5.9 Further differentiation questions for IB Maths AI

Write $f(x)$ in the form $a{x}^p+b{x}^q$, where $a,b,p,q\in \mathbb{Q}$.

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

Find the gradient of the graph of $y=f(x)$ at the point where $x=8$.

Write down the greatest angle between the rod and the vertical, in radians. Hence find this angle in degrees.

Find the length of the arc travelled by the bob as it swings from one extreme position to the other.

Find the period of the model.

Find all the times in the first 3 seconds at which $\theta =0.15$.

The horizontal displacement of the bob from its lowest position is $x=1.2\sin \theta$ metres. Find the greatest value of $x$.

Find ${\theta }^{\mathrm{\prime }}(t)$.

The speed of the bob is $1.2\mid {\theta }^{\mathrm{\prime }}(t)\mid$ ${\text{m s}}^{-1}$. Find the greatest speed of the bob and the first time at which it occurs.

Suggest one reason why this model may not be suitable for large values of $t$.

Write down a differential equation for $T$, using a positive constant $k$.

By solving your differential equation, show that $T=22+A{\mathrm{e}}^{-kt}$, where $A$ is a constant.

Write down the value of $A$.

After 20 minutes the core temperature of a loaf is 61 °C.

Show that $k=0.0320$, correct to three significant figures.

A loaf can be sliced once its core temperature has fallen to 35 °C.

Use the model to find how long after leaving the oven a loaf can be sliced. Give your answer to the nearest minute.

Find the rate at which the core temperature is changing at the moment the loaf leaves the oven.

The head baker, Jonas, thinks that $k$ should be estimated from more than one reading. He records the core temperature of a loaf every 10 minutes.

Complete the table, giving each value of $y$ correct to three decimal places.

Find the equation of the regression line of $y$ on $t$.

Hence find Jonas's estimates for $k$ and for $A$.

Write down the value of the coefficient of determination, $R^2$, for the regression in part (f)(ii), and interpret it in context.

On busy days the ovens warm the cooling room. Jonas models the room temperature as $22+0.15t$ °C and uses the differential equation

Use Euler's method with a step length of 5 minutes to estimate the core temperature of a loaf after 40 minutes on a busy day.

On a busy day, the measured core temperature of a loaf after 40 minutes is 45.2 °C. Suggest how Jonas could use Euler's method to obtain a more accurate estimate.

Show that $C^{\mathrm{\prime }}(t)=3{\mathrm{e}}^{-0.25t}(4-t)$.

Hence find the maximum concentration of the painkiller in Kamala's blood, and the time at which it occurs.

Find $C^{\mathrm{\prime }}(10)$.

Interpret your answer to part (c) in context.

Write down the depth of water at high tide and at low tide.

Find the time between two consecutive high tides. Give your answer in hours and minutes, correct to the nearest minute.

Find the time of the first high tide after midnight. Give your answer to the nearest minute.

The ferry Corsaire needs a depth of at least 6 m to dock at the jetty. Find the length of time between midnight and noon during which the Corsaire can dock. Give your answer to the nearest minute.

Find $D^{\mathrm{\prime }}(t)$.

Hence write down the greatest rate at which the depth of water decreases, and find the first time after midnight at which this occurs.

Suggest one reason why the model may not be suitable for predicting the depth of water a month later.

Find the equation of the regression line of $P$ on $v$.

Write down the coefficient of determination, $R^2$, for this linear model.

Calculate the residuals of the linear model at $v=4$, $v=8$ and $v=12$, and hence explain why a linear model is not appropriate, despite the high value of $R^2$.

Tomasz next considers a power model $P=a{v}^b$.

Show that this model gives a linear relationship between $\ln P$ and $\ln v$.

Use your graphic display calculator to find the values of $a$ and $b$.

For a model $P=f(v)$, the sum of square residuals is $S{S}_{res}={\sum }_{i=1}^9(P_i-f(v_i){)}^2$.

Find the value of $S{S}_{res}$ for the power model.

Physical theory suggests that the power output is proportional to the cube of the wind speed, so Tomasz now considers the model $P=k{v}^3$. He chooses $k$ to minimize

Show that $S(k)$ is minimized when $k={\displaystyle \frac{\sum v_i^3{P}_i}{\sum v_i^6}}$.

Hence find the value of $k$.

For the model $P=k{v}^3$, $S{S}_{res}=516$, correct to three significant figures.

Tomasz notices that this one-parameter model has a smaller value of $S{S}_{res}$ than the power model in part (b), which has two parameters. Explain why this is possible, and state which of the two models fits the data better.

On one day, the wind speed at the site $t$ hours after midnight is modelled by $v=7+3\sin \left(\frac{\pi t}{12}\right)$, for $0\le t\le 24$.

The energy, in kWh, produced by the turbine in one day is ${\int }_0^{24}P\mathrm{d}t$. Use the model $P=k{v}^3$ to find the energy produced that day.

The turbine's maximum power output is 1500 kW. Comment on using the model $P=k{v}^3$ to predict the power output when the wind speed is 15 m s⁻¹.