For a function smooth over the interval a \leq x \leq a+h , what does the expression $$ \frac{f(a+h) \ - \ f(a)}{h} $$ represent? (I:1)

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BlackTom AI · Accepted Answer

We start by understanding what the expression (f(a+h) - f(a)) / h represents for a function that is smooth on [a, a+h].
Option a: The integral from a to a+h. This would correspond to the accumulated area under the curve, not to a quotient of differences. The given expression is a difference quotient, not an integral, so this choice describes a different concept entirely.
Option b: The second derivative at x = a. The second derivative involves the rate of change of the first derivative, typically expressed as f''(a) or by a limit of a difference of slopes. The provided quotient uses f(a+h) and f(a) directly, divided by h, which does not measure curvature or second-order behavior at a. Therefore this option is incorrect.
Option c: The average rate of change over the interval [a, a+h]. This quotient precisely measures how much f changes on average per unit increase in x across the interval from a to a+h, and it aligns with the standard definition of the average rate of change: [f(a+h) − f(a)] / h. This choice matches the concept accurately.
Option d: The instantaneous rate of change at x = a. The instantaneous rate of change is the derivative f'(a), which is the limit of (f(a+h) − f(a)) / h as h → 0. Since the expression given uses a finite h and averages the change over the interval, it does not pin down the instantaneous slope at a unless h tends to 0. Hence this option is not correct in general.
In summary, the quotient given is best described as measuring how much f changes on average over the interval from a to a+h, rather than the integral, the second derivative, or the instantaneous rate at a (without taking a limit as h → 0).

For a function smooth over the interval a \leq x \leq a+h , what does the expression $$ \frac{f(a+h) \ - \ f(a)}{h} $$ represent? (I:1)

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