Final answer:
The student's question involves computing a locally linear approximation for the number of people at a venue at a given time t+1. The options provided represent different rates of change: some show an increase in people, while others show a decrease. Without additional data regarding the venue's dynamics at time t, it is not possible to definitively select the correct approximation.
Step-by-step explanation:
The question involves finding a locally linear approximation for the number of people, n(t), at a venue at time t. Given that there are 600 people at the venue at time t, to approximate the number of people at time t+1, we must consider the rate of change of n with respect to t. In the options provided, this rate is represented by the coefficient of (t - value).
If we assume that the rate is constant, then we can use the first-order Taylor approximation, which is essentially the equation of the tangent line at a specific point. Hence, the linear approximation of n around time t could be given as n(t+1) = n(t) + n'(t)(t+1 - t), where n'(t) is the derivative of n with respect to t at time t.
For options:
- Option a: The rate of change is -3, which would mean that 3 people are leaving per unit of time starting at time t = 5.
- Option b: The rate of change is +3, indicating that 3 people are arriving per unit of time starting at time t = 5.
- Option c: The rate of change is -5, which implies that 5 people are leaving per unit of time starting at time t = 3.
- Option d: The rate of change is +5, suggesting that 5 people are arriving per unit of time starting at time t = 3.
Without additional context or data about the direction (increase or decrease) and the specific time at which this change begins, it's impossible to choose one option over the others as the correct linear approximation.