Practice AP Calculus BC questions on using equal-step Euler approximations to estimate points on a differential-equation solution curve.
Let y=f(x) be the particular solution to the differential equation dxdy=(3−x)y2 with initial condition f(1)=-1.
Use Euler's method, starting at x=1 with two steps of equal size, to approximate f(1.4).
Show the work that leads to your answer.
D Use Euler's method, starting at x=1 with two steps of equal size, to approximate f(1.4). Show the
work that leads to your answer.
| f(1.2)≈f(1)+(1.2−1)⋅dxdy(1,−1)=−1+0.2(2)=−0.6 |
|---|
| f(1.4)≈f(1.2)+(1.4−1.2)⋅dxdy(1.2,−0.6)≈−0.6+0.2(3−1.2)(−0.6)2=−0.4704 |
An approximation of f(1.4) is -0.47.
Scoring Notes for Part D
- To earn P8, a response must demonstrate the first step of Euler's method, with the correct initial
condition, correct step size, and correct (or imported) expression for the derivative.
Note: Any subsequent error in simplification or rounding will not affect the scoring for P8.
- The two steps of Euler's method may be explicit expressions or may be presented in a table. For
example:
| x | y | dxdy⋅Δx( or dxdy⋅0.2) |
|---|---|---|
| 1 | -1 | 0.4 |
| 1.2 | -0.6 | 0.1296 |
| 1.4 | -0.4704 |
Note: In the presence of a correct answer, a table does not need to be labeled to earn both P8 and P9.
In the presence of no answer or an incorrect answer, such a table must be correctly labeled to earn
P8.
- A response of −0.6+0.2(3−1.2)(−0.6)2 earns P9, regardless of any subsequent errors in
simplification or rounding.
- A response that imports an incorrect value for f′(1) from part A or part B is eligible to earn P9 with