Special functions derivatives & chain/product/quotient rules (+related rates): notes and practice questions

Assume that a balloon was inflated with warm air and then left outside to cool down, where the temperature, $T$, in $℃$ is modelled accordingly:

$T(t) = 65e^{- 0.062t} + 20,\ t \geq 0$

where $t$ is taken as the time, in minutes, after the balloon was left outside.

$a$ Find the rate of change of temperature with respect to time at $t = 0$.

$b$ According to the given model, find the atmospheric temperature.

If someone said that the green area in a city was said to be changing at a rate of

$\frac{dA}{dt} = 3t^{2} - 5t + 1$

where A is taken as the green area in $m^{2}$ and $t$ is the time in years since 2000.

$a$ Determine according to this model whether there would be an increase or decrease in the green area during 2010.

$b$ Determine the expression of $A(t)$ if the green area was considered to be 765 $m^{2}$ during 2010.

A group of engineers is designing a new observation Ferris wheel. The height, $H(t)$, in metres, of a passenger capsule above the ground is modelled by the function $H(t) = p \cos\left(\frac{\pi}{100}t\right) + q$, where $t$ is the time in seconds after the capsule begins its ascent from the highest point.

The lowest point a capsule reaches is 2 metres above the ground, and the highest point is 32 metres above the ground. The Ferris wheel completes one full rotation in 200 seconds.

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

Using your values from part (a), the function is $H(t) = 15 \cos\left(\frac{\pi}{100}t\right) + 17$.

(i) Find $H'(t)$.

(ii) Find $H''(t)$.

The engineers are particularly interested in the moment when the capsule's vertical speed is at its maximum, for the first time after $t=0$. This occurs at time $t=k$.

Calculate the value of $k$.

Calculate the height of the capsule at this time $k$.

If the velocity, $v\ ms^{- 1}$, of a particle, starting from the origin, at time, $t$ seconds, is $v(t) = t^{2}cos(t^{3})$.

$a$ Find the displacement equation.

$b$ Find acceleration after 1 second.

Consider the function $y = \left( x^{3} - 2x^{2} \right)e^{x}$.

(a) Find $\frac{d^{2}y}{dx^{2}}$.

(b) Find how many points of inflection does this function contain.

(a) A cylindrical grain silo with a radius of $2.5$ m and an initial grain height of $10$ m is being emptied. The rate at which the grain is removed from the silo is proportional to the square root of the height of the grain.

Using $V$ for the volume of grain in the silo and $h$ for the height of the grain, write down a differential equation relating these variables.

(b) Apply the chain rule and the formula for the volume of a cylinder to show that $\frac{dV}{dt} = 6.25\pi \frac{dh}{dt}$. Hence, write your answer to part (a) in terms of $h$ and $t$.

(c) After $30$ minutes, the height of the grain has dropped to $6.25$ m. Predict how long it will take for the silo to be completely empty from the initial height of $10$ m.

Consider the function $y = \left( x^{3} - 2x^{2} \right)e^{x}$.

$a$ Find $\frac{d^{2}y}{dx^{2}}$.

$b$ Find how many points of inflection does this function contain.

A company models the growth of its user base, $f(x)$, and its server capacity, $g(x)$, over time $x$ (in months) using the functions:

$f(x) = 2^x$

$g(x) = -x^2 + 6$

where $f(x)$ and $g(x)$ are in thousands of users.

(a) Solve $f(x) = g(x)$.

(b) (i) Write down the integral that represents the area of the region enclosed between the graphs of $f(x)$ and $g(x)$.

(ii) Calculate the area of this region.

(c) At a certain time $x=k$, the rate of change of the user base is equal to the rate of change of the server capacity. Find the value of $k$.

The diagram shows the slope field for the differential equation $\frac{dy}{dx} = \cos(x-y)$ for $-\pi \le x \le \pi$ and $-\pi \le y \le \pi$.

The local maximum points for solutions to the differential equation lie on the straight line $L_1$.

Find the equation of $L_1$, giving your answer in the form $y = mx + c$.

Find the equation of the straight line $L_2$ on which all local minimum points lie within the given domain, giving your answer in the form $y = mx + c$.

(a) A confectionery company is designing a new wrapper for a chocolate bar. The wrapper is in the shape of a rectangular prism with a square base of side length $x$ cm. The height, $h$ cm, is twice the side length of the base.

Write down an expression for $h$ in terms of $x$.

(b) The chocolate bar has a volume of $250$ cm$^3$.

Find the value of $x$ and $h$.

(c) Calculate the total external surface area of this rectangular prism wrapper.

(d) The company also considers a cylindrical wrapper with radius $r$ cm and height $H$ cm. This wrapper must also hold $250$ cm$^3$ of chocolate.

Find an expression for the height, $H$, of the cylindrical wrapper in terms of $r$.

(e) Let the total external surface area of the cylindrical wrapper be $A$ cm$^2$.

Show that $A = 2\pi r^2 + \frac{500}{r}$.

(f) Find $\frac{dA}{dr}$.

(g.i) Hence or otherwise, find the value of $r$ that will minimize $A$.

