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 f(t)=2t2f(t) = 2t^2 on the interval [1,4][1,4], and we are tasked with dividing this interval into nn subintervals and finding the upper sum using right endpoints.

Step 1: Define aa, bb, and Δt\Delta t

  • aa is the left endpoint of the interval: a=1a = 1.
  • bb is the right endpoint of the interval: b=4b = 4.
  • The width of each subinterval, Δt\Delta t, is given by: Δt=ban=41n=3n.\Delta t = \frac{b - a}{n} = \frac{4 - 1}{n} = \frac{3}{n}.

Step 2: Set up the upper sum using right endpoints

For the upper sum, we evaluate f(t)f(t) at the right endpoints of each subinterval. The right endpoint for the ii-th subinterval is given by: ti=a+iΔt=1+i3n.t_i = a + i \Delta t = 1 + i \cdot \frac{3}{n}.

We then compute the value of f(ti)f(t_i) for each right endpoint: f(ti)=2(ti)2=2(1+3in)2.f(t_i) = 2(t_i)^2 = 2\left(1 + \frac{3i}{n}\right)^2.

Step 3: Write the upper sum

The upper sum is the sum of the areas of rectangles with height f(ti)f(t_i) and width Δt\Delta t. This gives us the following sum: Upper Sum=i=1nf(ti)Δt=i=1n2(1+3in)23n.\text{Upper Sum} = \sum_{i=1}^{n} f(t_i) \Delta t = \sum_{i=1}^{n} 2\left(1 + \frac{3i}{n}\right)^2 \cdot \frac{3}{n}.

Conclusion

Thus, the upper sum is: Upper Sum=i=1n2(1+3in)23n.\text{Upper Sum} = \sum_{i=1}^{n} 2\left(1 + \frac{3i}{n}\right)^2 \cdot \frac{3}{n}.

Would you like any details about specific steps or additional clarifications?

Related questions:

  1. How do you simplify this sum further?
  2. How would the result change if we used left endpoints instead of right endpoints?
  3. What is the exact value of the integral of f(t)f(t) over the interval [1,4][1, 4]?
  4. How does the value of the upper sum behave as nn \to \infty?
  5. How would the problem change if we had f(t)=3t3f(t) = 3t^3 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