$3x-\dfrac92x^{2}+\dfrac92x^{3}$
   • C$3x+\dfrac92x^{2}+9x^{3}$
   • D$3x-\dfrac92x^{2}+9x^{3}$

   (b)

   By differentiating your series from part (a), find the first three terms of the Maclaurin series for $\dfrac{1}{1+3x}$.

   [1 mark]

   • A$1-3x+9x^{2}$
   • B$3-9x+27x^{2}$
   • C$1+3x+9x^{2}$
   • D$x-\dfrac32x^{2}+3x^{3}$

   (c)

   Use the three terms found in part (a) to find an approximation for $\ln1.3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $p(x)=e^{x}\sin x$.

   (a)

   Using the Maclaurin series for $e^{x}$ and $\sin x$, show that the Maclaurin series for $p(x)$, up to and including the term in $x^{3}$, is $x+x^{2}+\dfrac13x^{3}$.

   [3 marks]

   (b)

   Hence find an approximate value for $\displaystyle\int_0^{0.2}e^{x}\sin x\,dx$. Give your answer correct to three significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $y=f(x)$ satisfies the differential equation $\dfrac{dy}{dx}=x+y^{2}$, with $y=1$ when $x=0$.

   (a)

   (i) Show that $\dfrac{d^{2}y}{dx^{2}}=1+2y\dfrac{dy}{dx}$.
   (ii) Find an expression for $\dfrac{d^{3}y}{dx^{3}}$ in terms of $y$, $\dfrac{dy}{dx}$ and $\dfrac{d^{2}y}{dx^{2}}$.
   (iii) Hence find the Maclaurin series for $y$, up to and including the term in $x^{3}$.

   [6 marks]

   (b)

   (i) Use your series to find an approximation for $f(0.1)$, giving your answer correct to four significant figures.
   (ii) Use your series to find $\displaystyle\lim_{x\to0}\frac{y-1-x}{x^{2}}$.
   (iii) A numerical solver gives $f(0.5)=2.235$ to four significant figures. Use your series to find an approximation for $f(0.5)$, and explain why this approximation is much less accurate than your approximation for $f(0.1)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions