Limits & derivative definition, increasing / decreasing functions: notes and practice questions

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 botanical garden features a winding path for visitors. The path can be modelled by the function $f(x) = x^3 - 3x^2 - 9x + 5$. All distances in the garden are in kilometres.

A new straight maintenance path needs to be constructed. This path will start at point P on the visitor path, which has coordinates $(2, -17)$.

(a) Using your graphic display calculator, find the value of $f'(2)$.

(b) Find the equation of the line normal to $f(x)$ at point P.

(c) The maintenance path connects point P to another point Q on the visitor path, where the normal line intersects the path again. For safety regulations, point Q must have a positive x-coordinate. Determine the length of this new maintenance path.

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 $y = mx + c$.

The concentration of a reactant A, in mg/L, in a chemical reaction over time $t$ (in minutes) is modelled by the function:

$C(t) = -t^3 + 15t^2 - 48t + 100$, for $0 \le t \le 10$.

(a) Find the coordinates of the local minimum point of the concentration.

(b) Find the coordinates of the local maximum point of the concentration.

(c) Find the set of values of $t$ for which the concentration of reactant A is above $100$ mg/L.

A high-speed drone is being tested, and its acceleration, $a$, is modelled by the function $a(t) = \frac{dv}{dt} = -0.8t + 20$, where $v$ is the speed of the drone in m/s and $t$ is the time in seconds, for $0 \le t \le 30$ seconds.

Determine whether the speed of the drone is increasing or decreasing at $t = 28$ seconds.

It is observed that when $t = 5$ seconds, the speed of the drone is $90$ m/s.

Find an expression for the function $v(t)$.

(a) When the profit is zero, find the possible number of units produced, $x$.

(b) Determine the positive values of profit, $P$, for which there is only one positive value of $x$ (units produced).

A chemical reaction is monitored in a laboratory. The concentration of a specific compound, C, in milligrams per litre (mg/L), changes over time, t, in minutes.

The rate of change of the compound's concentration is modelled by

$\frac{\mathrm{d}C}{\mathrm{d}t} = -0.5t + 10, \quad t \ge 0$

At 8 minutes, the concentration of the compound is 70 mg/L.

Find an expression for C in terms of t.

The experiment continues for an extended period.

Describe how the compound's concentration changes if the time elapsed is between 25 minutes and 35 minutes. Justify your answer.

(a) An architect is designing a decorative archway for a park entrance. The cross-section of one half of the archway is modeled. The archway is symmetrical about the y-axis.

The architect models the base section of the archway as a straight line passing through the points $(0, 2)$ and $(2, 4)$, where all units are in metres.

Find the equation of the line passing through these two points.

(b) The architect initially models the curved upper section of the archway using the following measured points:

$(2, 4)$, $(4, 5)$, $(5.5, 3)$, and $(7, 0)$.

(i) Find the equation of the least squares regression quadratic curve for these four points.

(ii) By considering the gradient of this curve when $x = 2$, explain why it may not be a good model for the archway.

(c) The architect decides that a better model for the curved section would be a quadratic curve with a maximum point at $(4.5, 5.5)$ and that passes through the endpoint $(7, 0)$.

Find the equation of this new quadratic model.

(d) Believing this to be a better model for the archway, the architect wants to estimate the volume of the solid generated by rotating this half-archway about the x-axis.

(i) Write down an expression for this estimate of the volume as a sum of two integrals.

(ii) Find the value of this estimate.

The rate of change of the volume of water in a reservoir, in thousands of cubic meters per year, is given by the equation:

$\frac{\mathrm{d}V}{\mathrm{d}t} = 6t^2 - 20t$

where $V$ is the volume of water in thousands of cubic meters and $t$ is the time in years since the start of monitoring.

(a) Determine whether the volume of water in the reservoir is increasing or decreasing when $t=2$ years.

(b) One year after the start of monitoring, the volume of water in the reservoir was 5 thousand cubic meters. Find an expression for $V(t)$, the volume of water in the reservoir at time $t$, for $t \ge 0$.

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 $k$.

A company is designing a new cylindrical storage tank. The cost of manufacturing, in thousands of dollars, is modelled by the function $C(r) = r^2 + \frac{100}{r}$, where $r$ is the radius of the tank in metres, and $r > 0$.

(a) Write down the equation of the vertical asymptote of $C(r)$.

(b) Find $C'(r)$.

(c) Determine the interval in which $C(r)$ is decreasing.

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.

A company's daily production cost, $C(x)$, in thousands of dollars, for producing $x$ units of a specialized component, is modelled by the function $C(x) = x^2 + \frac{54}{x}$, for $x > 0$.

