DP Mathematics: Analysis and Approaches Questionbank

Assuming the Maclaurin series for $\cos x$ and $\ln (1+x)$, show that the Maclaurin series for $\cos \left(\ln \right(1+x\left)\right)$ is

$1-\frac{1}{2}{x}^2+\frac{1}{2}{x}^3-\frac{5}{12}{x}^4+\dots$

By differentiating the series in part (a), show that the Maclaurin series for $\sin \left(\ln \right(1+x\left)\right)$ is $x-\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\dots$ .

Hence determine the Maclaurin series for $\tan \left(\ln \right(1+x\left)\right)$ as far as the term in $x^3$.

Use the Maclaurin series for $\ln (1+x)$ to write down the first three non-zero terms of the Maclaurin series for $f\left(x\right)$.

Hence find the first three non-zero terms of the Maclaurin series for $\frac{x}{1+x^2}$.

Use your answer to part (a)(i) to write down an estimate for $f\left(0.4\right)$.

Use the Lagrange form of the error term to find an upper bound for the absolute value of the error in calculating $f(0.4)$, using the first three non-zero terms of the Maclaurin series for $f\left(x\right)$.

With reference to the Lagrange form of the error term, explain whether your answer to part (b) is an overestimate or an underestimate for $f(0.4)$.

Let $g\left(x\right)={\text{e}}^{mx}, m\in \mathrm{\mathbb{Q}}$.

Consider the function $h$ defined by $h\left(x\right)=f\left(x\right)\times g\left(x\right)$ for $x>-1$.

It is given that the $x^2$ term in the Maclaurin series for $h\left(x\right)$ has a coefficient of $\frac{7}{4}$.

Find the possible values of $m$.

By using the Maclaurin series for $\cos x$ and the result from part (a), show that the Maclaurin series for $\text{sec} x$ up to and including the term in $x^4$ is $1+\frac{x^2}{2}+\frac{5x^4}{24}$.

By using the Maclaurin series for $\text{arctan} x$ and the result from part (b), find $\underset{x\to 0}{\lim }\left(\frac{x\text{\hspace{0.05em}arctan} 2x}{\text{sec} x-1}\right)$.

Write down the first three terms of the binomial expansion of $(1+t{)}^{-1}$ in ascending powers of $t$.

Find the value of $a$ and the value of $b$.

State the restriction which must be placed on $x$ for this expansion to be valid.

Hence or otherwise, determine the value of $\underset{x\to 0}{\lim }\left[\frac{{\left(x^2{\mathrm{e}}^x-x^2\right)}^3}{x^9}\right]$.

Prove by mathematical induction that $\frac{d^n}{d{x}^n}\left(x^2{\mathrm{e}}^x\right)=\left[x^2+2nx+n\left(n-1\right)\right]{\mathrm{e}}^x$ for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Hence or otherwise, determine the Maclaurin series of $f\left(x\right)=x^2{\mathrm{e}}^x$ in ascending powers of $x$, up to and including the term in $x^4$.

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

Using the result from part (c), find the Maclaurin series for $g\left(x\right)$ up to and including the $x^4$ term.

Find the first two derivatives of $f\left(x\right)$ and hence find the Maclaurin series for $f\left(x\right)$ up to and including the $x^2$ term.

Using the Maclaurin series for $\text{arctan}x$ and ${\text{e}}^{3x}-1$, find the Maclaurin series for $\text{arctan}\left({\text{e}}^{3x}-1\right)$ up to and including the $x^3$ term.

Hence, or otherwise, find $\underset{x\to 0}{\text{lim}}\frac{f\left(x\right)-1}{\text{arctan}\left({\text{e}}^{3x}-1\right)}$.

Find the perimeter of a regular hexagon, of side length, $x$ units, inscribed in a circle of radius 1 unit.

Show that the coefficient of $x^3$ in the Maclaurin series for $f\left(x\right)$ is zero.

Use an appropriate Maclaurin series expansion to find $\underset{n\to \mathrm{\infty }}{\text{lim}}{P}_i\left(n\right)$ and interpret this result geometrically.

Consider a square of side length, $x$ units, inscribed in a circle of radius 1 unit. By dividing the inscribed square into four isosceles triangles, find the exact perimeter of the inscribed square.

Consider an equilateral triangle ABC of side length, $x$ units, inscribed in a circle of radius 1 unit and centre O as shown in the following diagram.

The equilateral triangle ABC can be divided into three smaller isosceles triangles, each subtending an angle of $\frac{2\pi }{3}$ at O, as shown in the following diagram.

Using right-angled trigonometry or otherwise, show that the perimeter of the equilateral triangle ABC is equal to $3\sqrt{3}$ units.

The inequality found in part (h) can be used to determine lower and upper bound approximations for the value of $\pi$.

Determine the least value for $n$ such that the lower bound and upper bound approximations are both within 0.005 of $\pi$.

Show that $P_c\left(n\right)=2n\text{tan}\left(\frac{\pi }{n}\right)$.

Use the results from part (d) and part (f) to determine an inequality for the value of $\pi$ in terms of $n$.

Show that $P_i\left(n\right)=2n\text{sin}\left(\frac{\pi }{n}\right)$.

By writing $P_c\left(n\right)$ in the form $\frac{2\text{tan}\left(\frac{\pi }{n}\right)}{\frac{1}{n}}$, find $\underset{n\to \mathrm{\infty }}{\text{lim}}{P}_c\left(n\right)$.

By substituting $x=0$, find the value of $a_0$.

Differentiate the equation obtained part (b) and hence, find the first four terms in a power series for ${\left(1+x\right)}^{-2}$.

Hence, by recognising the pattern, deduce the first four terms in a power series for ${\left(1+x\right)}^{-n}$, $n\in {\mathbb{Z}}^{+}$.

Repeat this process to find the first four terms in a power series for ${\left(1+x\right)}^{-3}$.

Hence, write down the first four terms in what is called the Extended Binomial Theorem for ${\left(1+x\right)}^q\text{,}q\in \mathbb{Q}$.

Hence, using integration, find the power series for $\text{arctan}x$, giving the first four non-zero terms.

Write down the power series for $\frac{1}{1+x^2}$.

Expand ${\left(1+x\right)}^5$ using the Binomial Theorem.

Consider the power series $1-x+x^2-x^3+x^4-...$

By considering the ratio of consecutive terms, explain why this series is equal to ${\left(1+x\right)}^{-1}$ and state the values of $x$ for which this equality is true.

By differentiating both sides of the expression and then substituting $x=0$, find the value of $a_1$.

Repeat this procedure to find $a_2$ and $a_3$.

By finding a suitable number of derivatives of $f$, find the first two non-zero terms in the Maclaurin series for $f$.