2.14 Odd, even and self-inverse functions questions for IB Maths AA

Show that $g$ is an odd function.

Hence find the exact value of ${\displaystyle {\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}(x^2\sin x+\cos 2x)dx}$.

Show that $f$ is an even function.

Use your graphic display calculator to solve the equation $f(x)=0.3$ for $0\le x\le 2\pi$.

Hence write down the solutions of $f(x)=0.3$ for $-2\pi \le x<0$.

Use your graphic display calculator to find the value of $I_1$, the value of $I_2$ and the value of $I_3$.

Suggest an exact value for $I_n$.

Show that $\sin ({\displaystyle \frac{\pi }{2}}-x)=\cos x$ and $\cos ({\displaystyle \frac{\pi }{2}}-x)=\sin x$.

By using the substitution $u={\displaystyle \frac{\pi }{2}}-x$, show that

Hence prove that your suggestion in part (b) is correct for all $n\in {\mathbb{Z}}^{+}$.

Now consider a function $f$ that is continuous for $0\le x\le a$, where $a>0$, and such that $f(x)+f(a-x)\mathrm{\ne }0$ for $0\le x\le a$.

Show that

Hence find the value of ${\displaystyle {\int }_0^4\frac{\sqrt{x}}{\sqrt{x}+\sqrt{4-x}}dx}$.

Now let $g$ be an even function that is continuous for $-a\le x\le a$.

Show that

Hence find the exact value of ${\displaystyle {\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}\frac{\cos x}{1+e^x}dx}$.

Show that $f(2\pi -x)=-f(x)$.

Hence describe the symmetry of the graph of $y=f(x)$.

Find the $x$-intercepts of the graph of $f$.

Show that $f^{\mathrm{\prime }}(x)=2(2\cos x-1)(\cos x+1)$.

Find the coordinates of the local maximum point on the graph of $f$.

Hence write down the coordinates of the local minimum point on the graph of $f$.

Find the total area of the regions enclosed by the graph of $f$ and the $x$-axis.

Write down the value of ${\displaystyle {\int }_{\frac{\pi }{2}}^{\frac{3\pi }{2}}f(x)dx}$.

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 $y=g(x)$.

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 ${\displaystyle {\int }_0^{10\pi }\mathrm{\mid }f(x)\mathrm{\mid }dx}$.