Final Answer:
a) The integral of velocity, ∫v(t) dt, represents the displacement of the particle over the interval 0 < t < 6.
b) The acceleration function, a(t), can be determined by taking the derivative of the velocity function v(t).
c) The position function, x(t), can be found by integrating the velocity function v(t) over the interval 0 < t < 6 with appropriate initial conditions.
d) The initial value problem can be solved by integrating the velocity function to find the position function and applying initial conditions to determine any constants involved.
Step-by-step explanation:
a) Evaluating the integral of velocity (∫v(t) dt) from 0 to 6 gives the displacement of the particle over the given time interval. This integral represents the area under the velocity-time graph and provides information about how far the particle has traveled along the x-axis.
b) To find the acceleration function, differentiate the velocity function v(t) with respect to time. The resulting function, denoted as a(t), represents the rate of change of velocity with respect to time, giving insight into how the particle's velocity is changing along the x-axis.
c) The position function x(t) can be determined by integrating the velocity function v(t) with respect to time, considering the initial conditions. By evaluating ∫v(t) dt with appropriate initial position information (like x(0)), the position function can be established, describing the particle's location along the x-axis at any given time.
d) Solving the initial value problem involves integrating the velocity function to obtain the position function and applying initial conditions. By substituting the given initial values (such as initial position or velocity) into the general solution obtained from the integration, any constants can be determined to provide the specific solution for the particle's motion along the x-axis within the given time frame.