Question

question 8
estimate the instantaneous rate of change at ( x = 3 )
your estimate needs to be within 10% of the exact answer.
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Explanation:

Step1: Identify points near \( x = 3 \)

We can use the slope of the secant line between \( x = 2 \) and \( x = 3 \) to approximate the instantaneous rate of change at \( x = 3 \). From the graph, at \( x = 2 \), the \( y \)-value is \( 6 \), and at \( x = 3 \), the \( y \)-value is \( 14 \) (approximate from the grid).

Step2: Calculate the slope (rate of change)

The formula for the slope (rate of change) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( x_1 = 2 \), \( y_1 = 6 \), \( x_2 = 3 \), \( y_2 = 14 \). So, the slope is \( \frac{14 - 6}{3 - 2} = \frac{8}{1} = 8 \). (We can also check with another interval, but this gives a reasonable estimate within 10% of the exact value, likely.)

Answer:

\( 8 \) (or a value close to 8, depending on more precise graph reading, but 8 is a good estimate here)