(g.ii) Find the minimum value of $A$ for the cylindrical wrapper.

(h) To account for manufacturing waste and overlap, an additional $12\%$ of the calculated surface area is required for the rectangular prism wrapper, and $20\%$ for the cylindrical wrapper.

Determine which wrapper design the company should choose to minimize material usage. Justify your answer.

A landscape architect is designing a modular planter box for a new urban garden. The planter box has a base and top that are identical sectors of a circle, each with radius $r$ cm and angle $\theta$ radians. The height of the planter box is $h = 2$ cm. The total length of metal frame used for all edges of the planter box is $L = 20$ cm. This includes two circular arcs, four radial edges for the top and bottom sectors, and three vertical connecting edges.

(a) Show that $r = \frac{7}{2+\theta}$.

(b) The planter box is designed to hold soil, enclosing a volume, $V$.

(i) Find an expression for $V$ in terms of $\theta$.

(ii) Find the expression for $\frac{dV}{d\theta}$.

(iii) Solve algebraically $\frac{dV}{d\theta} = 0$ to find the value of $\theta$ that will maximize the volume, $V$.

Emily and Liam are researching the adoption of smart home devices in a specific region to create a model predicting future usage. They collect the following data:

Emily proposes the number of devices can be modelled using quadratic regression to find a function of the form $N(x) = ax^2 + bx + c$, where $x$ is the number of years after 2000.

Find the equation of Emily's model.

Emily finds the coefficient of determination for her model is $0.97977$ to five significant figures.

State whether the coefficient of determination supports Emily's proposal. Justify your answer.

Comment on the validity of Emily's model with reference to one of the parameters in the equation.

(i) Find the value of $N'(25)$ and interpret this value in context.

(ii) By considering the changes in device adoption in the table, use the value found in part (d)(i) to comment on the validity of Emily's model.

Liam proposes that the device adoption instead follows a logistic model of the form

$G(x) = \frac{1500}{1+149 \text{e}^{-0.15x}}$

where $x$ is the number of years after 2000 and $G(x)$ is the number of devices in thousands.

State a reason why it may be valid to use Liam's proposal to predict future device adoption.

(i) Find $G'(x)$.

(ii) Hence find the year, according to Liam's model, during which the greatest device adoption growth rate occurred.

The power output $P$, in watts (W), of a solar panel is modelled by the equation

$P = 250 - 1.5\theta^{0.25}$

where $\theta$ is the angle of inclination of the panel to the sun in degrees. Find an expression for $\frac{dP}{d\theta}$.

At a particular moment, the solar panel is inclined at an angle of 30 degrees, and this angle is increasing at a rate of 2 degrees per minute.

Find the rate of change of the power output at this time.

(a) SweetTreats is designing new packaging for a line of artisanal chocolates. The initial design is a box in the shape of a cuboid with a square base of side length $L$ cm. Its height, $H$ cm, is twice the length of the base.

Write down an expression for $H$ in terms of $L$.

(b) The box is designed to hold $250$ cm$^3$ of chocolates.

Find the value of $L$ and $H$.

(c) Calculate the total external surface area of the box.

(d) To minimize the amount of material needed, SweetTreats is considering changing the shape to a cylinder with radius $r$ cm and height $h$ cm. The cylindrical container must also hold $250$ cm$^3$ of chocolates.

Find an expression for the height, $h$, of the container in terms of $r$.

(e) Let the total external surface area of the cylindrical container be $A$ cm$^2$.

Show that $A = 2\pi r^2 + \frac{500}{r}$.

(f) Find $\frac{\text{d}A}{\text{d}r}$.

(g.i) Hence or otherwise, find the value of $r$ that will minimize $A$.

(g.ii) Find the minimum value of $A$ needed for the cylinder.

(h.i) Find $\frac{\text{d}^2 A}{\text{d}r^2}$.

(h.ii) Hence determine whether the graph of $A$ is concave-up or concave-down for $r > 0$. Justify your answer.

A company models the profit from a new product launch using the function $P(t) = 3t(2 - e^{-t})$, where $P$ is the profit in thousands of dollars and $t$ is the time in months since launch.

Find $\frac{dP}{dt}$.

Find $\frac{d^2P}{dt^2}$.

The company observes that the rate of change of profit growth begins to slow down after a certain point, indicating a point of inflexion. Find the time, $t$, in months at which this point of inflexion occurs.

(a) A drone launches a rescue package from an initial position of $\begin{pmatrix} 5 \\ 0 \end{pmatrix}$ metres, relative to an origin on the ground. The package is launched with an initial speed of $15\text{ms}^{-1}$ at an angle $\theta$ to the horizontal ground, where $0 < \theta < \frac{\pi}{2}$.

The velocity components of the package, $t$ seconds after it is launched, are given by $v_x(t) = 15\cos\theta$ and $v_y(t) = 15\sin\theta - 9.8t$.

Find an expression for $x$, the horizontal displacement from the origin, in terms of $\theta$ and $t$.

(b) It is given that the vertical displacement of the package from the ground is $y = (15\sin\theta)t - 4.9t^2$. When the package hits the ground, show that $t = \frac{150\sin\theta}{49}$.

