10.1.6 - Worked-Out Example
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Practice Questions
Test your understanding with targeted questions
What value of \( k_1 \) do we compute in the first iteration if \( x_0 = 0 \) and \( y_0 = 1 \)?
💡 Hint: Evaluate the function at the initial point.
If the initial step size is \( h = 0.1 \), what is the predicted value \( y^* \) after the first iteration?
💡 Hint: Use the formula for prediction with the calculated slope.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What is the main advantage of the Modified Euler's Method over the standard Euler's Method?
💡 Hint: Think about how multiple data points affect estimates.
True or False: The Modified Euler's Method requires two function evaluations for each step.
💡 Hint: Recall how we calculated both slopes.
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Challenge Problems
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Given the differential equation \( \frac{dy}{dx} = y - x^2 + 1 \) with an initial condition \( y(0) = 0.5 \) and step size \( h = 0.2 \), calculate \( y(0.2) \) using the Modified Euler’s Method.
💡 Hint: Remember to calculate both slopes at each step just like in our example.
Discuss how the error in the Modified Euler's Method can accumulate across iterations. Illustrate it with an example of taking too large of a step size.
💡 Hint: Think about how estimates differ when you leap over more points on the function.
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