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Chapter 7: Problem 134

Let \(f(t)\) be the velocity in meters/second of a car at time \(t\) in seconds. Give an integral for the change of position of the car (a) For the time interval \(0 \leq t \leq 60\). (b) In terms of \(T\) in minutes, for the same time interval.

Short Answer

Expert verified
(a) \( \int_{0}^{60} f(t) \, dt \); (b) \( \int_{0}^{60T} f(t) \, dt \).

Step by step solution

01

Understanding the context and integration

To find the change in position given the velocity function, use the concept of integration. The change in position of the car from time \(t = a\) to \(t = b\) can be found using the definite integral of the velocity function \(f(t)\) over the interval \([a, b]\).
02

Setting up the integral for part (a)

For part (a), the time interval is \(0 \leq t \leq 60\). Hence, the change of position \(s(60) - s(0)\) is given by the integral:\[ \int_{0}^{60} f(t) \, dt \]
03

Setting up the integral for part (b) in terms of T

For part (b), express the integral with the upper limit in terms of \(T\), where \(T\) is given in minutes. Notice that \(T\) minutes is equivalent to \(60T\) seconds. Therefore, the integral for the change of position between \(t = 0\) and \(t = 60T\) seconds is:\[ \int_{0}^{60T} f(t) \, dt \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Velocity Function
The velocity function, denoted as \(f(t)\), represents how fast a car is moving over time in meters per second. This function is crucial because it not only tells us the speed at each moment but also helps us calculate the total distance traveled by the car.
The velocity function is the rate of change of the position with respect to time. In other words, it answers the question: How quickly is the car moving? When we know the velocity function over time, we can determine how far the car has traveled by integrating this function over the desired time interval.
Interpreting the Change of Position
The concept of change of position refers to the total displacement of the car over a given period. It helps us quantify how far the car has moved from its starting point, regardless of its path. Calculating the change of position requires integrating the velocity function over a specified time interval.
This integral provides a single number that represents the car's total displacement. It's important to note that this involves considering both the magnitude and the direction of the car's movement since velocity includes these elements. The change of position is synonymous with the "net distance" traveled, accounting for any direction changes during the time interval.
Exploring Integration and Its Role
Integration, in the context of calculus, is a mathematical tool used to find the accumulated change over a period. With respect to the velocity function, integration allows us to determine the total change in position by calculating the area under the curve of the velocity function graph.
If the velocity is constant, integration is straightforward and involves simple multiplication. However, if the velocity changes over time, integration becomes a powerful tool to sum up all the incremental distance changes. The definite integral of the velocity function \(f(t)\) from \(t = a\) to \(t = b\) is given by:
  • \(\int_{a}^{b} f(t) \, dt\)
This integral calculates the exact change in position over the specified interval.
Dividing the Time Interval
The time interval in this scenario specifies the period over which we want to measure the car's positional change. It's defined between two points, usually denoted as \(a\) and \(b\), where \(a\) is the starting time, and \(b\) is the ending time.
For this exercise, the time interval of interest in part (a) is from 0 to 60 seconds, which directly measures the car's displacement over one minute.
In part (b), we express the interval in terms of \(T\), where \(T\) is minutes. Since each minute has 60 seconds, for \(T\) minutes, the time interval becomes \([0, 60T]\) seconds. This flexibility allows us to analyze the car's motion over various durations with ease.

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