Final answer:
The function with the larger magnitude derivative has the greater rate of change, while the function with the larger constant term as the y-intercept when the independent variable is zero has the greater initial value.
Step-by-step explanation:
To determine which function has the greater rate of change, we need to compare the derivatives of each function since the derivative represents the rate at which the function's value is changing. For position functions, the derivative gives us the velocity, which is the rate of change of position with respect to time. If we have two position functions, such as x1(t) and x2(t), the function with the larger magnitude of the derivative (velocity) at a given time has a greater rate of change.
As for the initial value of a function, it is the y-intercept or the value of the function when its independent variable (commonly t for time) is zero. If we are comparing two functions such as f(t) = at + b and g(t) = ct + d, the function with the larger constant term (b or d) has the greater initial value at t=0.