Find the Maclaurin series for $h(x)$ up to and including the term in $x^5$.

Use your series to estimate $h(0.4)$.

Find the Maclaurin series for $I(x)$ up to and including the term in $x^{11}$.

Use your series to estimate $I(1)$.

Determine the Maclaurin series for $f(x)$ up to and including the term in $x^3$.

Hence find $f'''(0)$.

Find $y'(0)$, $y''(0)$, $y'''(0)$ and $y^{(4)}(0)$.

Hence find the Maclaurin series for $y$ up to and including the term in $x^4$.

Find the possible values of $a$.

Find the Maclaurin series for $f(x)$ up to and including the term in $x^4$.

Find the Maclaurin series for $f(x)$ up to and including the term in $x^3$.

Hence find

Show that $\tan x=x+\dfrac{x^3}{3}+\dfrac{2x^5}{15}+\cdots$.

Using this series, estimate the smallest positive value of $x$ for which $\tan(2x)-2x=0.100$.

Find $y''(0)$ and $y'''(0)$.

Find the Maclaurin series for $y$ up to and including the term in $x^6$.

Find the Maclaurin series for $L(x)$ up to and including the term in $x^6$.

Hence find $\displaystyle \lim_{x\to 0}\frac{\ln(\cos x)+\frac{x^2}{2}+\frac{x^4}{12}}{x^6}$.

Find the Maclaurin series for $p(x)$ up to and including the term in $x^4$.

Using this series, estimate the smaller positive solution (the solution close to $x=0$) of $\arctan(e^x-1)=0.500$.

Write down this polynomial and state the interval of convergence of the corresponding infinite series.

Use a GDC to determine the largest positive value of $x$ for which $|\ln(1-2x)-P_3(x)|=0.005$.

Hence find the Maclaurin series for $f(x)$ up to and including the term in $x^3$.

Hence determine $\lim_{x\to 0}\frac{f(x)-2x+3x^2}{x^3}$

Hence find the Maclaurin series for $G(x)$ up to and including the term in $x^7$.

Using the first four non-zero terms of this series, find an approximate value for $G(1)$ as a single fraction.

Hence determine $\lim_{x\to0}\frac{\dfrac{x}{\sin x}-1-\dfrac{x^2}{6}}{x^4}$

Using the series from part (a), approximate $\int_0^1 \frac{f(x)-1}{x^2}\,\mathrm{d}x$

Justify why every odd derivative of $f$ at $0$ is zero.

By writing $y=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5+\cdots$, determine the Maclaurin series for $y$ up to and including the term in $x^5$.

Use a GDC to determine the largest positive value of $x$ for which

Hence state the interval, symmetric about zero, on which $P_5(x)$ approximates $\sin x$ with absolute error less than $0.001$.

Explain why the Maclaurin series for $\sin x$ itself converges for all real $x$, but the polynomial $P_5(x)$ does not give a uniformly accurate approximation for all real $x$.

Determine the Maclaurin series for $y$ up to and including the term in $x^4$.

Use this series to estimate $y(0.2)$.

Show that the Maclaurin series for $F(x)$, up to and including the term in $x^3$, is $e\left(1-\dfrac{x}{2}+\dfrac{11x^2}{24}-\dfrac{7x^3}{16}+\cdots\right)$.

Hence find $\displaystyle \lim_{x\to 0}\frac{(1+x)^{1/x}-e}{x}$.

Use the series from part (a) to estimate $\pi$.

The next term in the series for $\arcsin x$ is $\frac{35x^9}{1152}$. You may use that, for the terms of this series evaluated at $x=\frac12$, the ratio of each term to the preceding term is less than $\frac14$. Use this to find an upper bound for the error in your estimate in part (b), and state whether the estimate is an underestimate or an overestimate.

Hence find the Maclaurin series for $y$ up to and including the term in $x^5$.

Using this series, approximate $\displaystyle\int_0^1 y\,\mathrm{d}x$.

Hence determine $\lim_{x\to0}\frac{e^{\sqrt3 x}\cos x-1-\sqrt3 x-x^2}{x^4}$

Hence find the Maclaurin series for $y$ up to and including the term in $x^7$.

Using this series, find an approximate value for $y(1)$ as a single fraction.

Hence determine $\lim_{x\to0}\frac{\ln(1+\sin x)-x+\frac{x^2}{2}-\frac{x^3}{6}+\frac{x^4}{12}}{x^5}$

Using this series, approximate $\displaystyle\int_0^1 y\,\mathrm{d}x$.

Let $M(x)=\sqrt{1-x^2}\,L(x)$. Find the Maclaurin series for $M(x)$ up to and including the term in $x^5$.

Hence determine $\lim_{x\to0}\frac{M(x)-2x+\frac{x^3}{3}}{x^5}$

Use $P_6(x)$ to find a polynomial approximation for

Using the approximation from part (b), determine the two positive values of $a$ for which $I(a)=0.240$.

Using your series, estimate the smallest positive solution of $\ln(1+\sin x)=0.250$.

Hence determine the exact value of

Hence determine the Maclaurin series for $y$ up to and including the term in $x^6$.

Using the polynomial found in part (b), estimate the first positive value of $x$ for which $y=0$.