5.6 Chain, product and quotient rules questions for IB Maths AA

Show that $r={\displaystyle \frac{24}{2+\theta }}$.

Show that the area, $A$ square metres, of the garden bed is given by $A={\displaystyle \frac{288\theta }{(2+\theta {)}^2}}$.

Find the two values of $\theta$ for which the area of the garden bed is 32 m${}^2$.

Show that ${\displaystyle \frac{\mathrm{d}A}{\mathrm{d}\theta }}={\displaystyle \frac{288(2-\theta )}{(2+\theta {)}^3}}$.

Hence, find the maximum possible area of the garden bed, justifying that it is a maximum.

Show that $h(t)=5.5-1.5\cos \left({\displaystyle \frac{\pi t}{6}}\right)$.

Write down the period of $h$.

Write down the maximum height and the minimum height of the buoy above the seabed.

Find the total time during the 24 hours for which the buoy is more than 6.2 m above the seabed.

Find the maximum rate of increase of the height of the buoy, and the first time at which it occurs.

Find the values of $t$ for which the height of the buoy is increasing at a rate of 0.5 m per hour.

Use this definition to show that the derivative of $x^2$ is $2x$.

Show that $(x+h{)}^3-x^3=h[(x+h{)}^2+(x+h)x+x^2]$.

Hence find the derivative of $x^3$ from first principles.

Let $n\in {\mathbb{Z}}^{+}$, $n\ge 2$, and let $a,b\in \mathbb{R}$. By expanding the right-hand side, show that

Hence prove that the derivative of $x^n$ is $n{x}^{n-1}$, for $n\in {\mathbb{Z}}^{+}$, $n\ge 2$.

Let $f$ and $g$ be polynomials and let $P(x)=f(x)g(x)$.

Show that $P(x+h)-P(x)=f(x+h)[g(x+h)-g(x)]+g(x)[f(x+h)-f(x)]$.

You may assume that ${\displaystyle \underset{h\to 0}{\lim }f(x+h)=f(x)}$ for any polynomial $f$.

Hence show that $P^{\mathrm{\prime }}(x)=f(x)g^{\mathrm{\prime }}(x)+g(x)f^{\mathrm{\prime }}(x)$.

Use the result of part (f) to show that the derivative of ${[f(x)]}^2$ is $2f(x)f^{\mathrm{\prime }}(x)$.

Use mathematical induction to prove that, for all $n\in {\mathbb{Z}}^{+}$, the derivative of ${[f(x)]}^n$ is $n{[f(x)]}^{n-1}{f}^{\mathrm{\prime }}(x)$.

Let $q(x)={(x^2-3x+1)}^5$. Use the result of part (g)(ii) to find $q^{\mathrm{\prime }}(2)$, and check your answer using your GDC.

Find the exact height of the seat when $t=9$.

Write down the maximum height of the seat and the value of $t$ at which it occurs.

Find the values of $t$ for which $h(t)=29$.

Hence, find the length of time during one revolution for which the seat is more than 29 m above the ground.

Find the rate at which the height of the seat is changing when $t=2$.

Find the values of $t$ at which the seat is descending at a rate of ${\displaystyle \frac{3\pi }{4}}$ metres per minute.

Write down the maximum height of cabin P and the first time at which it reaches this height.

Let $d(t)=h_Q(t)-h_P(t)$. Show that $d(t)=10\cos \left({\displaystyle \frac{\pi t}{6}}\right)+10\sqrt{3}\sin \left({\displaystyle \frac{\pi t}{6}}\right)$.

By expanding $20\sin ({\displaystyle \frac{\pi t}{6}}+{\displaystyle \frac{\pi }{6}})$, show that $d(t)=20\sin ({\displaystyle \frac{\pi t}{6}}+{\displaystyle \frac{\pi }{6}})$.

Write down the greatest value of $d(t)$, and find the first time at which it occurs.

Find the length of time, during the first revolution, for which cabin Q is more than 15 m higher than cabin P.

Find the rate of change of $d$ when $t=5$.

Consider the equation $x^3-3x+c=0$, where $c\in \mathbb{R}$.

By sketching the graph of $y=x^3-3x+c$, or otherwise, write down the number of distinct real roots of the equation when $c=1$.

Write down the number of distinct real roots of the equation when $c=2$.

Write down the number of distinct real roots of the equation when $c=3$.

Factorise $x^3-3x+2$ completely.

Let $p(x)$ be a polynomial with a repeated root $x=a$, so that $p(x)=(x-a{)}^{2}g(x)$, where $g(x)$ is a polynomial.

Show that $p(a)=0$ and $p^{\mathrm{\prime }}(a)=0$.

Consider now the equation $x^3+px+q=0$, where $p,q\in \mathbb{R}$ and $p\mathrm{\ne }0$. Suppose that this equation has a repeated root $x=a$.

Use part (c) to show that $3a^2+p=0$.

Show that $a=-{\displaystyle \frac{3q}{2p}}$.

Hence show that $4p^3+27q^2=0$.

Verify that the equation in part (b) satisfies this condition.

Find the values of $k$ for which the equation $x^3-12x+k=0$ has a repeated root. For each of these values of $k$, find all the roots of the equation.

By using the substitution $x=y-2$, find the values of $c$ for which the equation $x^3+6x^2+3x+c=0$ has a repeated root.

Give your answers in the form $m\pm n\sqrt{3}$, where $m,n\in \mathbb{Z}$.

Suppose now that $p<0$. The graph of $y=x^3+px+q$ has stationary points where $x=\pm s$, where $s=\sqrt{-{\displaystyle \frac{p}{3}}}$.

Show that the product of the $y$-coordinates of the two stationary points is $q^2+{\displaystyle \frac{4p^3}{27}}$.

Hence state the condition on $p$ and $q$ for the equation $x^3+px+q=0$ to have three distinct real roots. Justify your answer.