Final answer:
To analyze a bug's motion on the x-axis, we study its position, velocity, and acceleration functions. The acceleration function is the derivative of the velocity, and the position is found by integrating velocity. To determine when the bug stops or walks left, we analyze the velocity function for times when it is zero or negative, respectively.
Step-by-step explanation:
When we analyze the motion of a bug or a particle along the x-axis, we assess its position, velocity, and acceleration over time to understand its kinematic behavior. The velocity function often provides a means to deduce the acceleration by taking its derivative. Likewise, the position can be inferred by integrating the velocity function or evaluating it at specific times, assuming we have the initial conditions.
Acceleration is the time derivative of velocity, thus if the velocity function is v(t) = A + Bt⁻¹, the acceleration would be found by differentiating this expression with respect to time. Similarly, if given an acceleration function, velocity can be found by integrating the acceleration over time, and the position function by integrating the velocity.
The particle's position at the initial moment (when it first starts walking or moving) is commonly given by evaluating the position function at t = 0 or by the initial conditions provided. The times when the bug is stopped can be found by setting the velocity function equal to zero and solving for time t, which gives us the specific moments when the bug's velocity is zero, indicating it has stopped. To determine the intervals when the bug is walking to the left, we look for when the velocity is negative, as a negative velocity indicates motion in the negative x-direction.