Question

Writing to Learn In Example 2 (before rounding) we found the average temperature to be 65.17 degrees when we used the integral approximation, yet the average of the 13 discrete temperatures is only 64.69 degrees. Considering the shape of the temperature curve, explain why you would expect the average of the 13 discrete temperatures to be less than the average value of the temperature function on the entire interval.

Short answer

The average of the 13 discrete temperatures is less than the integral approximation because, considering the shape of the temperature curve, the integral approximation captures any increase in the curve that the discrete temperatures might miss. So if the curve is generally increasing, its integral approximation value will likely be higher than the discrete average value.

Step-by-step solution

Understanding the difference between integral approximation and discrete average

The integral approximation gives the average of the continuous function over an interval, taking into account every single point within that interval. Meanwhile, when calculating the discrete average of 13 different points, only those 13 specific temperatures are taken into account and averaged. Therefore, the discrete average often misses data between these points, leading to less accuracy.

Analyzing the curve shape

If the curve is increasing or decreasing in the part that is not covered by the 13 discrete temperatures, the integral approximation will capture that detail but the discrete average won't. An average of discrete temperatures might tend to be smaller if the function is increasing because some of the 'peaks' might not be captured by discrete points.

Application to the temperature curve

Considering the shape of the temperature curve, if the curve is mostly increasing, the integral approximation (which captures the entire curve) will generally be higher than the average of the 13 discrete temperatures (which might miss some of the peaks).