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Cogito

AP Calculus BC · Chapter 8 · Lesson 1

Practice Worksheet

Euler's Method

Following a slope field numerically.

12 problems · about 22 minutes · FUN-7.C

NameDate

Figure — use these to answer the problems

Euler steps along these segments in straight lines. Smaller steps hug the true curve more closely.
  • The equationdy/dx = a·y
  • Through(0, 1)
k, the constant1
Starting x0
Starting y1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. y = 3, slope 4, step 0.5. What is the new y?

    Answer

  2. Why does Euler underestimate a concave-up solution?

    • The tangent line lies below a concave-up curve.
    • Because of rounding.

  3. y = 2, slope 3, step 0.5. New y?

    Answer

  4. y = 5, slope 2, step 1. New y?

    Answer

  5. y = 10, slope −3, step 0.5. New y?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Does a smaller step size improve accuracy? 1 yes, 0 no.

    Answer

  2. On a concave-up solution, does Euler over- or underestimate? 1 over, 2 under.

    Answer

  3. Going from x = 0 to x = 2 with step 0.5. How many steps?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Put one Euler step in order.

    Write 1 to 4 in the boxes to put these in order.

    • Evaluate the slope at the current point.
    • Multiply that slope by the step size.
    • Add the result to the current y.
    • Advance x by the step size and repeat.

  2. y = 4, slope 6, step 0.25. New y?

    Answer

  3. The First Step

    dy/dx = y with y = 1 and a step of 0.5. What is the new y?

    Answer

  4. The Second Step

    From y = 1.5 with slope 1.5 and step 0.5. What is the new y?

    Answer