Write down the equation of the vertical asymptote of $C(x)$.

Find $C'(x)$, the marginal cost function.

Determine the interval for $x$ where the production cost $C(x)$ is increasing.

The rate of change of pollution, $\frac{dP}{dt}$ (in tonnes per day), in a protected lake is modelled by $\frac{dP}{dt} = 1 - 0.1t$, where $t$ is the time in days since monitoring began, for $0 \le t \le 15$.

(i) Find the value of $\frac{dP}{dt}$ at $t = 4$ days.

(ii) Interpret the meaning of your answer to part (a) (i) in context.

Use $\frac{dP}{dt}$ to find the value of $t$ when the pollution in the lake reaches its maximum level.

Two days after monitoring began, the pollution in the lake was $5.5$ tonnes.

Find an expression for $P$ in terms of $t$, for $0 \le t \le 15$.

Hence, find the maximum pollution level in the lake.

A second source of pollution, from a nearby factory, is modelled by $Q = 3(1.15)^t$, where $Q$ is the pollution in tonnes and $t$ is the time in days, for $0 \le t \le 15$.

Write down the initial pollution from this factory.

Each day, the pollution from the factory increases by $p$ %.

Find the value of $p$.

Find the value of $t$ when the pollution from the initial lake source and the factory are the same.

A spherical ice sculpture is melting in a gallery. Initially, the sculpture has a radius of 30 cm. This information is illustrated in the following diagram.

The gallery curator predicts that, as the ice sculpture melts, its radius will decrease at a constant rate of 0.5 cm per hour.

According to this model, find

(i) the radius of the ice sculpture, 10 hours after it begins melting.

(ii) the volume of the ice sculpture, 10 hours after it begins melting. Give your answer to one decimal place.

Let the function $V(t)$ represent the volume of the ice sculpture, $\text{cm}^3$, $t$ hours after it begins melting. $V(t)$ is given by

$V(t) = 113000 - 5000t + 150t^2 - 1.5t^3$, for $0 \le t \le 20$.

Find $V'(t)$.

Find the rate of change of the volume of the ice sculpture at $t = 10$ hours.

State one reason why the radius of the ice sculpture may not always decrease at a constant rate.

A chemical spill has contaminated a section of a river. Environmental engineers are monitoring the concentration of a particular pollutant. Let $P(t)$, measured in milligrams per litre (mgL$^{-1}$), be the concentration of the pollutant, $t$ days after a new batch of pollutant is introduced. The rate at which the pollutant naturally degrades or is flushed away is modelled as directly proportional to its concentration, leading to the differential equation

$\frac{dP}{dt} = -kP$, where $k \in \mathbb{R}^+$.

The initial concentration is $P_0$ mgL$^{-1}$, $P_0 > 0$.

By solving the differential equation, show that $P = P_0 e^{-kt}$.

For the remainder of this question, you will consider this pollutant where it is known that $k = 0.15$. The first significant spill occurs at time $t = 0$ and it is assumed that before this there is no pollutant present in the river.

Find the time, in days, for this pollutant to reach 10% of its initial concentration.

The pollutant is added to the river every $T$ days due to regular discharges, and in constant amounts, such that the concentration of the pollutant is increased by an amount $P_0$ mgL$^{-1}$. To simplify the model, it is assumed that each time the pollutant is added, the concentration in the river increases instantaneously.

Show that the concentration of the pollutant is $P_0(1+e^{-0.15T} + e^{-0.30T})$ immediately after the third discharge is given.

Immediately after the $n^{th}$ discharge is given, the concentration of the pollutant is

$P_0(1+e^{-0.15T} + e^{-0.30T} + ... + e^{-0.15(n-1)T})$.

Show that this concentration can be expressed as $P_0\left(\frac{1-e^{-0.15nT}}{1-e^{-0.15T}}\right)$.

After the river has been subjected to these discharges for a long time, it is required to keep the pollutant concentration within a particular range to ensure ecological safety.

Let $H_n$ be the highest concentration of the pollutant in the river for the interval $(n-1)T < t < nT$.

Let $L_n$ be the lowest concentration of the pollutant in the river for the interval $(n-1)T < t < nT$.

This is shown in the following graph.

$H_\infty$ is defined as $\lim_{n\to\infty} H_n$ and $L_\infty$ is defined as $\lim_{n\to\infty} L_n$.

Find, in terms of $P_0$ and $T$, an expression for

$H_\infty$.

$L_\infty$.

Show that

$H_\infty - L_\infty = P_0$.

$\frac{1}{0.15} \ln\left(\frac{H_\infty}{L_\infty}\right) = T$.

