Math Problem Statement
Let � : � → � f:R→R be the function defined by � ( � )
� 2 f(x)=x 2 , �
( 0 , 0.5 , 1 , 1.6 , 2 ) P=(0,0.5,1,1.6,2) and �
( 0 , 0.6 , 1.2 , 1.7 ) T=(0,0.6,1.2,1.7). What is � ( � , � , � ) R(f,P,T)?
Solution
The expression typically represents a Riemann sum, which is used to approximate the area under a curve defined by a function over a certain interval. Here’s a step-by-step solution for calculating given the information:
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Function Definition: The function is defined over , meaning it maps real numbers to real numbers.
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Partition : The partition divides the interval into subintervals:
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Sample Points : The sample points indicate the points in each interval where we evaluate the function :
- in
- in
- in
- in
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Calculate the Riemann Sum : The Riemann sum for this partition is computed by summing up the products of at the sample points and the widths of the corresponding subintervals defined by : where is the width of each subinterval and is the sample point in that subinterval.
- Interval : , , so
- Interval : , , so
- Interval : , , so
- Interval : , , so
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Compute the Sum:
Final Answer:
Would you like further details on this calculation?
- What if the function changes to a different polynomial function?
- How would changing the sample points affect the Riemann sum?
- What does this Riemann sum approximate in terms of area under the curve?
- How does the partition size affect the accuracy of the Riemann sum?
- Could we use this approach to estimate the integral as the partition becomes finer?
Tip: For more accurate area approximations, increase the number of intervals in the partition , which reduces each subinterval’s width.
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Math Problem Analysis
Mathematical Concepts
Riemann Sum
Integration
Partitioning Intervals
Quadratic Functions
Formulas
Riemann Sum formula: R(f, P, T) = Σ f(T_i) * Δx_i
Theorems
Riemann Sum Approximation
Suitable Grade Level
Grades 11-12
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