List of basic formulas related to approximate calculations using differential:

  1. 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.
  2. 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.
  3. Approximation for small changes:
    If ( \Delta x ) is small, then:
    [
    f(x + \Delta x) \approx f(x) + \Delta y
    ]
  4. 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.
  5. 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 ).

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