position function
[position function|position function] The position function s(t) describes a particle's location along a line at each time t.
Definition
position function The position function s(t) describes a particle's location along a line at each time t. Assuming differentiability, the Mean <a href='/en/entity/value-theorem'>Value Theorem</a> applies to conclude that at some time c ∈ (0, 1), the instantaneous speed equals the average speed of 45 mph.
Mechanism
position function The Mean <a href='/en/entity/value-theorem'>Value Theorem</a> ensures there exists a point c in (0, 5/2) where the derivative s'(c) equals the average rate of change. Differentiating the position function s(t) yields s'(t) = -32t, confirming that the instantaneous velocity at time c matches the mean value.
Effects
position function affects the calculation of position over time, with its derivative representing velocity. The Mean <a href='/en/entity/value-theorem'>Value Theorem</a> guarantees a point c in the interval (0, 5/2) where the instantaneous velocity equals the average velocity. This function's continuity and differentiability ensure the existence of such a point.