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A velocity function in calculus represents how fast an object is moving at any point in time. For Carl Lewis's 100-meter dash record, the velocity function is the derivative of the given distance function. In our exercise, the initial distance function is denoted as:

• \( y_1(x) = 11.274 \left[x - 1.06\left(1-e^{-x / 1.06}\right)\right] \)

This function models the distance he ran over time. By differentiating it with respect to time, we obtain the velocity function \( y_2(x) \). This derived function tells us how fast Lewis was running at every second during the race.

This mathematical process reflects a fundamental concept in calculus where the derivative of a function gives us the rate of change of the quantity, in this case, speed. Using a graphing calculator or computer software is crucial here to precisely compute and visualize this derivative, helping us understand the dynamics of Lewis's sprint.

Graphing is a powerful analytical tool in calculus that helps visualize functions and their derivatives. In this exercise, by graphing both the distance function \( y_1(x) \) and the velocity function \( y_2(x) \), we gain insights into the motion of Carl Lewis during the 100-meter dash.

• The window for these graphs is set to an x-axis range from 0 to 9.93 seconds, corresponding to the race's duration.
• The y-axis is set from 0 to 100 meters or meters per second, reflecting the possible distances and speeds.

Through this visualization, we can trace the velocity curve to examine how Lewis's speed changes through each moment of the race. When you observe the graph, it's easier to identify key points, such as when the velocity peaks or how quickly certain speeds are reached, enhancing your understanding of the problem.

The maximum speed calculation involves identifying the highest value of the velocity function \( y_2(x) \). To find this maximum is to determine Carl Lewis's peak performance during his sprint.By using the graph tracing technique, you observe the velocity graph to find its pinnacle point, confirming the problem statement that said his maximum speed was approximately 11.27 meters per second. Using a graphing tool makes this straightforward by allowing you to follow the curve visually until you reach the top.

Furthermore, determining how quickly Lewis reached a speed of 10 meters per second, which is 95% of his maximum, helps understand his acceleration at the start of the race. By finding the x-value where \( y_2(x) = 10 \), you can ascertain the time he took to reach this crucial speed. This aspect of the exercise showcases the utility of calculus in analyzing real-world motion by using derivatives to describe dynamic changes in velocity.