Calculate the average rate of change of the given function...

Calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. HINT [See Example 1.] $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{f ( x )} & 3 & 5 & 2 & -1 \\ \hline \end{array} $$ Interval: $[1,3]$

Calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. HINT [See Example 1.]
$$
\begin{array}{|c|c|c|c|c|}
\hline \boldsymbol{x} & 0 & 1 & 2 & 3 \\
\hline \boldsymbol{f ( x )} & 3 & 5 & 2 & -1 \\
\hline
\end{array}
$$
Interval: $[1,3]$

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Key Concepts

Interval

An interval in mathematics specifies a set of input values over which a function is analyzed. In the context of calculating the average rate of change, the interval provides the endpoints that are used to evaluate the function and determine how its values change from one end to the other.

Average Rate of Change

The average rate of change of a function measures how much the function's output changes per unit change in the input over a specified interval. It is computed by taking the difference between the function's values at the endpoints of the interval and dividing by the difference between the corresponding input values. This concept is analogous to calculating the slope of the secant line that connects the two points on the function.

Difference Quotient

The difference quotient is the formula (f(b) - f(a)) / (b - a), where a and b are two distinct inputs. It is the fundamental method used to compute the average rate of change over an interval and is a precursor to the concept of derivatives in calculus.

Units of Measurement

Considering units of measurement is important when computing rates of change, as the resulting value will have units that are a ratio of the function's output unit to its input unit. This ensures that the rate is not only numerically correct but also meaningful in the context of the quantities involved.

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