The following table gives approximate values of the average...

The following table gives approximate values of the average annual atmospheric rate of increase in carbon dioxide $\left(\mathrm{CO}_{2}\right)$ each decade since $1960,$ in parts per million (ppm). Estimate the total increase in atmospheric $\mathrm{CO}_{2}$ between 1964 and 2013 .

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Key Concepts

Rate of Change

This concept refers to how a quantity, such as atmospheric CO2 concentration, increases or decreases over time. It represents the derivative of the quantity with respect to time and is central to determining how much the quantity has changed over a given period.

Average Rate

In many practical scenarios, especially when dealing with empirical data, a rate may be given as an average over a certain time period. This simplifies calculations by assuming a constant rate over that interval, which allows for the estimation of total change by multiplying the average rate by the duration of the period.

Summation of Incremental Changes

When different time intervals have different rates of change, the overall change is estimated by calculating the change for each interval (by multiplying the average rate by the interval’s length) and then summing these changes. This technique is analogous to using the area under a curve in a piecewise fashion.

Integration of a Rate Function

In a more formal mathematical sense, if the rate of change of a quantity is given as a function of time, the total change over an interval can be exactly determined by integrating this function over the time period. This process accumulates all the infinitesimal changes to yield the net change.

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