2.10 Solving equations questions for IB Maths AA

Expand $(a+h{)}^3$.

Show that the gradient of the chord $[\mathrm{P}\mathrm{Q}]$ is $3a^2+3ah+h^2$.

Hence write down an expression, in terms of $a$, for the gradient of the curve at P.

Find the coordinates of the points on the curve where the gradient is $12$.

The tangent to the curve at $(2,8)$ meets the curve again at the point R. Find the coordinates of R.

Now consider the tangent to the curve at a general point $\mathrm{P}(a,a^3)$, where $a\mathrm{\ne }0$. This tangent meets the curve again at the point S.

Show that the gradient of the curve at S is four times the gradient of the curve at P.

Show that the volume of water, $V$ cm${}^3$, is given by $V={\displaystyle \frac{\pi h^3}{27}}$.

The glass contains $150$ cm${}^3$ of water. Find the depth of the water.

Let $\alpha$ be the angle between the side of the glass and its vertical axis. Show that $\sin \alpha ={\displaystyle \frac{1}{\sqrt{10}}}$.

A solid ball of radius $1.5$ cm is dropped into the glass. The ball rests against the side of the glass, with its centre on the axis of the glass.

Show that the centre of the ball is $1.5\sqrt{10}$ cm above the vertex of the glass.

Find the new depth of the water, assuming the ball is completely under the water.

Verify that the ball is completely under the water.

Find the volume of water that must be added for the water to reach the rim of the glass.

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 $f^{\mathrm{\prime }}(x)$ can be written as $3(x-1{)}^2+2$.

Hence explain why $f$ is increasing for all values of $x$.

Using your answer to part (b), explain why the equation $f(x)=0$ has exactly one real solution, and show that this solution lies between $0$ and $1$.

Show that $\cos \mathrm{C}\widehat{\mathrm{A}}\mathrm{B}={\displaystyle \frac{5x-80}{3x}}$.

Find the value of $x$ for which $\mathrm{C}\widehat{\mathrm{A}}\mathrm{B}=90\mathrm{°}$.

Find the set of possible values of $x$.

Show that the area of the enclosure, $E$ m${}^2$, is given by $E=20\sqrt{-x^2+50x-400}$.

Find the maximum area of the enclosure and the value of $x$ for which it occurs.

Find the values of $x$ for which the area of the enclosure is $200$ m${}^2$.

Show that $f^{\mathrm{\prime }}(x)=-e^{-x}(x^2-5x+7)$.

Show that $x^2-5x+7>0$ for all values of $x$.

Hence show that $f$ is decreasing for all values of $x$.

Find the range of $f$ for $0\le x\le 2$.

It is given that ${\displaystyle \underset{x\to \mathrm{\infty }}{\lim }f(x)=0}$.

Find the set of values of $k$ for which the equation $f(x)=k$ has a solution with $x\ge 0$.

Show that the equation $f(x)=x$ has exactly one solution, and that this solution lies between $0$ and $2$.