Speedometer readings for a motorcycle at 12 -second...

Speedometer readings for a motorcycle at 12 -second intervals are given in the table. (a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals. (b) Give another estimate using the velocities at the end of the time periods. (c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain. $$\begin{array}{|c|c|c|c|c|c|c|}\hline t(s) & {0} & {12} & {24} & {36} & {48} & {60} \\ \hline v(f t / s) & {30} & {28} & {25} & {22} & {24} & {27} \\ \hline\end{array}$$

Speedometer readings for a motorcycle at 12 -second intervals are given in the table.
(a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals.
(b) Give another estimate using the velocities at the end of the time periods.
(c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain.
$$\begin{array}{|c|c|c|c|c|c|c|}\hline t(s) & {0} & {12} & {24} & {36} & {48} & {60} \\ \hline v(f t / s) & {30} & {28} & {25} & {22} & {24} & {27} \\ \hline\end{array}$$

Key Concepts

Definite Integral as an Accumulated Quantity

In calculus, a definite integral is used to calculate the total accumulation of a quantity, such as distance traveled, by integrating a rate of change like velocity over a specific time interval. This concept links the area under a curve to a physical quantity, providing powerful insights in physics and engineering.

Riemann Sum

A Riemann sum is a method for approximating the value of a definite integral by summing the products of function values at selected sample points and the widths of subintervals. This technique is foundational in numerical integration, bridging the gap between discrete data and continuous analysis.

Left-Endpoint Approximation

This method of Riemann sum estimation uses the value of the function at the beginning of each subinterval to approximate the area under the curve. It is particularly useful when data points are available at the start of each time interval, and it often provides an underestimate or overestimate depending on whether the function is increasing or decreasing.

Right-Endpoint Approximation

The right-endpoint approximation is another Riemann sum method that uses the function's value at the end of each subinterval to estimate the integral. Like the left-endpoint method, its accuracy depends on the behavior of the function over the interval, and it can serve as either an upper or lower estimate depending on the context of the function's variation.

Upper and Lower Estimates in Numerical Integration

In numerical integration, recognizing whether an approximation is an upper or lower estimate is critical. For a monotonically decreasing function, the left-endpoint method typically produces an overestimate, while for a monotonically increasing function, it produces an underestimate. Understanding these properties helps in evaluating the potential error and refining the approximation technique.

Transcript

00:01 So in this problem, what we are trying to do is to figure out an estimate for how far the motorcycle has traveled over the first 60 seconds or minute of its journey.

00:14 We are going to do this through using approximating rectangles known as remon sums.

00:21 And we are going to look for an overestimate and an underestimate of this distance.

00:28 So the first thing that i've done here is i have drawn a representation of the information on the table.

00:39 And so the first thing i've done is i've drawn a graph.

00:47 Okay.

00:49 At 12 seconds, we're at a velocity of 28 feet per second.

00:53 At 36 seconds, our velocity is 22 feet per second and so on.

00:57 So i'm going to go ahead and connect these data points.

01:00 Notice we are decreasing, then we are slightly increasing.

01:06 So now we must estimate the distance.

01:10 First, we are doing this using the velocities at the beginning of the time intervals.

01:16 So if you notice, we have five time intervals, 0 to 12 seconds, 12 to 24, 24, 24, 36, 36, 46, and 48 to 60.

01:24 If we were to use the beginning of the time intervals, what we are using, at a 0, 12, 24, 36, and 48 second marks.

01:34 So, in order to approximate the distance traveled, we are going to find the area under the curve.

01:41 So my second step was to choose my endpoints, and i did so carefully because we are using left endpoints.

01:56 With five intervals, we are choosing the five endpoints on the left.

02:00 So now, starting at zero, i'm going to go up until i touch my curve, then i go over to my next end point, and go down.

02:12 And starting at 12, i'm going to go right up until i touch my curve, over and down.

02:17 24, i'm going to go right up, over and down, and so on.

02:26 Now, i have five rectangles.

02:28 The width of each of these rectangles is 12, and the height of this rectangle, the height of this rectangle, here is going to be the y value of the rectangle which is 30 and here i'll call this rectangle a rectangle b has a height of 28 rectangle c as a height of 25 d of 22 and e of 24 notice we are not using that 27 so the area of each rectangle is going to be calculated as the width which is 12 times the height which is 30 so it'll be 12 times 30, 12 times 28, 12 times 25, and so on.

03:14 So my third step is going to be to draw the rectangles and calculate area.

03:20 So draw and calculate area of rectangles...

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