The derivative A′(x) equals f(x) because moving the endpoint from x to x+h adds a thin strip whose area increment is approximately f(x)h. Dividing this increment by h and taking the limit as h approaches zero yields the local average height, which continuity ensures approaches the exact endpoint height f(x).
Conditions: The function f is continuous at x.; A(x) is defined as the accumulation of f from a fixed lower limit to x.
The derivative A′(x) equals f(x) because moving the endpoint from x to x+h adds a thin strip whose area increment is approximately f(x)h. Dividing this increment by h and taking the limit as h approaches zero yields the local average height, which continuity ensures approaches the exact endpoint height f(x).
Conditions: The function f is continuous at x.; A(x) is defined as the accumulation of f from a fixed lower limit to x.
Continuity ensures that the local average height of the function over a small interval approaches the exact height at the endpoint as the interval width shrinks to zero. This property is crucial for proving that the derivative of the accumulation function A(x) is exactly the integrand f(x), as it allows the replacement of the average value with the point value in the limit.
Conditions: The function f is continuous at the point x.; The interval width h approaches zero.
Continuity ensures that the local average height of the function over a small interval approaches the exact height at the endpoint as the interval width shrinks to zero. This property is crucial for proving that the derivative of the accumulation function A(x) is exactly the integrand f(x), as it allows the replacement of the average value with the point value in the limit.
Conditions: The function f is continuous at the point x.; The interval width h approaches zero.