The table shows world average daily oil consumption...

\begin{equation} \begin{array}{l}{\text { The table shows world average daily oil consumption from }} \\ {1985 \text { to } 2010 \text { measured in thousands of barrels per day. }} \\ {\text { (a) Compute and interpret the average rate of change from }} \\ {1990 \text { to } 2005 . \text { What are the units? }}\end{array} \end{equation}\begin{equation} \begin{array}{l}{\text { (b) Estimate the instantaneous rate of change in } 2000 \text { by }} \\ {\text { taking the average of two average rates of change. }} \\ {\text { What are its units? }}\end{array} \end{equation}

\begin{equation}
\begin{array}{l}{\text { The table shows world average daily oil consumption from }} \\ {1985 \text { to } 2010 \text { measured in thousands of barrels per day. }} \\ {\text { (a) Compute and interpret the average rate of change from }} \\ {1990 \text { to } 2005 . \text { What are the units? }}\end{array}
\end{equation}\begin{equation}
\begin{array}{l}{\text { (b) Estimate the instantaneous rate of change in } 2000 \text { by }} \\ {\text { taking the average of two average rates of change. }} \\ {\text { What are its units? }}\end{array}
\end{equation}

Key Concepts

Derivative Approximation Using Finite Differences

A common technique for approximating the instantaneous rate of change is to use finite differences. This method involves calculating the average rate of change over a very small interval around the point of interest, often by averaging two average rates computed over short intervals. This approach is useful when an exact derivative is difficult to compute or when working with empirical data.

Units in Rate Calculations

When calculating rates of change, it is important to consider the units involved. The units of the rate are determined by dividing the units of the quantity by the units of the independent variable. This ensures that the interpretation of the rate is meaningful in context, and the units remain consistent with the dimensional analysis.

Average Rate of Change

The average rate of change is a measure of how a quantity changes between two values of an independent variable. It is computed by taking the difference in the dependent variable values and dividing by the difference in the independent variable values. This concept is analogous to the slope of the secant line connecting two points on a curve and provides an overall rate over an interval.

Instantaneous Rate of Change

The instantaneous rate of change represents how a quantity is changing at a specific point. It is the limit of the average rate of change as the interval approaches zero, which is conceptually related to the derivative of a function at a point. This measure gives the instantaneous speed of change at that particular moment or value of the independent variable.

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