5.6 Stationary points questions for IB Maths AI

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

Find the coordinates of the two points on the graph of $f$ where the gradient is zero.

Determine the nature of the stationary point with $x=-1$. Justify your answer.

Find ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}}$.

Hence find the maximum height of the tunnel.

The table shows some heights of the roof.

Find the value of $p$ and the value of $q$.

Use the trapezoidal rule with the values in the table to estimate the area of the cross-section of the tunnel.

(i) Write down an integral for the area of the cross-section of the tunnel.

(ii) Hence find the area of the cross-section.

Find the percentage error in the estimate from part (d).

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.

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

Use your answer to part (a) to find the coordinates of the local minimum point of the graph of $f$.

Write down the range of $f$.

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

Hence find the coordinates of the two stationary points on the northern edge.

Find the coordinates of the most northerly point of the plot of land. Justify your answer.

Find the area of the plot of land.

A straight footpath is to be built along the tangent to the northern edge at the point where $x=20$.

Find the equation of the tangent to the curve $y=f(x)$ at $x=20$.

Write down the distance from $O$ to the point where the footpath meets the western edge.

Write down the value of $d$ and a system of three equations satisfied by $a$, $b$ and $c$.

Find the values of $a$, $b$ and $c$.

Use calculus to find the maximum internal radius of the decanter and the height at which it occurs.

Find the internal radius at the narrowest point of the neck.

Find the capacity of the decanter, in litres.

Before fitting the curve, Lucía measured the internal radius of a prototype every 4 cm.

Use the trapezoidal rule with the cross-sectional areas $\pi r^2$ at these heights to estimate the capacity of the prototype, in cm³.

Find the percentage difference between your answers to parts (e) and (f)(i), taking the answer to part (e) as the true value.

The decanter stands upright and 1 litre of wine is poured into it.

Find the depth of the wine.

Wine is poured into the empty decanter at a constant rate of $50{\text{ cm}}^3{\text{ s}}^{-1}$.

Find the rate at which the depth of the wine is increasing when the depth is 10 cm.

Lucía also makes a larger version of the decanter in which every length is multiplied by 1.2.

Find the capacity of the larger decanter, in litres.

Give one reason why the capacity found in part (e) may be different from the volume of wine the decanter actually holds.