explain why the average change from the start of a period to the end is also zero?
If you're talking about trig functions, a period is how long the function takes to return to its starting value. At that time, f(x) has not changed, so the average change is zero.
The average change from the start of a period to the end is zero because it is determined by calculating the difference between each individual value and the start value, and then taking the mean of these differences. When we consider the entire period, the positive differences are canceled out by the negative differences, resulting in an overall sum of zero. This means that, on average, there is no net change from the start to the end of the period.
To understand why the average change from the start of a period to the end is zero, we need to consider the concept of average and how it relates to change.
In any given period, let's say we have a set of values that represent some measure or quantity at different points in time. To calculate the average change, we need to determine the difference between each value and the starting value, and then calculate the mean of these differences.
Here's a step-by-step breakdown of the process:
1. Start by identifying the initial value at the beginning of the period.
2. Next, determine the difference between each value and the initial value. This means subtracting the initial value from each subsequent value.
3. Sum up all these differences.
4. Finally, divide the sum by the total number of values to calculate the average change.
Now, let's consider why the average change from the start of a period to the end is zero. The key insight lies in understanding that for any given period, the difference between the initial value and any subsequent value can be positive or negative, depending on whether the subsequent value is larger or smaller than the initial value.
Since the calculation of the average change involves adding up all these differences, the positive and negative changes will cancel each other out. This is because for every positive change, there will be an equal and opposite negative change.
As a result, the sum of the differences will be zero. Dividing this sum by the number of values will also yield zero, indicating that the average change from the start to the end of the period is zero.
In simpler terms, if you calculate the average change by considering all the differences, the positive and negative changes will balance each other out, resulting in a zero average change from the start to the end of the period.