Find the difference between the upper and lower estimates of the distance traveled at velocity 25−t2 on the interval 1≤t≤4 for 1000 subdivisions.

Question

BlackTom AI · Accepted Answer

We start by identifying the velocity function given: v(t) = 25 − t^2 over the interval [1, 4].

First, note that v'(t) = −2t is negative for t > 0, so the function is decreasing on [1, 4].

For a decreasing function, the upper Riemann sum U using a uniform partition uses left endpoints, while the lower sum L uses right endpoints. The difference U − L for a partition of n subintervals of width Δt is:
U − L = Δt · [f(t0) − f(tn)], where t0 = a and tn = b and Δt = (b − a)/n.

Here, a = 1, b = 4, n = 1000, so Δt = (4 − 1)/1000 = 3/1000 = 0.003.
f(1) = 25 − 1^2 = 24, and f(4) = 25 − 4^2 = 25 − 16 = 9.

Plugging into the formula:
U − L = Δt · [f(1) − f(4)] = 0.003 · (24 − 9) = 0.003 · 15 = 0.045.

Thus, the difference between the upper and lower estimates of the distance traveled on this interval with 1000 subdivisions is 0.045.

Note: If answer options were present, we would map this calculation to the corresponding choice, but in this case no options were provided to compare against.

MAT136H1 S LEC0101 Online Practice 1

Find the difference between the upper and lower estimates of the distance traveled at velocity 25−t2 on the interval 1≤t≤4 for 1000 subdivisions.

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MAT136H1 S LEC0101 Online Practice 1

Find the difference between the upper and lower estimates of the distance traveled at velocity 25−t2 on the interval 1≤t≤4 for 1000 subdivisions.

View Explanation

Log in for full answers

Similar Questions

Match the type of estimation technique shown in the pictures with the correct label. 1: ____ 2: ____ 3: ____ 4: ____

Express the following integral as a limit of Riemann sums:

Suppose we want to find the area under the curve (pictured below) on the interval using a Riemann sum with rectangles. Which of the following provides the most accurate approximation of the area under the curve on the given interval?

Suppose we want to find the area under the curve (pictured below) on the interval using a Riemann sum with rectangles. Which of the following provides the most accurate approximation of the area under the curve on the given interval?

Suppose represents the velocity of an object on an interval . Using the Riemann sum approximation, , to approximate the area under the curve estimates which of the following?

You should see 2 equations in this question. Does the 2nd equation render correctly in your browser? It should look the same as the image in this box: ∫ 0 1 𝑥 2 𝑑 𝑥 = lim 1 𝑛 ( 1 2 𝑛 2 + 2 2 𝑛 2 + . . . + ( 𝑛 2 𝑛 2 ) ) = lim ∑ 𝑛 𝑚 𝑥 × 𝑛 2 2 3

You should see 2 equations in this question. Does the 2nd equation render correctly in your browser? It should look the same as the image in this box: ∫ 1 0 x2dx=lim 1 n ( 12 n2 + 22 n2 +...+( n2 n2 ))=lim∑ n m x × n2 23

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Match the type of estimation technique shown in the pictures with the correct label. 1: 2: 3: 4: