Final answer:
The differential at a point is found by taking the derivative of the position function with respect to time to find velocity, then substituting the given x and t values to calculate the change in position for a small change in time.
Step-by-step explanation:
The differential of a function represents how the function's value changes in response to a small change in its input value. In physics, particularly when dealing with motion, it's common to analyze position as a function of time. The differential would thus be the velocity, which is the derivative of the position function with respect to time. To estimate a change in position, Δx, given a small change in time, Δt, you would use the differential.
Steps to find the differential and estimate change:
- Find the derivative of the position function with respect to time to get velocity.
- Substitute the given x (position) and t (time) values into the velocity function to find the slope at that point.
- Use this slope (velocity) to multiply by the small change in time to estimate the small change in position.
The slope is the key in estimating the change, and since it's given as constant in this scenario, any two points on the graph could be used to find the slope.