105–106. {Use of Tech} Races The velocity function and initial position of Runners A and B are given. Analyze the race that results by graphing the position functions of the runners and finding the time and positions (if any) at which they first pass each other.


A : v(t) = sin t; s(0) = 0 B. V(t) = cos t; S(0) = 0

Verified step by step guidance

1. Step 1: Understand the problem. You are given the velocity functions of two runners, A and B, as v_A(t) = sin(t) and v_B(t) = cos(t), along with their initial positions s_A(0) = 0 and s_B(0) = 0. The goal is to analyze the race by graphing their position functions and determining when and where they first pass each other.
2. Step 2: Recall that the position function s(t) is the integral of the velocity function v(t). For Runner A, integrate v_A(t) = sin(t) to find s_A(t). Similarly, for Runner B, integrate v_B(t) = cos(t) to find s_B(t). Use the initial conditions s_A(0) = 0 and s_B(0) = 0 to determine the constants of integration.
3. Step 3: Perform the integration. For Runner A, integrate sin(t) to get s_A(t) = -cos(t) + C_A. Use the initial condition s_A(0) = 0 to solve for C_A. For Runner B, integrate cos(t) to get s_B(t) = sin(t) + C_B. Use the initial condition s_B(0) = 0 to solve for C_B.
4. Step 4: Graph the position functions s_A(t) and s_B(t) over a suitable interval of time. This will help visualize the motion of the runners and identify any points where their positions intersect.
5. Step 5: Solve for the time t at which the runners first pass each other by setting s_A(t) = s_B(t). This involves solving the equation -cos(t) = sin(t). Once you find the time t, substitute it back into either position function to find the corresponding position where they pass each other.

Key Concepts

Velocity and Position Functions

In calculus, velocity functions describe how the position of an object changes over time. For Runners A and B, their velocity functions are given as v(t) = sin(t) and v(t) = cos(t), respectively. To find their positions, we need to integrate these velocity functions, which will yield the position functions s(t) for each runner. Understanding the relationship between velocity and position is crucial for analyzing their movements.

Integration

Integration is a fundamental concept in calculus that allows us to find the accumulated value of a function over an interval. In this context, we will integrate the velocity functions of Runners A and B to determine their position functions. The integral of a velocity function gives the position function, which is essential for determining when and where the runners meet during the race.

Graphing Functions

Graphing functions is a vital tool in calculus for visualizing the behavior of mathematical relationships. By plotting the position functions of Runners A and B, we can visually analyze their movements and identify the points where they intersect, indicating when they pass each other. Understanding how to interpret graphs will help in determining the time and positions at which the runners meet.