Cogito
AP Calculus BC · Chapter 8 · Lesson 1
Euler's Method
Following a slope field numerically.
12 problems · about 22 minutes · FUN-7.C
What this lesson teaches
The student applies Euler's method to approximate a solution to a differential equation.
- Euler's method steps along tangent lines through a slope field.
- New y is old y plus slope times step size.
- Concave up underestimates and concave down overestimates.
Warm Up
Straightforward practice. Get the method working first.
5 problemsy = 3, slope 4, step 0.5. What is the new y?
Answer 5
Why 5.
Why does Euler underestimate a concave-up solution?
Answer The tangent line lies below a concave-up curve.
Why The tangent sits below the curve.
y = 2, slope 3, step 0.5. New y?
Answer 3.5
Why 2 + 1.5.
y = 5, slope 2, step 1. New y?
Answer 7
Why 5 + 2.
y = 10, slope −3, step 0.5. New y?
Answer 8.5
Why 10 − 1.5.
Build It Up
The same ideas with more to keep track of.
3 problemsDoes a smaller step size improve accuracy? 1 yes, 0 no.
Answer 1
Why It follows the curve more closely.
On a concave-up solution, does Euler over- or underestimate? 1 over, 2 under.
Answer 2
Why The tangent lies below the curve.
Going from x = 0 to x = 2 with step 0.5. How many steps?
Answer 4
Why 2 ÷ 0.5.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut one Euler step in order.
Answer 1. Evaluate the slope at the current point. 2. Multiply that slope by the step size. 3. Add the result to the current y. 4. Advance x by the step size and repeat.
Why The slope is found before anything is added.
y = 4, slope 6, step 0.25. New y?
Answer 5.5
Why 4 + 1.5.
The First Step: dy/dx = y with y = 1 and a step of 0.5. What is the new y?
Answer 1.5
Why 1.5.
The Second Step: From y = 1.5 with slope 1.5 and step 0.5. What is the new y?
Answer 2.25
Why 2.25.