Final answer:
The question is related to finding increasing and decreasing intervals in a function using calculus methods, specifically through derivatives and their signs over an interval. The integral is another calculus concept used to find the area under a curve between two points.
Step-by-step explanation:
The student is querying about how to find increasing and decreasing intervals using various mathematical concepts, which relate to Calculus. In calculus, to define whether the function is increasing or decreasing, the derivative of the function is analyzed.
If the derivative of a function is positive over an interval, the function is increasing on that interval. Conversely, if the derivative is negative, the function is decreasing on that interval.
Solution A involves a step-by-step computation to find this, either by manually calculating the derivative or by using mathematical software. Solution B would entail using the computational capabilities of a calculator like TI-83, 83+, or 84 to determine the derivative and analyze its sign over specified intervals to determine where the function is increasing or decreasing.
The reference to the integral indicates a separate concept in calculus which concerns finding the area under a curve. The mention of 'limits of integration a and b' is related to the definition of the upper and lower bounds of that area. When integrating a curve, these limits must be established to calculate the definite integral, which represents things such as the total displacement given a velocity-time graph (Column A, part a).