Maclaurin series

(a) Using the Maclaurin series for $e^x$ and $\sin x$, find the series expansion of $e^x\sin x$ up to and including the term in $x^3$. [5]

(b) Hence estimate $\displaystyle\int_0^{0.5}e^x\sin x\,dx$, giving your answer as a fraction. [2]

(a) Using the standard series for $\cos x$ and $\ln(1+u)$, show that the Maclaurin series for $\ln(\cos x)$ begins $-\dfrac{x^2}{2}-\dfrac{x^4}{12}$. [5]

(b) Hence find $\displaystyle\lim_{x\to0}\dfrac{\ln(\cos x)+\frac{x^2}{2}}{x^4}$. [2]

(a) Use the Maclaurin series for $e^x$ to find the first four terms of the Maclaurin series for $e^{-2x}$. [2]

(b) Hence find the Maclaurin series for $(1+3x)e^{-2x}$ up to and including the term in $x^2$. [2]

Let $f(x)=\tan x$, $-\dfrac{\pi}{2}

(a) Find $f'(x)$, $f''(x)$ and $f'''(x)$. [3]

(b) Hence show that the Maclaurin series for $\tan x$ up to and including the term in $x^3$ is $x+\dfrac{x^3}{3}$. [2]

(c) Use your answer to part (b) to find the Maclaurin series for $\sec^2 x$ up to and including the term in $x^2$. [2]

(a) Find the Maclaurin series for $(1+x)^{-\frac12}$ up to and including the term in $x^3$. [2]

(b) Hence find the Maclaurin series for $\dfrac{1}{\sqrt{1-x^2}}$ up to and including the term in $x^6$. [2]

(c) By integrating your answer to part (b), find the first four non-zero terms of the Maclaurin series for $\arcsin x$. [2]

(d) Use $x=\dfrac12$ in your answer to part (c) to find an approximation for $\pi$, giving your answer to 4 decimal places. [1]

(a) Use the Maclaurin series for $\ln(1+x)$ to show that $$\ln\left(\frac{1+x}{1-x}\right)=2x+\frac{2x^3}{3}+\frac{2x^5}{5}+\dots$$ [3]

(b) State the set of values of $x$ for which this expansion is valid. [1]

(c) Find the value of $x$ for which $\dfrac{1+x}{1-x}=2$. [1]

(d) Hence, using the three terms given in part (a), find an approximation for $\ln 2$, giving your answer as a fraction in its lowest terms. [1]

(e) Find the percentage error in this approximation. [2]

(a) Write down the Maclaurin series for $\sin 2x$ up to and including the term in $x^3$. [1]

(b) Find the value of the constant $a$ for which $\displaystyle\lim_{x\to0}\frac{\sin 2x-ax}{x^3}$ exists. [2]

(c) For this value of $a$, use Maclaurin series to find $\displaystyle\lim_{x\to0}\frac{\sin 2x-ax}{x(1-\cos x)}$. [3]

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

(a) Show that $\dfrac{d^3y}{dx^3}=2\left(\dfrac{dy}{dx}\right)^2+2y\dfrac{d^2y}{dx^2}$. [2]

(b) Hence find the Maclaurin series for $y$ up to and including the term in $x^3$. [4]

(c) Use your series to estimate $f(0.1)$, giving your answer to 4 significant figures. [1]

(d) Explain why this estimate would be less reliable for $f(0.8)$. [1]

Let $y=e^{\arctan x}$.

(a) Show that $(1+x^2)\dfrac{dy}{dx}=y$. [2]

(b) By differentiating the equation in part (a) twice, show that $$(1+x^2)\frac{d^3y}{dx^3}+4x\frac{d^2y}{dx^2}+2\frac{dy}{dx}=\frac{d^2y}{dx^2}.$$ [3]

(c) Hence find the Maclaurin series for $y$ up to and including the term in $x^3$. [4]

The diagram shows the graph of $y=\cos x$ (solid) and the graph of $y=P_2(x)$ (dashed), where $P_2(x)$ is the Maclaurin polynomial of degree 2 for $\cos x$.

(a) Write down $P_2(x)$. [1]

(b) The Maclaurin polynomial of degree 4 for $\cos x$ is $P_4(x)$. Calculate the percentage error when $P_4(1)$ is used to approximate $\cos 1$. [2]

(c) With reference to the diagram, describe how the accuracy of $P_2(x)$ as an approximation to $\cos x$ changes as $|x|$ increases. [1]

(d) Find the largest interval $-k[2]