Let d(t) be the shortest distance between Earth and the nearest other planet as a function of time t
(where t is in years and t = 0 is noon on January 1, 1000 A.D., say). Is d continuous? Why or why not?
To determine whether the function d(t) is continuous, we need to examine its properties and check for any potential discontinuities.
In this scenario, we have a function d(t) that represents the shortest distance between Earth and the nearest other planet as a function of time t. Since this function involves physical distances, it is reasonable to assume that it is continuous, but let's analyze it further.
Continuous functions share several key characteristics:
1. The function is defined at every point within its domain.
2. The limit of the function exists at each point in its domain.
3. The function's graph is a single, unbroken curve, with no jumps, holes, or infinite oscillations.
In the case of d(t), let's consider these characteristics:
1. The function is defined for any given time t, as long as it falls within the specified time range. Since we are assuming time t is in years, any real number within the given range is a valid input, and hence d(t) is defined for every point within its domain.
2. The limit of the function exists at each point in its domain. Since the function represents the shortest distance between two celestial bodies, it is unlikely to have any sudden disruptions or jumps that prevent the limit from existing. As time progresses, the distance between planets may change due to their orbits, but these changes occur gradually and smoothly. Therefore, we can expect the limit of d(t) to exist at every point in its domain.
3. The function's graph, representing the shortest distance between Earth and the nearest other planet, is expected to be a continuous curve without jumps, holes, or infinite oscillations. While we cannot provide a specific graph without further information, the nature of planetary orbits suggests gradual and smooth changes in distance. This continuity ensures that the graph will not exhibit any sudden interruptions, jumps, or infinite oscillations.
Based on these considerations, it is reasonable to conclude that the function d(t) representing the shortest distance between Earth and the nearest other planet is continuous.