Unit 8 - Advanced Calculus (HL only)
Required Prior Knowledge
Questions
a) Find the equation of the tangent to \(f\left(x\right)=x^{3}\) at the point \(\left(1,1\right) \).
b) If \(u_{n+1}=2u_{n}-3\) and \(u_{1}=4\) determine \(u_{5}\), the fifth term in the sequence.
Solutions
Get Ready
Questions
Suppose we have the differential equation \(\frac{dy}{dx}=3x^{2}\) with initial condition \(y\left(1\right)=1\).
a) By integration, what is the particular solution?
b) Now suppose we want to find the value of \(y\left(2\right)\). In this case we can evaluate our solution at \(x=2\). What value do we get?
If we were unable to integrate the DE to find a solution, we would not know this value. But we can still approximate it using the method below.
c) Now find the equation of the tangent of \(y\left(x\right)\) at the point \(\left(1,1\right)\), our initial condition. Evaluate this equation at \(x=2\).
How does this relate to the graph below?
Solutions
Now suppose we go half way to \(x=2\), and then find the equation of the line passing through that point, but with the gradient of \(y\left(x\right)\) at \(x=1.5\).
d) Evaluate this line at \(x=2\).
Consider how the graph below shows this situation to help you.
Now suppose we split the move to \(x=2\) into \(4\) pieces, at each stage finding the equation of the line through the end point of the last line that is parallel to the tangent of \(y\left(x\right)\) at that \(x\) value.
e) Evaluate this line at \(x=2\).
Consider how the graph below shows this situation to help you.
What happens as we increase the number of steps we take (or decrease the distance between each point)?
Suppose we did not know the original function was \(y=x^{3}\), but only the original differential equation \(\frac{dy}{dx}=3x^{2}\). We could still work out the equations of all the tangents as we have the points and the gradients. In this way we can approximate the value of \(y\left(2\right)\) using the tangents.
At each stage we ‘recalculate’ the direction in order to stay closer to the actual path (that we can’t see!).
That is, we do not know the red line in the graphs above, and are using the straight lines to approximate it with information from the differential equation.
Notes
Euler’s Method is a numerical method for approximating a value of a first order differential equations of the form $$\frac{dy}{dx}=f\left(x,y\right)$$with \(y\left(x_{0}\right)=y_{0}\).
It does this using the gradient of the function, adjusting the gradient to the gradient of the function at equal steps. This adjustment, brings the approximation closer to the actual value by making the steps fit the original function better.
The long process above can be simplified into a pair of recursive formulae to determine the points at each stage.
We can do this using the recursive relations below$$x_{n+1}=x_{n}+h\\y_{n+1}=y_{n}+hf\left(x_{n},y_{n}\right)$$where \(\frac{dy}{dx}=f\left(x,y\right)\).
Examples and Your Turns
Example
Use Euler’s Method to find an approximation of \(y\left(2\right)\) with step \(h=0.2\) for the differential equation$$\frac{dy}{dx}=3x^{2},\quad y\left(1\right)=1$$
Example
Use Euler’s Method to find an approximation of \(y\left(\right)\) with step \(h=0.2\) for the differential equation$$\frac{dy}{dx}+2y=2-e^{-4x},\quad y\left(0\right)=1$$
Verify that \(y=1+\frac{1}{2}e^{-4x}-\frac{1}{2}e^{-2x}\) is a solution to the differential equation.
Hence determine the percentage error of your approximation.
Your Turn
Use Euler’s Method to find an approximation of \(y\left(1\right)\) with step \(h=0.1\) for the differential equation$$\frac{dy}{dx}=1+y,\quad y\left(0\right)=1$$With reference to the concavity of the function at \(\left(0,1\right)\), determine whether this approximation is an overestimate or an underestimate.
Your Turn
Use Euler’s Method to find an approximation of \(y\left(0.5\right)\) with step \(h=0.1\) for the differential equation$$\frac{dy}{dx}=y^{2}-x,\quad y\left(0\right)=1$$With reference to the concavity of the function at \(\left(0,1\right)\), determine whether this approximation is an overestimate or an underestimate.
Your Turn
Use Euler’s Method to find an approximation of \(y\left(0.5\right)\) with step \(h=0.1\) for the differential equation$$\frac{dy}{dx}=\sin\left(x+y\right)-e^{x},\quad y\left(0\right)=4$$With reference to the concavity of the function at \(\left(0,4\right)\), determine whether this approximation is an overestimate or an underestimate.
Notes
If a function is concave up at the known point, then Euler’s Method will lead to an underestimate.
If a function is concave down at the known point, then Euler’s Method will lead to an overestimate.
Euler’s Method has featured in the blockbuster film Hidden Figures, the true story of the African-American female mathematicians behind the success of some of the earliest space missions. Below you can see a brief scene from the film where Euler’s Method is mentioned.
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
What happens to our approximation as \(h\to 0\)?
How is the value of \(h\) linked to the number of steps in the process?
If the curve is concave up, will Euler’s method always underestimate or overestimate the true value? Why?
Common Mistakes / Misconceptions
Doing the calculations manually is likely to lead to algebraic errors. This is a calculator question, and you need to be able to use your calculator to do this quickly. You should write down all the values in the table you generate.
The most common mistake is not using the correct function for \(f\) in the equation \(y_{n+1}=y_{n}+hf\left(x_{n},y_{n}\right)\). Recall that \(f\left(x,y\right)=\frac{dy}{dx}\) which is the original differential equation.
Connecting This to Other Skills
To understand the concept of Euler’s Method, you need to understand Tangents (4.13) and Equations of Straight Lines (2.1).
To be able to use the formulae for Euler’s Method, you need to understand recursive formulae for Sequences and Series (1.3).
We are going to see exact methods to solve these differential equations (8.6, 8.7 and 8.8) and you are often asked to solve them both ways and compare.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?