Question

What is the role of the derivative in determining increasing or decreasing intervals, and how can one use a calculator to analyze these intervals?

a) The derivative helps identify increasing intervals by being positive, and decreasing intervals by being negative.
b) The derivative helps identify increasing intervals by being negative, and decreasing intervals by being positive.
c) The derivative helps identify increasing intervals by having values greater than a threshold, and decreasing intervals by having values below the threshold.
d) The derivative is not relevant for determining increasing or decreasing intervals.

Answer 1

The derivative indicates increasing intervals by being positive and decreasing intervals by being negative. It represents the slope of the tangent line at a point on a function. Calculators can be used to determine the sign of the derivative and identify the intervals of increase or decrease.

Step-by-step explanation:

The role of the derivative in determining increasing or decreasing intervals is pivotal. The correct answer to the question is a), which states that the derivative helps identify increasing intervals by being positive, and decreasing intervals by being negative. When the derivative of a function is positive on an interval, it indicates that the function is increasing on that interval. Conversely, a negative derivative indicates that the function is decreasing. Derivatives represent the slope of the tangent line to the function at any given point. A positive slope means the line moves up as the x-value increases, indicating an increase in function. Meanwhile, a negative slope indicates a decline. To analyze these intervals using a calculator, one can calculate the derivative at several points along the function to determine the sign of the derivative and thus identify intervals of increase or decrease.