Question

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estimate the instantaneous rate of change at ( x = 3 )
4
your estimate needs to be within 10% of the exact answer.
question help: video

Explanation:

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

From the graph, at \( x = 2 \), the \( y \)-value is around 6, and at \( x = 3 \), the \( y \)-value is around 14.

Step2: Calculate the average rate of change (slope)

The formula for the average rate of change between \( x = a \) and \( x = b \) is \( \frac{f(b) - f(a)}{b - a} \). Here, \( a = 2 \), \( b = 3 \), \( f(2) = 6 \), \( f(3) = 14 \). So, \( \frac{14 - 6}{3 - 2} = \frac{8}{1} = 8 \). (We can also check with a smaller interval, but this gives a reasonable estimate for the instantaneous rate of change at \( x = 3 \) as the function is increasing and the slope is steepening, but the average rate from \( x = 2 \) to \( x = 3 \) is a good approximation here.)

Answer:

8