5.13 Limits and L’Hôpital’s rule questions for IB Maths AA

Show that ${\displaystyle {\int }_0^b\frac{1}{x^2+4}\mathrm{d}x=\frac{1}{2}\arctan \left(\frac{b}{2}\right)}$, where $b>0$.

Hence, by considering ${\displaystyle \underset{b\to \mathrm{\infty }}{\lim }{\int }_0^{b}f(x)\mathrm{d}x}$, show that $k={\displaystyle \frac{4}{\pi }}$.

Write down the mode of $X$.

Show that the median of $X$ is $2$.

By using the identity for $\tan 2\theta$, show that $\tan \left({\displaystyle \frac{\pi }{8}}\right)=\sqrt{2}-1$.

Hence find the exact value of the interquartile range of $X$.

Show that $\mathrm{E}(X)$ does not exist.

Show that $\mathbf{a}$ and $\mathbf{v}$ are not perpendicular for any value of $t$.

Show that $1+t^2+t^4=(1+t+t^2)(1-t+t^2)$.

Hence show that $\cos \theta ={\displaystyle \frac{\sqrt{1+t+t^2}}{\sqrt{3(1-t+t^2)}}}$.

Find the values of $t$ for which $\theta ={\displaystyle \frac{\pi }{6}}$.

Find the exact value of ${\displaystyle \underset{t\to \mathrm{\infty }}{\lim }\theta }$.

Find the maximum value of $\theta$ and the value of $t$ at which it occurs.

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.

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 $m\in \mathbb{N}$.

Show that $g$ is an even function.

By considering limits, show that the graph of $y=g(x)$ has a horizontal asymptote and state its equation.

Find the coordinates of the maximum point on the graph of $y=g(x)$.

Hence write down the range of $g$.

Solve the equation $g(x)=1$.

Find the area of the region enclosed by the graph of $y=g(x)$, the $x$-axis and the lines $x=-2$ and $x=2$.

The region described in part (f) is rotated $2\pi$ about the $x$-axis. Find the volume of the solid formed.

Find the coordinates of the points of inflexion on the graph of $y=g(x)$.

Show that ${\displaystyle {\int }_0^1{f}_n(x)\mathrm{d}x=1}$ for all $n\in {\mathbb{Z}}^{+}$.

Consider the case $n=3$.

Find $\mathrm{E}(X_3)$.

Find the median of $X_3$, and hence state whether the median of $X_3$ is greater than or less than $\mathrm{E}(X_3)$.

By using the substitution $u=1-x$, show that $\mathrm{E}(X_n)={\displaystyle \frac{1}{n+1}}$.

Show that $\mathrm{V}\mathrm{a}\mathrm{r}(X_n)={\displaystyle \frac{n}{(n+1{)}^2(n+2)}}$.

Show that the median of $X_n$ is $m_n=1-2^{-\frac{1}{n}}$.

Show that, for $n\in {\mathbb{Z}}^{+}$, the inequality $\mathrm{E}(X_n)>m_n$ is equivalent to ${(1+{\displaystyle \frac{1}{n}})}^n>2$.

By using the binomial theorem, prove that $\mathrm{E}(X_n)>m_n$ for all integers $n\ge 2$.

Use l'Hôpital's rule to find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }n{m}_n}$.

Hence find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{m_n}{\mathrm{E}(X_n)}}$.