5.7 The second derivative questions for IB Maths AA

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.

Solve the differential equation.

Apply Euler's method, with a step length of $0.25$, to estimate the value of $y$ at $x=1$.

Show that, with a step length of ${\displaystyle \frac{1}{n}}$, where $n\in {\mathbb{Z}}^{+}$, Euler's method gives the estimate ${(1+{\displaystyle \frac{1}{n}})}^n$ for the value of $y$ at $x=1$.

By considering the concavity of the solution curve, explain why every estimate in part (c) is less than $e$.

By considering $\ln {(1+{\displaystyle \frac{1}{n}})}^n$, use l'Hôpital's rule to show that ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{1}{n})}^n=e}$.

Let $E(n)=e-{(1+{\displaystyle \frac{1}{n}})}^n$ be the error in the estimate.

Use your GDC to find $E(n)$ and $nE(n)$ for $n=10,20,40$ and $80$. Give your answers correct to four significant figures.

Hence describe the effect on the error of halving the step length, and suggest the limit of $nE(n)$ as $n\to \mathrm{\infty }$.

Use the Maclaurin series for $\ln (1+x)$ to show that $n\ln (1+{\displaystyle \frac{1}{n}})=1-{\displaystyle \frac{1}{2n}}+{\displaystyle \frac{1}{3n^2}}-\dots$, and hence show that $E(n)\approx {\displaystyle \frac{e}{2n}}$ for large $n$.

Use the approximation $E(n)\approx {\displaystyle \frac{e}{2n}}$ to estimate the least value of $n$ for which the error is less than $0.001$.

Use your GDC to find the least value of $n$ for which $E(n)<0.001$.

Find $f(x)$.

Find the coordinates of the stationary points of the curve.

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

Hence determine the nature of each stationary point.

The curve $y=f(x)+k$, where $k\in \mathbb{R}$, meets the $x$-axis at exactly two points.

Find the possible values of $k$.

Find the area of the region bounded by the curve $y=f^{\mathrm{\prime }}(x)$ and the $x$-axis.

Let $k=0$. Find the values of $x$ for which $f$ is increasing.

Let $k=0.1$. Find the values of $x$ for which $f$ is decreasing.

For general $k$, show that $f^{\mathrm{\prime }\mathrm{\prime }}(x)=(x-2)e^{-x}$.

Hence show that the least value of $f^{\mathrm{\prime }}(x)$ is $k-e^{-2}$.

Find the set of values of $k$ for which $f$ is increasing for all $x\in \mathbb{R}$.

Show that $f^{\mathrm{\prime }}(x)={\displaystyle \frac{2-\ln x}{2x\sqrt{x}}}$.

The graph of $f$ has a local maximum at the point P. Find the coordinates of P.

It is given that $f^{\mathrm{\prime }\mathrm{\prime }}(x)={\displaystyle \frac{3\ln x-8}{4x^2\sqrt{x}}}$.

Use $f^{\mathrm{\prime }\mathrm{\prime }}(x)$ to show that P is a local maximum.

The graph of $f$ has a point of inflexion at Q. Find the $x$-coordinate of Q, justifying that it is a point of inflexion.

Write down the $x$-intercept of the graph of $f$.

Hence show that $\ln x\le {\displaystyle \frac{2}{e}}\sqrt{x}$ for all $x>0$.