Solved examples of Euler's Method in high school calculus

📚 Quick Study Guide

• 🔢 Euler's Method is a numerical technique for approximating the solution to an ordinary differential equation (ODE) with a given initial value.
• 📝 The formula for Euler's Method is: $y_{i+1} = y_i + h \cdot f(x_i, y_i)$, where $h$ is the step size.
• 📈 $y_{i+1}$ is the approximation of the solution at $x_{i+1}$.
• 📌 $y_i$ is the approximation of the solution at $x_i$.
• 📐 $f(x_i, y_i)$ is the value of the derivative at the point $(x_i, y_i)$.
• ⏱️ Smaller step sizes generally lead to more accurate approximations, but require more computation.
• 💡 To apply Euler's Method:
  1. Define the differential equation $dy/dx = f(x, y)$ and the initial condition $y(x_0) = y_0$.
  2. Choose a step size $h$.
  3. Iteratively apply the formula $y_{i+1} = y_i + h \cdot f(x_i, y_i)$ to find approximate values of $y$ at subsequent points $x_{i+1} = x_i + h$.

Practice Quiz

1. What is the main purpose of Euler's Method?
1. A) To find the exact solution of a differential equation.
2. B) To approximate the solution of a differential equation.
3. C) To find the derivative of a function.
4. D) To find the integral of a function.


2. What does 'h' represent in the Euler's Method formula?
1. A) The exact solution.
2. B) The derivative.
3. C) The step size.
4. D) The initial value.


3. Given the differential equation $dy/dx = x + y$ and initial condition $y(0) = 1$, what is the value of $f(x_0, y_0)$?
1. A) 0
2. B) 1
3. C) 2
4. D) 3


4. If $y_i = 2$, $h = 0.1$, and $f(x_i, y_i) = 3$, what is the value of $y_{i+1}$ using Euler's Method?
1. A) 2.1
2. B) 2.2
3. C) 2.3
4. D) 2.4


5. What happens to the accuracy of Euler's Method as the step size 'h' decreases?
1. A) Accuracy decreases.
2. B) Accuracy increases.
3. C) Accuracy remains the same.
4. D) Accuracy fluctuates randomly.


6. Euler's Method is a first-order method. What does this imply about its error?
1. A) The error is proportional to $h^2$.
2. B) The error is proportional to $h$.
3. C) The error is zero.
4. D) The error is exponential.


7. For the differential equation $dy/dx = 2x$ with $y(1) = 1$ and $h = 0.5$, what is the approximate value of $y(1.5)$ using Euler's method?
1. A) 1.5
2. B) 2.0
3. C) 2.5
4. D) 3.0