A slowing race Starting at the same time and place, Abe and Bob race, running at velocities u(t) = 4 / (t + 1) mi/hr and v(t) = 4e^(−t/2) mi/hr, respectively, for t ≥ 0.
b. Find and graph the position functions of both runners. Which runner can run only a finite distance in an unlimited amount of time?

Verified step by step guidance

1. Identify the position functions for Abe and Bob by integrating their velocity functions with respect to time, since position is the integral of velocity: \(s(t) = \int u(t) \, dt\) and \(r(t) = \int v(t) \, dt\).
2. For Abe, integrate \(u(t) = \frac{4}{t + 1}\): set up the integral \(s(t) = \int \frac{4}{t + 1} \, dt\). Recognize this as a natural logarithm integral.
3. For Bob, integrate \(v(t) = 4e^{-t/2}\): set up the integral \(r(t) = \int 4e^{-t/2} \, dt\). Use substitution to handle the exponential decay term.
4. Apply the initial condition that both runners start at the same place at time \(t=0\), so \(s(0) = 0\) and \(r(0) = 0\), to solve for the constants of integration in both position functions.
5. Analyze the behavior of both position functions as \(t \to \infty\) to determine which runner covers a finite distance and which can run indefinitely. This involves evaluating the limits of \(s(t)\) and \(r(t)\) as \(t\) approaches infinity.

Key Concepts

Velocity and Position Relationship

Velocity is the rate of change of position with respect to time. To find the position function from a given velocity function, you integrate the velocity over time. This process accumulates the total distance traveled starting from the initial position.

Definite and Indefinite Integration

Integration is used to find the position function from velocity. An indefinite integral gives a general antiderivative plus a constant, while a definite integral calculates the net change over a specific interval. Proper integration techniques are essential to handle functions like rational and exponential expressions.

Limits and Infinite Time Behavior

Analyzing the limit of the position function as time approaches infinity helps determine if a runner covers a finite or infinite distance. If the position approaches a finite value, the runner can only run a limited distance despite unlimited time. This concept involves understanding improper integrals and convergence.