Euler's Method – AP Calculus BC

Learning Objectives

Numerical Approximation: When a differential equation is impossible to solve algebraically, we iterate tiny local tangent lines to follow the true curve step by step.

The Algorithm: Starting at $(x_0, y_0)$, we step forward by $y_{new} = y_{old} + \frac{dy}{dx} \cdot \Delta x$. We re-evaluate the derivative at every new stop.

Truncation Error: Because tangent lines only point straight, concave curves will bend away, creating cumulative drift/error the larger $\Delta x$ gets.

Tags

ApproxEuler's MethodLimitsError

       MATH ALGORITHM

       $y_{n} = y_{n-1} + f'(x_{n-1}, y_{n-1}) \cdot \Delta x$

Select Differential Equation $f'(x,y)$

Step Size Control ($\Delta x$)

$\Delta x$ = 1.00

Evaluation Box $(x=3.0)$

Initial Condition: $(0, 1)$

ITERATION VS ANALYTICAL SOLUTION

True Analytical Value: -

Euler Approx Value: -

Absolute Error: -

Number of Steps $N$ -

Error Over/Under: Notice that when the true solution curve is CONCAVE UP, the local tangent lines fall entirely *under* the curve, forcing Euler's method into a systematic UNDERESTIMATE.