5.19 Maclaurin series questions for IB Maths AA

Show that $f^{\mathrm{\prime }}(x)=\sqrt{2}{\mathrm{e}}^x\sin (x+{\displaystyle \frac{\pi }{4}})$.

Prove by mathematical induction that $f^{(n)}(x)=2^{\frac{n}{2}}{\mathrm{e}}^x\sin (x+{\displaystyle \frac{n\pi }{4}})$, for all $n\in {\mathbb{Z}}^{+}$.

Hence find the first four non-zero terms of the Maclaurin series for $f(x)$.

Hence find ${\displaystyle \underset{x\to 0}{\lim }\frac{{\mathrm{e}}^x\sin x-x-x^2}{x^3}}$.

Find, in terms of $m$, the coefficient of $x^{4m+1}$ in the Maclaurin series for $f(x)$, where $m\in \mathbb{N}$.

Find the first three terms of the Maclaurin series for $(1-u{)}^{-\frac{1}{2}}$.

Hence show that $E\approx {\displaystyle \frac{1}{2}}m{v}^2+{\displaystyle \frac{3m{v}^4}{8c^2}}$.

A particle moves at speed $v=0.4c$.

Find the percentage error when the approximation in part (b) is used to find the kinetic energy of the particle.

Show that the Maclaurin series for $\cos x$ up to and including the term in $x^4$ is $1-\frac{x^2}{2}+\frac{x^4}{24}$.

Hence find ${\displaystyle \underset{x\to 0}{\lim }\frac{1-\cos x}{x^2}}$.

Verify your answer to part (b) using L’Hôpital’s rule.

Find $f^{\mathrm{\prime }}(x)$, $f^{\mathrm{\prime }\mathrm{\prime }}(x)$ and $f^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(x)$.

Use mathematical induction to prove that $f^{(n)}(x)={\displaystyle \frac{n!}{(2-x{)}^{n+1}}}$ for all $n\in {\mathbb{Z}}^{+}$.

Show that ${\displaystyle \frac{f^{(n)}(0)}{n!}}={\displaystyle \frac{1}{2^{n+1}}}$.

Consider the series $S(x)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{x}{2^2}}+{\displaystyle \frac{x^2}{2^3}}+{\displaystyle \frac{x^3}{2^4}}+\cdots$, whose coefficients are the values found in part (c).

Show that $S(x)=f(x)$ for $\mathrm{\mid }x\mathrm{\mid }<2$.

It is given that $f(-2)={\displaystyle \frac{1}{4}}$. Show that the series $S(-2)$ does not converge, so $S(x)=f(x)$ does not hold at $x=-2$.

Use the Maclaurin series for $\sin x$ to find the first three non-zero terms of the Maclaurin series for ${\displaystyle \frac{\sin x}{x}}$.

Hence find an approximate value for ${\displaystyle {\int }_0^1\frac{\sin x}{x}\mathrm{d}x}$. Give your answer correct to six significant figures.

Use your GDC to find the percentage error in your answer to part (b).

For $a>0$, explain why $\arg (a+\text{i})=\arctan \left({\displaystyle \frac{1}{a}}\right)$.

Find $(2+\text{i})(3+\text{i})$.

Hence show that $\arctan \left({\displaystyle \frac{1}{2}}\right)+\arctan \left({\displaystyle \frac{1}{3}}\right)={\displaystyle \frac{\pi }{4}}$.

Let $a,b\in \mathbb{R}$, where $a>1$ and $b>1$.

Show that $(a+\text{i})(b+\text{i})=(ab-1)+(a+b)\text{i}$.

Show that $\arctan \left({\displaystyle \frac{1}{a}}\right)+\arctan \left({\displaystyle \frac{1}{b}}\right)={\displaystyle \frac{\pi }{4}}$ if and only if $(a-1)(b-1)=2$.

Hence find all the pairs of integers $a$ and $b$, with $1<a\le b$, such that $\arctan \left({\displaystyle \frac{1}{a}}\right)+\arctan \left({\displaystyle \frac{1}{b}}\right)={\displaystyle \frac{\pi }{4}}$.

Find $(3+\text{i})(7+\text{i})$.

Hence show that $\arctan \left({\displaystyle \frac{1}{3}}\right)+\arctan \left({\displaystyle \frac{1}{7}}\right)=\arctan \left({\displaystyle \frac{1}{2}}\right)$.

Hence show that ${\displaystyle \frac{\pi }{4}}=2\arctan \left({\displaystyle \frac{1}{3}}\right)+\arctan \left({\displaystyle \frac{1}{7}}\right)$.

Show that $(5+\text{i}{)}^4=476+480\text{i}$.

Hence show that $(5+\text{i}{)}^4(239-\text{i})=114244(1+\text{i})$.

Hence show that ${\displaystyle \frac{\pi }{4}}=4\arctan \left({\displaystyle \frac{1}{5}}\right)-\arctan \left({\displaystyle \frac{1}{239}}\right)$.

For small values of $x$, $\arctan x\approx x-{\displaystyle \frac{x^3}{3}}$.

Use this approximation and the result from part (e)(iii) to find an approximate value for $\pi$. Give your answer correct to five significant figures.

Using the same approximation with ${\displaystyle \frac{\pi }{4}}=\arctan 1$ gives $\pi \approx 2.67$. Suggest why the result from part (e)(iii) gives a much better approximation.