The profits of a small company for each of the first five...

The profits of a small company for each of the first five years of its operation are given in the following table: a. Plot points representing the profit as a function of year, and join them by as smooth a curve as you can. b. What is the average rate of increase of the profits between 2012 and $2014 ?$ c. Use your graph to estimate the rate at which the profits were changing in 2012 .

The profits of a small company for each of the first five years of its operation are given in the following table:
a. Plot points representing the profit as a function of year, and join them by as smooth a curve as you can.
b. What is the average rate of increase of the profits between 2012 and $2014 ?$
c. Use your graph to estimate the rate at which the profits were changing in 2012 .

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Key Concepts

Graphing Data

Graphing data involves plotting points on a coordinate system to represent the relationship between two variables. In the context of functions, each point corresponds to an input (often time) and an output (often profit or another measured quantity). Connecting these points with a smooth curve can help identify trends and patterns in the data even when the relationship is not strictly linear.

Average Rate of Change

The average rate of change of a function over an interval is a measure of how much the function's output changes per unit change in the input. It is calculated by taking the difference in the function's values at the endpoints of the interval and dividing it by the length of the interval. This concept is analogous to finding the slope of a line that connects two points on the graph of the function.

Instantaneous Rate of Change

The instantaneous rate of change of a function at a specific point is the derivative at that point, representing the slope of the tangent to the function's graph. It provides a precise measure of how fast the function is changing exactly at that moment, and can often be estimated by looking at the steepness of the curve in a small neighborhood around the point in question.

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