For the function y=3x2: (a) Find the average rate of change of y with respect to x over the interval [3,6]. (b) Find the instantaneous rate of change of y with respect to x at the value x=3. Average Rate of Change : Instantaneous Rate of Change at x=3:

The instantaneous rate of change of with respect to at the value is 18.

Step-by-step explanation:

a) Geometrically speaking, the average rate of change of with respect to over the interval by definition of secant line:

(1)

Where:

, - Lower and upper bounds of the interval.

, - Function exaluated at lower and upper bounds of the interval.

If we know that , and , then the average rate of change of with respect to over the interval is:

The average rate of change of with respect to over the interval is 27.

b) The instantaneous rate of change can be determined by the following definition:

(2)

Where:

- Change rate.

, - Function evaluated at and .

If we know that and , then the instantaneous rate of change of with respect to is:

The instantaneous rate of change of with respect to at the value is 18.

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