3.3.2 - Error in Newton-Cotes Formulas
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Practice Questions
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What type of polynomial does the Trapezoidal Rule use for numerical integration?
💡 Hint: What shape do trapezoids form?
What is the error rate of Simpson's Rule compared to the Trapezoidal Rule?
💡 Hint: Think about both methods' error orders.
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Interactive Quizzes
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What is the primary advantage of using Simpson's Rule over the Trapezoidal Rule?
💡 Hint: Think about how each rule approximates the curve.
True or False: The error of the trapezoidal rule becomes negligible as you approach zero for step size.
💡 Hint: Recall the error proportionality relationship.
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Challenge Problems
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Using f(x) = sin(x), calculate the area under the curve from 0 to π/2 using both Trapezoidal and Simpson's rules with n = 4 intervals. Analyze which method yields less error and why.
💡 Hint: Consider the behavior of sin(x) in multiple intervals.
A smooth function's integral is calculated using both methods. Outline the implications of choosing a method depending on the smoothness of the function and computational resources.
💡 Hint: Reflect on the characteristics of the functions involved.
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