Unit 8 - Advanced Calculus (HL only)
Required Prior Knowledge
Questions
Integrate the following
a) \(\int \frac{1}{1+y^{2}}dy \)
b) \(\int xe^{x}dx\)
c) Make \(y\) the subject of \(\ln \left|y\right|=x+c\)
d) Decompose \(\frac{1}{P\left(A-P\right)}\) into partial fractions
Solutions
Get Ready
Questions
The marginal costs of producing \(x\) urns per day is given by$$\frac{dC}{dx}=2.15-0.02x+0.00036x^{2}$$dollars per urn provided \(0\le x\le 120\).
The initial cost before production starts is \($185\).
Find the total cost of producing \(100\) urns per day.
Solutions
Notes
A separable differential equation is one of the form$$\frac{dy}{dx}=f\left(x\right)g\left(y\right)$$For example$$\begin{matrix}\frac{dy}{dx}=xy^{2}&\frac{dy}{dx}=y^{2}+3&\frac{dy}{dx}=\frac{x+2}{y^{3}}\end{matrix}$$We call them separable because we can rearrange them into the form$$\frac{1}{g\left(y\right)}dy=f\left(x\right)dx$$where we have separated the variables so they are on different sides of the equation.
We can then integrate both sides with respects to that variable to solve the equation.
Examples and Your Turns
Example
Solve the differential equation$$\frac{dy}{dx}=xy$$
Your Turn
Solve the differential equation$$\frac{dy}{dx}=4xy^{2}$$
Your Turn
Solve the differential equation$$\frac{dy}{dx}=\frac{1}{y^{2}}$$
Your Turn
Solve the differential equation$$\frac{dy}{dx}=x\left(1+y\right)e^{x}$$
Your Turn
Solve the differential equation$$\frac{dy}{dx}=xy^{2}+x$$ given that \(y\left(0\right)=1\)
Your Turn
Solve the differential equation$$\frac{dy}{dx}=e^{2y}\cos 3x$$given that \(y\left(0\right)=0\).
Your Turn
Solve the differential equation$$\left(1+x^{2}\right)\frac{dy}{dx}=1+y^{2}$$given that \(y\left(0\right)=2\).
Your Turn
Find the general solution to the differential equation$$\frac{dy}{dx}=e^{4x-3y}$$in the form \(y=f\left(x\right)\).
Notes
A common problem is one of the form$$\frac{dP}{dt}=kP$$This represents the situation where a population \(P\) is increasing (or decreasing) at a rate proportional to the population itself.
We can solve this general case:$$\frac{dP}{dt}=kP\\ \frac{1}{P} dP=k dt\\ \int \frac{1}{P} dP=\int k dt \\ \ln \left|P\right|=kt + c\\ P=e^{kt+c}\\ P=e^{kt}e^{c}\\ P=Ae^{kt}$$If the initial population is \(P_{0}\) we get:$$P=P_{0}e^{kt}$$
Examples and Your Turns
Example
A salmon farm has an initial population of \(80\). The population grows according to the differential equation$$\frac{dP}{dt}=\frac{1}{5}P$$ where \(t\) is the time in years.
a) Write an expression for \(P\) in terms of \(t\).
b) Find the salmon population after \(10\) years.
Your Turn
The rate of growth of a colony of bacteria, \(\frac{dP}{dt}\), is proportional to its current size, \(P\).
At \(t=0\) hours, the population of the bacteria is \(1000\).
a) Write a differential equation to model this population.
After one hour the colony has grown by \(20%\).
b) Find the constant of proportionality.
c) Determine how many hours it will take for the population of bacteria to grow to \(50 000\).
Notes
It is common for a population to grow exponentially at first, but then, due to limited resources such as food or shelter, for the population to level off at a maximum level.
This kind of growth model is known as logistic growth.
Some examples of scenarios that follow a logistic model are:
populations (bacteria in a petri dish, rabbits on an island)
viral infections (such as COVID)
adoption of new technologies (AI)
sign ups to new social media platforms
spread of rumours
A logistic growth model always has this shape.
Logistic growth is given by the differential equation$$\frac{dP}{dt}=kP\left(1-\frac{P}{A}\right)$$where \(A\) is the limiting maximum value of the population.
It is a separable differential equation.
Examples and Your Turns
Example
The population of an island is currently \(154\). Its expected growth rate is given by $$\frac{dP}{dt}=0.16P\left(1-\frac{P}{500}\right)$$where \(t\) is in years.
a) Write \(P\) as a function of \(t\).
b) Find the population after \(10\) years.
c) Find the time taken for the population to increase to \(480\).
d) What is the limiting population size?
Your Turn
The rate of growth of a giraffe population can be modelled by$$\frac{dP}{dt}=0.0002P\left(200-P\right)$$where \(t\) is in years. If there were \(30\) giraffes at the time scientists started studying the population, determine
a) How many years it would take for the giraffe population to reach \(150\).
b) The maximum the giraffe population can grow to.
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Why can we separate the \(\frac{dy}{dx}\) when separating the variables?
In logistic growth, for the differential equation \(\frac{dP}{dt}=kP\left(1-\frac{P}{A}\right)\), why is the limiting maximum population size equal to \(A\)? Consider what happens as \(P\to \A\).
Common Mistakes / Misconceptions
The most common mistake with all differential equations is choosing the correct method. Spotting that the original equation is separable is key to then being able to solve it. Remember that \(\frac{dy}{dx}=x+y\) is NOT separable.
If you spot it is separable, then the next most common mistake is to separate the variables incorrectly. For example, not factorising first, or forgetting to divide through by the function in \(y\).
A common mistake occurs in the simplification of \(e^{x+c}\). Remember this is \(e^{x+c}=e^{x}e^{c}=Ae^{x}\) and NOT \(e^{x}+c\).
And there is always the possibility of forgetting to include the \(+c\) in the first place. As with any indefinite integration, you need the constant of integration.
Connecting This to Other Skills
We are directly building on the ideas of Differential Equations (8.4) to see our first method for solving specific types of differential equations.
We require all our knowledge of Integration (Unit 5) in order to solve these equations.
We often need to use Laws of Indices (PK1) when simplifying expressions.
Logistic growth requires the use of Partial Fractions (1.19).
As we usually need to rewrite our final answer in a certain form, we use the algebraic techniques behind finding Inverse Functions (2.13).
After this we will see Homogenous Differential Equations (8.7) which required knowledge of this method. And then Integrating Factor Method (8.8) which is another method for solving differential equations that are NOT separable.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?