Practice Worked-out Example (10.1.6) - Modified Euler’s Method - Mathematics - iii (Differential Calculus) - Vol 4

Practice: Worked-Out Example

Lesson 10.1.6Modified Euler’s Method

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6 sample questions

Question 1.
What value of $k_1$ do we compute in the first iteration if $x_0 = 0$ and $y_0 = 1$ ?
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Evaluate the function at the initial point.
Question 2.
If the initial step size is $h = 0.1$ , what is the predicted value $y^*$ after the first iteration?
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Use the formula for prediction with the calculated slope.
3Question 3.
What is the main advantage of the Modified Euler's Method over the standard Euler's Method?
- It is easier to compute
- It uses only one slope
- It improves accuracy by averaging slopes
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Think about how multiple data points affect estimates.
4Question 4.
True or False: The Modified Euler's Method requires two function evaluations for each step.
- True
- False
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Recall how we calculated both slopes.
Question 5.
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.
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Remember to calculate both slopes at each step just like in our example.
Question 6.
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.
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Think about how estimates differ when you leap over more points on the function.

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