91Ó°ÊÓ

The percentage rate of change measures how much a function's value changes, relative to its initial value, when the variable changes. Imagine you are measuring how fast something grows or shrinks. This concept helps us put that change into a percentage format, making it easier to compare and understand. For example, in the exercise, we are calculating the percentage rate of change of the function at two different points. The formula for percentage rate of change is: \[\text{Percentage Rate of Change} = \frac{\triangle g}{g(1)} \times \frac{1}{\triangle p} \times 100\] This tells us that to find the percentage rate of change, we need to determine how much the function's output (\triangle g) changes in relation to the initial output value (g(1)), and also factor in the change in the variable (\triangle p).

Calculus is a branch of mathematics that studies how things change. It's all about understanding change and motion. In this context, we use calculus to help us figure out the rate at which something changes. There are two main ideas in calculus: - **Differentiation**: This tells us how a function changes at any given point. - **Integration**: This helps us find the total accumulation of a quantity. When finding the percentage rate of change, we are effectively using differentiation to study how our function changes as the input variable changes. This makes calculus essential when analyzing functions and their behavior over intervals.

Function analysis is a process used to study and understand functions. We look at how functions behave, find their values at specific points, and examine how they change. We also might explore attributes like: - **Domain**: The set of all possible inputs for the function. - **Range**: The set of all possible outputs. - **Intervals of increase and decrease**: Where the function goes up and down. - **Maximum and minimum points**: The highest and lowest points of the function. In the given exercise, we analyze the function g(p) to find its values at specific points (p=1 and p=11). We then calculate how it changes over that interval. This gives us a clear idea of the behavior of the function across these points.

A derivative represents the rate at which a function changes at any given point. It's like the function's way of telling us how it is changing at each specific point. Mathematically, the derivative of a function g(p) with respect to p is denoted as g'(p). In the given exercise, if we were to find the derivative of g(p)=\(\frac{5}{2p+3}\), it would give us the exact rate of change of the function at any point p. Here's how you'd get it: 1. Express the function as a quotient. 2. Use the quotient rule: \(\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\) Here, using it directly might seem complex, but it’s crucial to understand that derivatives give us a precise way to calculate rates of change, which underpins the exercise of finding percentage rates of change.