List of basic formulas related to approximate calculations using differential:
- Approximate change in function:
[
\Delta y \approx dy = f'(x) \cdot \Delta x
]
where ( \Delta y ) is the change in function, ( dy ) is the differential of the function, ( f'(x) ) is the derivative of the function at the point ( x ), and ( \Delta x ) is the change in the argument. - Approximate value of a function:
[
f(x + \Delta x) \approx f(x) + f'(x) \cdot \Delta x
]
This expression allows us to estimate the value of a function at a point ( x + \Delta x ) based on its value at a point ( x ) and its derivative. - Approximation for small changes:
If ( \Delta x ) is small, then:
[
f(x + \Delta x) \approx f(x) + \Delta y
] - Second differential (for more accurate approximations):
[
f(x + \Delta x) \approx f(x) + f'(x) \cdot \Delta x + \frac{f”(x)}{2} \cdot (\Delta x)^2
]
This expression takes into account not only the first derivative, but also the second, which allows improving the accuracy of the approximation. - Approximation error:
The error when using the differential can be estimated as:
[
E \approx \frac{f”(c)}{2} \cdot (\Delta x)^2
]
where ( c ) is the point between ( x ) and ( x + \Delta x ).