93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:
vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).
d. Which car ultimately gains the lead and remains in front?

Verified step by step guidance

1. Understand that the position of each car at time \( t \) is found by integrating its velocity function from 0 to \( t \). This gives the displacement \( s(t) \) for each car.
2. Write the position functions as integrals of the velocity functions: \( s_A(t) = \int_0^t \frac{88u}{u + 1} \, du \), \( s_B(t) = \int_0^t \frac{88u^2}{(u + 1)^2} \, du \), and \( s_C(t) = \int_0^t \frac{88u^2}{u^2 + 1} \, du \).
3. Evaluate or analyze the behavior of each position function as \( t \to \infty \) to determine which car travels the farthest distance over a long time, since the car with the greatest displacement will be in the lead ultimately.
4. Compare the limits or growth rates of \( s_A(t) \), \( s_B(t) \), and \( s_C(t) \) as \( t \to \infty \) to see which one dominates.
5. Conclude which car remains in front based on which position function grows fastest or has the greatest limit as time goes to infinity.

Key Concepts

Velocity and Position Relationship

Velocity is the rate of change of position with respect to time. To determine which car is ahead, we need to find each car's position by integrating its velocity function over time. The position function gives the displacement from the start, allowing comparison of their locations at any time.

Definite Integration for Displacement

Displacement over a time interval is found by integrating the velocity function from the start time to the given time. This process accumulates the total distance traveled, accounting for changes in velocity, and is essential to compare the cars' positions as time progresses.

Limits and Long-Term Behavior

Analyzing the limit of the position functions as time approaches infinity helps determine which car ultimately leads. Understanding how velocity and position behave for large time values reveals which car maintains the lead indefinitely.