proponents of the peak oil theory claim that worldwide production of petroleum will soon reach a maximum and start to decline. (there is considerable debate about how soon this will occur)
let P(t)= the world's rate of petroleum production t years after 2000. let T be the date at which petroleum production actually peaks. what is the sign of P'(T)? what is the sign of P"(T)?
To determine the sign of P'(T) and P"(T), we need to understand the definitions of P'(t) and P"(t).
P'(t) represents the first derivative of P(t), which measures the rate at which the world's petroleum production is changing over time. A positive value for P'(t) indicates an increasing rate of petroleum production, while a negative value indicates a decreasing rate.
P"(t) represents the second derivative of P(t), which measures the rate of change of the rate of petroleum production. A positive value for P"(t) indicates an increasing rate of increase (acceleration) in petroleum production, while a negative value indicates a decreasing rate of increase (deceleration).
In the context of the peak oil theory, proponents argue that worldwide petroleum production will reach a maximum and start to decline. Based on this information, we can infer the following:
1. At the point of peak oil production (T), the rate of petroleum production will stop increasing and start decreasing. Therefore, P'(T) will be negative, indicating a decline in production.
2. As petroleum production continues to decrease after the peak, the rate of decline will slow down. This means that the rate of decrease in petroleum production will itself decrease. Therefore, P"(T) will be positive, indicating a deceleration in the decline of production.
In summary:
- The sign of P'(T) will be negative, indicating a declining rate of petroleum production.
- The sign of P"(T) will be positive, indicating a deceleration in the decline of petroleum production.