Madison Finance opened two branches on September 1 ( t=0 )....

Madison Finance opened two branches on September 1 ( $t=0$ ). Branch $A$ is located in an established industrial park, and Branch $B$ is located in a fast-growing new development. The net rate at which money was deposited into Branch $A$ and Branch $B$ in the first 180 business days is given by the graphs of $f$ and $g$, respectively (see the figure). Which branch has the larger amount on deposit at the end of 180 business days? Justify your answer.

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Key Concepts

Area Under the Curve in Graphical Analysis

When a rate function is graphed, the area under the curve represents the total accumulation of the quantity over the given period. This concept is vital as it allows one to visualize and compare the cumulative deposits of each branch by comparing the areas under the curves of their respective deposit rate functions.

Definite Integration

Definite integration is the mathematical process used to determine the accumulation of a quantity over an interval given its rate of change. In deposit problems, integrating the net deposit rate over time yields the total amount deposited, making this concept essential for comparing cumulative effects.

Net Rate of Change

This concept refers to the rate at which a quantity changes over time. In this context, the net rate is represented by the functions showing how money is being deposited into each branch. Understanding net rates is crucial for analyzing how a quantity accumulates over a period.

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