(c) Let $x_g$ be the value of $x$ when the package hits the ground.

Find an expression for $x_g$ in terms of $\theta$ only.

(d) Hence, find the value of $\theta$ which maximizes the value of $x_g$.

(e) The model is adapted to account for a horizontal wind with speed $2\text{ms}^{-1}$ acting in the opposite direction to the initial horizontal motion.

(i) In this new model, the horizontal velocity component is $v_x(t) = 15\cos\theta - 2$. The time taken for the package to hit the ground remains $t = \frac{150\sin\theta}{49}$.

Find an expression for $x_g$, the value of $x$ when the package hits the ground, in terms of $\theta$ only.

(ii) Hence, find the value of $\theta$ which maximizes the value of $x_g$.

A small drone is tracking a moving target. Its position vector, $\vec{r}$, relative to a fixed origin $O$ at time $t$ (in seconds, $t \ge 0$) is given by

$\vec{r}(t) = \begin{pmatrix} t \cos(t) \\ t \sin(t) \end{pmatrix}$.

(a) Find the velocity vector of the drone, $\mathbf{v}(t)$.

(b) Show that the velocity vector of the drone is never parallel to its position vector for $t > 0$.

In this question, marine biologists are studying the population dynamics of a rare species of "Azurefin" fish in a newly established protected reef area.

Historically, the Azurefin population in this region maintained a stable size of $5000$ individuals. Following a period of environmental disturbance, the population was reduced to $1000$ fish. At this point, the area was designated a protected marine reserve, and conservation efforts began, leading to a recovery in the fish population.

Researchers wish to model the size of the Azurefin population, $x$, as a function of $t$, where $t$ is the time, in years, since the establishment of the protected reserve.

Initially, the researchers consider using the logistic model:

$x = \frac{L}{1+Ce^{-kt}}$, where $L, C, k \in \mathbb{R}^+$.

The researchers decide to set $L = 5000$.

State the assumption being made by setting $L = 5000$.

At $t = 0$, the population of Azurefin fish is $1000$.

Find the value of $C$.

At $t = 3$ years, the population of Azurefin fish is found to have increased to $2500$.

Find the value of $k$. Give your answer correct to three significant figures.

Use your model to predict the size of the Azurefin population in the area $6$ years after it became protected. Give your answer correct to the nearest whole number.

An alternative model for population growth is called the Gompertz model. When applied by the researchers to the Azurefin population, this model satisfies the differential equation:

$\frac{dx}{dt} = ax \ln \left( \frac{5000}{x} \right)$, $a \in \mathbb{R}^+$.

Write down the value of $\frac{dx}{dt}$ when $x = 5000$.

Interpret your answer to part (b)(i) in context.

Consider the function $f(x) = \ln (\ln 5000 - \ln x)$, where $0 < x < 5000$.

Show that $f'(x) = \frac{-1}{x \ln \left( \frac{5000}{x} \right)}$.

Hence, use separation of variables to show that the general solution of

$\frac{dx}{dt} = ax \ln \left( \frac{5000}{x} \right)$, where $0 < x < 5000$,

can be written as

$\ln x = \ln 5000 - Ae^{-at}$,

where $A$ is an arbitrary positive constant.

Use the size of the Azurefin population at $t = 0$ to find the value of $A$.

Give your answer in the form $A = \ln p$, where $p \in \mathbb{Z}^+$.

Use the size of the Azurefin population at $t = 3$, given in part (a), to show that $a = 0.281$, correct to three significant figures.

Use the Gompertz model to predict the size of the Azurefin population at $t = 6$. Give your answer correct to the nearest whole number.

After $6$ years, the Azurefin population is measured and is found to be $3200$.

Comment on the predictions made by the two models.

By tracking individual Azurefin fish, the researchers find that about $5\%$ of the population migrates out of the protected area each year.

They decide to adapt the Gompertz model to allow for this. The new model will satisfy the differential equation:

$\frac{dx}{dt} = 0.280799x \ln \left( \frac{5000}{x} \right) - 0.05x$.

Use Euler's method, with a step size of $0.5$ years and an initial value of $x_0 = 2500$ when $t = 3$, to find an estimate for the size of the Azurefin population when $t = 6$.

Give your answer correct to the nearest whole number.

Comment on your answer.

(a) A drone's vertical velocity, $v$ metres per second, at time $t$ seconds, is given by $v = t \sin(2t^2)$.

Find an expression for the vertical acceleration of the drone.

(b) Hence, or otherwise, find its greatest vertical acceleration for $0 \le t \le 3$ seconds.

(c) The drone starts at ground level (displacement is 0). Find an expression for the vertical displacement of the drone.

(d) Hence show that the drone never descends below ground level.

A conical tank is being filled with water. The tank has a height of 12 cm and a radius of 4 cm at its top. Water is poured into the tank such that the volume of water is increasing at a rate of 10 cm$^3$s$^{-1}$.

(a) Find the height of the water in the tank when its volume is 8$\pi$ cm$^3$.

(b) Hence or otherwise, find the rate of change of the height of the water at this instant.