Question

Use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ g(x)=x^{4}-x^{2}+2 ;[1,3] $$

Short answer

Once the calculations for average and instantaneous rates are completed, their comparison will help understand the behavior of the function within the interval [1,3]. It indicates if the function is increasing more rapidly near x=1 or x=3, or if it has a relatively consistent rate of increase or a point of inflection.

Step-by-step solution

Graphing the Function

Using a graphing utility, input the function \( g(x)=x^{4}-x^{2}+2 \) to obtain its graph. By observing the graph, one can get a visual idea of how the function behaves over the interval [1,3]

Calculating the Average Rate of Change

The average rate of change is calculated using the formula \( (g(b)-g(a))/(b-a) \), where a and b are the limits of the interval. In this case, a=1 and b=3. Substituting these values into the function and then into the average rate formula, we find the value of the average rate.

Determine the Instantaneous Rates of Change

The instantaneous rate of change can be found by taking the derivative of the function at the endpoints. The derivative of the function \( g(x)=x^{4}-x^{2}+2 \) is \( g'(x)=4x^{3}-2x \). Substituting the endpoints (1, 3) into this derivative provides the instantaneous rates of change at these points.

Comparison of Rates

The final step is to compare the average rate of change to the instantaneous rates at the end points. This comparison can show whether the function is increasing or decreasing on average.

Conclusion

If the average rate is closer to the instantaneous rate at x=1, this indicates that the function is increasing more rapidly near x=1. If it is closer to the instantaneous rate at x=3, then it is increasing more rapidly near x=3. If it is in between, the function could be increasing at a steady pace, or it could have a point of inflection within the interval.