It is known that this pollutant is considered safe if the long-term concentration never exceeds $0.50 \text{ mgL}^{-1}$ and ecologically effective if it never drops below $0.10 \text{ mgL}^{-1}$.

Hence, for this pollutant, find a suitable value for

$P_0$.

$T$.

For the values of $P_0$ and $T$ found in part (g), find the proportion of time for which the concentration of the pollutant is at least $0.10 \text{ mgL}^{-1}$ between the first and second discharges.

Suggest a reason why the environmental regulations might specify a different value for $T$ to that found in part (g)(ii).

The rate of profit, $P(t)$, in thousands of dollars per month, for a new product is modelled by the piecewise function

$P(t) = \begin{cases} P_1(t), & 0 \leq t \leq T \\ P_2(t), & t \geq T \end{cases}$

where $P_1(t) = -t^2 + 6t$ and $P_2(t) = -2t + 16$. For a smooth transition in the profit rate, it is required that $P_1(T) = P_2(T)$.

Find the value of $T$.

Show that $P_1'(T) = P_2'(T)$.

The total profit from the product at time $t=0$ is zero.

Find the time when the total profit returns to its initial position.

A small scientific probe is launched into a dense liquid to collect data. Its descent is affected by gravity, buoyancy, and liquid resistance. The direction downwards is taken to be positive.

Initially, a simple model for the probe's velocity, $v$ ms$^{-1}$, at time $t$ seconds, assumes constant effective acceleration due to gravity and buoyancy, $g'$ ms$^{-2}$, given by:

$\frac{dv}{dt} = g'$

When the probe enters the liquid at $s = 0$ m, its initial velocity is $v = 5$ ms$^{-1}$. The displacement from its initial position is $s$ metres.

(i) Use the chain rule to show that $\frac{dv}{dt} = v\frac{dv}{ds}$.

(ii) Assuming that $g'$ is a constant, solve the differential equation $v\frac{dv}{ds} = g'$ to find $v$ as a function of $s$.

(iii) Using $g' = 9.8$ ms$^{-2}$, determine whether the model predicts that the probe will reach a velocity of $15$ ms$^{-1}$ at some point before it reaches a depth of $s = 200$ m. Justify your answer.

To test the model $\frac{dv}{dt}=g'$, the probe conducted a trial descent, and data for $v$ against $t$ was recorded.

(i) If the model is correct, describe the shape of the graph of $v$ against $t$.

(ii) The observed data showed a graph where the velocity increased rapidly at first, then its rate of increase slowed down, eventually approaching a constant value. Use this observation to comment on the validity of the model in part (a).

An improved model considers liquid resistance, using

$\frac{dv}{dt} = g'-k'v^2$

where $k'$ is a positive constant. You are reminded that initially $s = 0$ and $v = 5$. You may assume that $g' - k'v^2 > 0$.

(i) By using $\frac{dv}{dt} = v\frac{dv}{ds}$, solve the differential equation to find $v$ in terms of $s$, $g'$ and $k'$.

The probe's engineers use the graph of $v$ against $t$ from the trial descent to estimate the value of $k'$.

(ii) The gradient $\frac{dv}{dt}$ is estimated to be $3.05$ ms$^{-2}$ when $v = 15$ ms$^{-1}$. Taking $g'$ to be $9.8$ ms$^{-2}$, use this information to show that the engineers found that $k' = 0.03$.

(iii) Hence, find the value of $v$ predicted by this model, as $s$ tends to infinity.

(iv) Find the upper bound for the velocity according to this model, given that $0 < s \le 200$. Give your answer to four significant figures.

(d) A more refined model for the probe's descent suggests that the rate of change of velocity with respect to displacement is given by $\frac{dv}{ds} = \frac{5000}{(1000-s)^2} - 0.0001 v^2$. Use Euler's method with a step length of $50$ m to estimate the value of $v$ when $s = 200$ m. Take the initial velocity $v = 5$ ms$^{-1}$ at $s = 0$ m.

(i) Suggest one improvement to the use of Euler's method which might increase the accuracy of the prediction of the model.

(ii) Suggest one factor not explicitly considered by the model in part (d) which might lead to a difference between the model's prediction and the data collected.

The concentration, $C$, of a certain chemical in a solution, measured in milligrams per litre (mg/L), $t$ hours after a reaction begins, is modelled by the function $C(t) = -0.5t^3 + \frac{B}{t}$, where $B$ is a constant.

The rate of change of the concentration at $t = 2$ hours is $0.75$ mg/L per hour.

Determine the value of $B$.

Show that the concentration of the chemical is decreasing at $t = 3$ hours.

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.