5.15 Derivatives of further functions questions for IB Maths AA

Show that $g(\pi -x)=g(x)$.

Let $I={\displaystyle {\int }_0^{\pi }xg(x)dx}$. By using the substitution $u=\pi -x$, show that

By using the substitution $v=\cos x$, show that ${\displaystyle {\int }_0^{\pi }g(x)dx={\int }_{-1}^1\frac{1}{1+v^2}dv}$.

Hence find the exact value of ${\displaystyle {\int }_0^{\pi }g(x)dx}$.

Hence find the exact value of ${\displaystyle {\int }_0^{\pi }\frac{x\sin x}{1+{\cos }^{2}x}dx}$.

By using a similar method, or otherwise, find the exact value of ${\displaystyle {\int }_0^{\pi }x{\sin }^{3}xdx}$.

Show that $\theta =\arctan \left({\displaystyle \frac{3}{x}}\right)-\arctan \left({\displaystyle \frac{1.2}{x}}\right)$.

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

Find ${\displaystyle \frac{d\theta }{dx}}$.

Hence show that the maximum value of $\theta$ occurs when $x=\sqrt{3.6}$.

The visitor walks directly towards the wall at a constant speed of 0.5 m s${}^{-1}$.

Find the rate of change of $\theta$ when the visitor is 4 m from the wall.

Find the range of values of $x$ for which $\theta >0.3$.

Find the exact value of $f\left(\frac{\pi }{6}\right)$.

Find the exact value of $f\left(\frac{5\pi }{6}\right)$.

Find the exact value of $f\left(\frac{7\pi }{6}\right)$.

Use your GDC to sketch the graph of $y=f(x)$ for $-\pi \le x\le 2\pi$. Clearly indicate the coordinates of any maximum and minimum points.

Show that $f(x)=\pi -x$ for $\frac{\pi }{2}\le x\le \frac{3\pi }{2}$.

Write down the period of $f$.

Hence write down an expression for $f(x)$ for $\frac{3\pi }{2}\le x\le \frac{5\pi }{2}$.

Show that $f$ is an odd function.

Write down an expression for $g(x)$ for $0\le x\le \pi$.

Show that $g(x)=2\pi -x$ for $\pi \le x\le 2\pi$.

Let $h(x)=f(x)+g(x)$, for $0\le x\le 2\pi$.

Find an expression for $h(x)$ on each of the intervals $0\le x\le \frac{\pi }{2}$, $\frac{\pi }{2}\le x\le \pi$, $\pi \le x\le \frac{3\pi }{2}$ and $\frac{3\pi }{2}\le x\le 2\pi$.

Find all the solutions of the equation $f(x)={\displaystyle \frac{x}{2}}$, for $x\ge 0$.

Show that, for $\cos x\mathrm{\ne }0$, $f^{\mathrm{\prime }}(x)={\displaystyle \frac{\cos x}{\mathrm{\mid }\cos x\mathrm{\mid }}}$.

Hence explain why $f$ is not differentiable at $x=\frac{\pi }{2}$.

Find the exact value of ${\displaystyle {\int }_0^{10\pi }\mathrm{\mid }f(x)\mathrm{\mid }dx}$.

Show that $k={\displaystyle \frac{3}{\pi }}$.

Find the median of $X$.

Find $\mathrm{E}(X)$, giving your answer in the form ${\displaystyle \frac{a\ln b}{\pi }}$, where $a,b\in {\mathbb{Z}}^{+}$.

Find the speed at which P moves along the shoreline when $x=300$.

Find the distance OP at the instant when P is moving along the shoreline at $1000$ m s${}^{-1}$.

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