Part 5/7:
Using direct substitution into the function provides us the exact change.
- Use Differentials (dy):
Calculate the derivative ( f'(x) ), which allows us to estimate changes quickly:
[
dy = f'(2) \cdot dx
]
Comparing ( \Delta y ) and ( dy ) reveals that while ( \Delta y ) is a precise value obtained through substitution, ( dy ) provides a close approximation, showcasing the practicality of differentials.
Example B: From ( x = 2 ) to ( x = 2.01 )
Repeating the procedure allows us to see how ( dy ) and ( \Delta y ) converge as the interval of change becomes smaller. As the calculations narrow down the values, we find that ( dy ) becomes an effective approximation method, particularly when direct computation becomes cumbersome.