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estimate the instantaneous rate of change at x = 3

Explanation:

Step1: Recall the concept of instantaneous rate of change

The instantaneous rate of change of a function at a point is the slope of the tangent line at that point. For a linear - like part of the curve near \(x = 3\), we can estimate the slope using two points.

Step2: Select two points near \(x = 3\)

Let's take the points \((2,7)\) and \((3,12)\) (by observing the graph).

Step3: Calculate the slope

The slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Here, \(x_1 = 2,y_1 = 7,x_2=3,y_2 = 12\). So \(m=\frac{12 - 7}{3 - 2}\).
\[m=\frac{5}{1}=5\]

Answer:

5