Construct tangent lines to visualize Euler's method and explain how the step size affects the approximation.
Use Euler's method to approximate a value on a solution curve given the curve's differential equation and a step size.
Understand how Euler's method improves a simple local linearization by re-calibrating slope.
Quick Lesson Plan
In today’s activity students build on their understanding of tangent line approximations from Unit 4 to arrive at Euler’s method for approximating solutions to a differential equation.
In questions 1-3 students use a tangent line at x=0 to approximate f(0.5) and f(1), reviewing the idea that tangent line approximations are more accurate at points close to the point of tangency. The rest of the questions on the first page lead students through the process of using an additional tangent line that would provide a recalibrated slope based on the differential equation. This is helpful because even though the new point of tangency is based on an estimate and thus introduces some error, the slope of the second tangent line is much closer to the slope of the actual curve at x=0.5 than the slope at x=0. Note that the new tangent line has a positive slope, indicating that the solution curve changed from decreasing to increasing.
Note how question 7 follows the same structure as many of our Calculus lessons: how do we improve an approximation? This is a big theme of the course and the underlying principle behind a derivative and an integral. The difference here is that Euler’s method will never get an exact value but simply an improved approximation. This is similar to when students use 50 rectangles instead of 5 to approximate the area under a curve.
You may wish to debrief the first page of the activity before having students move on to the second page.
On page 2, students use a Geogebra applet to get a better visualization of this process using smaller and smaller step sizes (and thus more and more tangent lines). As students are completing the table in question b, make sure they understand how to find the change in y. While students may just subtract the new y-value given in the applet from the old one, make sure students understand how this value is related to the slope of the tangent line and the step size.
In question 12, students think critically about the trade-off between accuracy and ease. Euler’s method by hand can be tedious, so students should consider what level of accuracy in the approximation is "good enough". Of course, technology helps make cumbersome calculations much faster and can be used to gain a higher level of precision in approximations.
What if we used this tangent line to predict f(2)? Would this be a good estimate?
Why is your tangent line in question 4 not actually tangent to the solution curve?
What’s the advantage of calculating a new tangent line near x=0.5?
How can you calculate the slope at any point for your tangent line equation?
How is the step size related to the number of steps?
Can you represent the change in y visually?
How is the change in y calculated?
The debrief of the first page should help students to see Euler’s method as an extension of their previous work in Unit 4 with local linear approximations. The idea is that by using multiple tangent lines with updated slopes, the approximation of the point on the solution curve will be closer to the actual value. We also introduce the terms “error” and “steps”. Using the word “iterations” is also helpful to refer to the number of steps or number of tangent lines used.
Note that for the purpose of this activity we draw in the solution curve so students get a sense of how good their approximation is. In reality, the power of Euler’s method is its ability to approximate solutions to differential equations that can’t be solved (and thus graphed) by hand.
While writing an equation of a line is old hat for students, the notation for the nth tangent line in its general form can be a bit clunky. We have tried to make it as accessible as possible by omitting some of the formal “previous term” notation and instead using “old” and “new”, but you may wish to write the equation differently.
Students may have to give an approximation using Euler’s method on the free response section or multiple choice section of the AP Calculus BC exam. In past years, this has rarely required more than 2 or 3 equal sized steps to keep the calculation burden at bay. See 2010 BC 5 part a and 2013 BC 5 part b for examples.