2.9 Exponential and logarithmic functions questions for IB Maths AA

Find the value of $k$.

Find the median of $X$.

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

Show that $\mathrm{V}\mathrm{a}\mathrm{r}(X)=\mathrm{E}(X)$.

Solve the equation $f(x)=2$, for $0\le x\le 2\pi$.

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

Hence, find the exact coordinates of the local maximum point and the local minimum point on the graph of $f$, for $0\le x\le 2\pi$.

Find the area of the region bounded by the curve $y=f(x)$, the lines $x=0$ and $x=2\pi$, and the $x$-axis.

Show that $f(x)f(x+\pi )=1$.

Hence, solve the equation $f(x)=2f(x+\pi )$, for $0\le x\le 2\pi$.

Use your GDC to solve the inequality $2^x>x^2$, for $x>0$.

Use your GDC to solve the inequality $3^x>x^3$, for $x>0$.

Show that, for $x>0$, the inequality $a^x>x^a$ is equivalent to ${\displaystyle \frac{\ln x}{x}}<{\displaystyle \frac{\ln a}{a}}$.

Let $g(x)={\displaystyle \frac{\ln x}{x}}$, for $x>0$.

Find the coordinates of the stationary point on the graph of $y=g(x)$ and show that it is a maximum.

Hence show that ${\mathrm{e}}^x\ge x^{\mathrm{e}}$ for all $x>0$, with equality only when $x=\mathrm{e}$.

Hence determine which is larger, ${\mathrm{e}}^{\pi }$ or ${\pi }^{\mathrm{e}}$, justifying your answer.

You are given that $g(x)<0$ for $0<x<1$, $g(1)=0$ and $g(x)\to 0$ as $x\to \mathrm{\infty }$.

Now let $1<a<\mathrm{e}$. Explain why the equation $g(x)=g(a)$ has exactly two solutions for $x>0$, one of which is $x=a$, and the other of which, $x=b$, satisfies $b>\mathrm{e}$.

Hence write down, in terms of $a$ and $b$, the solution of $a^x>x^a$ for $x>0$, where $1<a<\mathrm{e}$, and state the value of $b$ when $a=2$.

Suggest how the solution of $a^x>x^a$ changes when $a>\mathrm{e}$. Use part (b) to support your answer.

Find the set of values of $a>1$ for which $a^x>x^a$ is true for all $x>a$.

Show that $p={\displaystyle \frac{1}{\ln a}}$.

Show that the $y$-coordinate of P is $\mathrm{e}$ for all values of $a>1$.

For one particular value of $a$, the point P has $x$-coordinate $2$.

Find the exact value of $a$ and the exact gradient of the tangent at P for this value of $a$.

Find the value of $k$.

Find the probability that a seed travels more than $2$ metres.

Find the median distance travelled by a seed.

Let $z=y^{1-n}$. Show that the Bernoulli equation can be written as

Consider the differential equation ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}}-{\displaystyle \frac{y}{x}}=-y^2$, for $x>0$, where $y={\displaystyle \frac{1}{2}}$ when $x=1$.

Use part (a) to show that $z={\displaystyle \frac{1}{y}}$ satisfies ${\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x}}+{\displaystyle \frac{z}{x}}=1$.

Hence show that $y={\displaystyle \frac{2x}{x^2+3}}$.

Find the exact coordinates of the maximum point on the graph of $y={\displaystyle \frac{2x}{x^2+3}}$.

The logistic equation ${\displaystyle \frac{\mathrm{d}N}{\mathrm{d}t}}=rN(1-{\displaystyle \frac{N}{K}})$, where $r$ and $K$ are positive constants, models a population $N$ at time $t$. It is given that $N=N_0$ when $t=0$, where $0<N_0<K$.

Show that the logistic equation is a Bernoulli equation, stating the values of $n$, $P$ and $Q$.

Use the substitution $z={\displaystyle \frac{1}{N}}$ to show that $N={\displaystyle \frac{K{N}_0}{N_0+(K-N_0)e^{-rt}}}$.

Write down ${\displaystyle \underset{t\to \mathrm{\infty }}{\lim }N}$.

Consider the differential equation ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}}+y=e^x\sqrt{y}$, where $y=1$ when $x=0$.

Show that the substitution $z=\sqrt{y}$ gives ${\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x}}+{\displaystyle \frac{1}{2}}z={\displaystyle \frac{1}{2}}{e}^x$.

Hence find $y$ in terms of $x$.

Verify that $y=0$ also satisfies the differential equation in part (d), and suggest why the method of part (a) does not produce this solution.