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The table shows a speedometer readings at 10-second intervals during a 1-minute period for a car racing at the Daytona International Speedway in Florida.

(a) Estimate the distance the race car traveled 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.

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03:03

Frank Lin

Calculus 1 / AB

Chapter 5

Integrals

Section 1

Areas and Distances

Integration

Oregon State University

Harvey Mudd College

University of Nottingham

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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The table shows a speedome…

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The table shows speedomete…

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Speedometer readings for a…

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8. Speedometer readings fo…

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The velocity of a car was …

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A car moves along a straig…

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A car comes to a stop five…

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(II) The position of a rac…

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Distance traveled The acco…

So in this problem were given The speedometer readings at 10 2nd intervals during a one minute period for a racing car. And so we have that table of those. Okay. Were used to estimate the distance the car traveled. Using the velocities at the beginning of intervals that has to do it again using the loss at the end of the intervals. And then we are asked uh our estimates in estimates in parts A and B. Upper and lower estimates. Well let's think about this for a minute. If this is our graph of our data, think of something like this. Okay. And we draw in these rectangles that are going to use the left side of the curve like so okay, be like this right all the time. That means that the laughter of the beginning. Okay, the beginning are going to be the lower sums and the ending ones then are going to be the upper sums our estimates. Okay, so that's how that's going to work for us. So let's look at the data now. So we are given this set of data right here. I just piped it into a little spreadsheet real quick right here, Time from 0 to 60 seconds and our velocity of the speedometer and miles per second. His shows are 18, 168 and so on. Okay, So I know each of these intervals is 10 seconds. So whatever delta T is 10. So then taking the 10 times the beginning. So I mean is this first one b 10 times the 1 82.9. Is this the 1829? And so the next 10 it becomes the 168 would be that one? Okay. That gives us the lower sums of 87, 79. Course, this is in miles. And then the upper sums, yeah. Would be the ones on the ending right over here. Okay. And that 10 miles. Okay? We can see the difference between the two and we see which ones are the lower and which ones are the upper sums.

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