5.10 The second derivative questions for IB Maths AI

Find the number of students who have heard the rumour after 6 hours.

Find the time at which the rumour is spreading most quickly, and write down the number of students who have heard the rumour at that time.

Explain why a logistic model is more suitable than an exponential model in this context.

Write down the number of registered users that the model predicts in the long term.

Show that $C=31$.

Find the value of $k$.

Find the number of registered users predicted after 6 years.

Show that $U^{\mathrm{\prime }}(t)=\frac{80000Ck{\mathrm{e}}^{-kt}}{{(1+C{\mathrm{e}}^{-kt})}^2}$.

Find the time at which the number of registered users is increasing most rapidly, and the rate of increase at that time.

The council claims that the scheme will have more than 90 000 registered users in the long term. Comment on this claim.

Use the trapezoidal rule with all the values in the table to estimate the distance between the two stops.

The tram company's engineer, Mireille, models the velocity by

Find the distance between the stops according to this model.

Find the percentage difference between your answer to part (a) and your answer to part (b)(i), taking the value in part (b)(i) as the true value.

Use differentiation to find the maximum velocity of the tram according to Mireille's model.

Find the maximum acceleration of the tram according to the model, and the time at which it occurs.

For passenger comfort, the magnitude of the acceleration of a tram must never exceed 1.3 m s⁻².

Show that Mireille's model satisfies this condition.

Find the time at which the tram has travelled half of the distance between the stops.

On a new route, two stops are 900 m apart. Mireille uses a model of the same form,

where $T$ seconds is the journey time and $c$ is a positive constant.

Show that $c={\displaystyle \frac{10800}{T^4}}$.

Show that the greatest magnitude of the acceleration during the journey is ${\displaystyle \frac{10800}{T^2}}$.

Hence find the shortest journey time on the new route that satisfies the comfort condition in part (e).

Find the maximum velocity of the tram on this journey.

State one reason why Mireille's model may not be suitable for the new route.

Find the mass of a chick when it hatches.

Write down the mass that the chicken approaches in the long term, according to the model.

Find the age of the chicken when its mass reaches 2 kg.

Find the age of the chicken when it is gaining mass most quickly, and its mass at that time.

Show that $C(v)=0.02v^{\frac{3}{2}}+72v^{-\frac{1}{2}}$.

Find the cost per kilometre when the drone flies at 20 km h⁻¹.

Find $C^{\mathrm{\prime }}(v)$.

Use your answer to part (c) to show that the cost per kilometre is least when $v=\sqrt{1200}$.

Find $C^{\mathrm{\prime }\mathrm{\prime }}(v)$ and hence justify that $v=\sqrt{1200}$ gives a minimum.

Find the minimum cost of a 45 km delivery flight. Give your answer to the nearest thousand KRW.

Show that $h={\displaystyle \frac{500}{\pi r^2}}-{\displaystyle \frac{2r}{3}}$.

Show that the total cost, $C$ USD, of building the silo is given by

Explain why the model is only valid for $0<r<6.20$.

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

Find the value of $r$ and the value of $h$ that minimize the cost, and find this minimum cost to the nearest dollar.

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

A local planning regulation states that the total height of the silo, including the roof, must not exceed 9 m.

Show that the radius must satisfy ${\displaystyle \frac{500}{\pi r^2}}+{\displaystyle \frac{r}{3}}\le 9$.

Find the least value of $r$ that satisfies the regulation.

Explain why the cheapest silo that satisfies the regulation has the radius found in part (g)(ii), and find its cost to the nearest dollar.

Find the percentage increase in the minimum cost caused by the regulation.

Suggest one further cost that the cooperative should include in the model before deciding on the radius of the silo.