Math Problem Statement
.Let 𝑓(𝑡) = 2t^2 and consider f(t) on the interval [1,4]. Then, if we divide the interval into n subintervals, what are the values of a, b, and delta t? Using the right endpoints, find the upper sum. This will have n’s in the answer—no need to simplify into one fraction.
Solution
We are given the function on the interval , and we are tasked with dividing this interval into subintervals and finding the upper sum using right endpoints.
Step 1: Define , , and
- is the left endpoint of the interval: .
- is the right endpoint of the interval: .
- The width of each subinterval, , is given by:
Step 2: Set up the upper sum using right endpoints
For the upper sum, we evaluate at the right endpoints of each subinterval. The right endpoint for the -th subinterval is given by:
We then compute the value of for each right endpoint:
Step 3: Write the upper sum
The upper sum is the sum of the areas of rectangles with height and width . This gives us the following sum:
Conclusion
Thus, the upper sum is:
Would you like any details about specific steps or additional clarifications?
Related questions:
- How do you simplify this sum further?
- How would the result change if we used left endpoints instead of right endpoints?
- What is the exact value of the integral of over the interval ?
- How does the value of the upper sum behave as ?
- How would the problem change if we had instead?
Tip:
When dividing an interval into subintervals, always check if you're using left, right, or midpoint endpoints to ensure accurate sums.
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Math Problem Analysis
Mathematical Concepts
Integral Approximation
Riemann Sum
Quadratic Functions
Formulas
Delta t = (b - a) / n = 3 / n
f(t) = 2t^2
Upper sum = Σ[2(1 + (3i/n))^2 * (3/n)] for i=1 to n
Theorems
Riemann Sum
Suitable Grade Level
Grades 11-12, Early College
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