5.3 Differentiating polynomials questions for IB Maths AA

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

Find the coordinates of the two stationary points of the graph of $f$.

Find the equation of the tangent to the graph of $f$ at the point where $x=4$.

Find the angle of inclination of the road at the point where $x=50$, giving your answer in degrees.

Find the values of $x$ at which the angle of inclination of the road is ${10}^{\circ }$.

Consider the case $p(x)=2x^3-5x^2+4x-7$.

Find $p^{\mathrm{\prime }}(x)$, $p^{\mathrm{\prime }\mathrm{\prime }}(x)$ and $p^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(x)$.

Find $p(0)$, $p^{\mathrm{\prime }}(0)$, $p^{\mathrm{\prime }\mathrm{\prime }}(0)$ and $p^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(0)$. Hence verify that ${\displaystyle \frac{p^{(k)}(0)}{k!}}$ is equal to the coefficient of $x^k$ in $p(x)$, for $k=0,1,2,3$.

Now consider the general polynomial $p$. Show that $p^{(k)}(0)=k!a_k$, for $k=0,1,\dots ,n$.

Hence write $p(x)$ in terms of $x$ and the values $p^{(k)}(0)$, for $k=0,1,\dots ,n$.

Use part (c) with $p(x)=(1+x{)}^n$ to show that the coefficient of $x^k$ in the expansion of $(1+x{)}^n$ is ${\displaystyle \frac{n!}{k!(n-k)!}}$.

Fix a value of $x$, and let $q(h)=p(x+h)$, a polynomial in $h$ of degree $n$.

By applying part (c) to $q$, show that

Hence show that ${\displaystyle \underset{h\to 0}{\lim }{\displaystyle \frac{p(x+h)-p(x)}{h}}=p^{\mathrm{\prime }}(x)}$.

Let $p(x)=x^4-2x^3$. Use the result in part (e) with $x=1$ to express $p(1+h)$ as a polynomial in $h$. Hence find the exact value of $p(1.1)$, and check your answer using your GDC.

The polynomial $p$ is said to have a repeated root at $x=a$ if $p(x)=(x-a{)}^{2}r(x)$ for some polynomial $r$.

Show that $p$ has a repeated root at $x=a$ if and only if $p(a)=0$ and $p^{\mathrm{\prime }}(a)=0$.

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

Show that $h={\displaystyle \frac{45}{r^2}}-{\displaystyle \frac{2r}{3}}$.

The external surface of the tank, including the base, is $S$ m${}^2$. Show that $S={\displaystyle \frac{5}{3}}\pi r^2+{\displaystyle \frac{90\pi }{r}}$.

Find ${\displaystyle \frac{\mathrm{d}S}{\mathrm{d}r}}$.

Find the value of $r$ for which $S$ has a minimum value.

Use the second derivative to justify that this value of $r$ gives a minimum value of $S$.

Find the minimum value of $S$, and the corresponding height of the cylinder.

Find the value of $a$ and the value of $b$.

Interpret, in context, the value of $a$.

Use the regression line to estimate the number of tickets sold when the ticket price is 42 euros.

The promoter uses the regression line to model the number of tickets sold when the ticket price is $x$ euros. The revenue from ticket sales is $R(x)$ euros.

Write down an expression for $R(x)$ in terms of $x$.

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

Hence find the ticket price that maximises the revenue. Justify that this price gives a maximum.

Find the maximum revenue.

The next concert is held at a venue with 1200 seats, so at most 1200 tickets can be sold. The promoter continues to use the model.

Find the ticket price that maximises the revenue at this venue, and the corresponding revenue. Justify your answer.

Show that $f(1+h)=2+h-h^2$.

The point $\mathrm{Q}(1+h,f(1+h))$, where $h\mathrm{\ne }0$, also lies on the graph of $f$.

Show that the gradient of the chord $[\mathrm{P}\mathrm{Q}]$ is $1-h$.

Hence find the equation of the tangent at P.