Question

What does the term "increasing on the interval" refer to in calculus?

a) The function is rising continuously over a specific range.
b) The function's derivative is positive over a given range.
c) The function's derivative is negative over a specific interval.
d) The function is at its maximum value over a given range.

Answer 1

In calculus, a function is said to be 'increasing on an interval' when its derivative is positive over that interval, which means that as the input values of the function increase, so do the output values within that range.

Step-by-step explanation:

The term “increasing on the interval” in calculus refers to a situation where a function's output values become larger as the input values (often x-values) also become larger, within a certain range of the input values. This is seen graphically as the function rising from left to right over the specified interval.

Importantly, this concept is closely associated with the function's derivative, which will be positive over that interval. In calculus, the slope of the tangent line to the curve represents the derivative at a particular point, so if the curve is increasing, the slopes of the tangent lines are positive.

As a result, option b) The function's derivative is positive over a given range is the correct interpretation of a function increasing on an interval.