#Accumulation of Change

#Table of Contents

1. Introduction to Accumulation of Change
2. Connection with Graphs
3. Sign of Rate of Change
4. Units of Accumulation of Change
5. Worked Example
6. Practice Questions
7. Glossary
8. Summary and Key Takeaways

#Introduction to Accumulation of Change

#What is an Accumulation of Change?

An accumulation of change refers to the total change in a quantity over a given interval, based on its rate of change.

For example, if a strawberry harvester gathers 0.5 kilograms of strawberries for each meter of strawberry plants (rate of change = 0.5 kilograms per meter), then over 6 meters, the accumulation of change is: $$0.5 \times 6 = 3 \text{ kilograms of strawberries}$$

The accumulation of change tells you the amount that a quantity changes by, not its total value.

To find a total value for the quantity, a boundary value (a known value at a specific point) is required. For instance, if 23 kilograms of strawberries have already been harvested, after another 6 meters of harvesting: $$23 + 3 = 26 \text{ kilograms of strawberries}$$

#Connection with Graphs

#Graphical Representation

If you have a graph of the rate of change function for a quantity, the area between the rate of change function and the x-axis over a given interval represents the accumulation of change over that interval.

#Methods to Calculate Areas

There are different methods to calculate such areas:

• Simple Geometry: Using areas of rectangles, triangles, trapezoids, semicircles, etc.
• Definite Integrals: For more complex shapes, a definite integral is used. For example: $${\int }_{a}^{b} f(x) , dx$$ can calculate the area between the x-axis and the graph of $y = f(x)$ from $x = a$ to $x = b$.

Refer to the 'Properties of Definite Integrals' study guide for more details.

#Sign of Rate of Change

#Positive and Negative Intervals

• Positive Rate of Change:
  • The graph will be above the x-axis.
  • The accumulation of change will be positive.
  • The quantity will be increasing.
• Negative Rate of Change:
  • The graph will be below the x-axis.
  • The accumulation of change will be negative.
  • The quantity will be decreasing.

#Units of Accumulation of Change

#Calculating Units

The units for an accumulation of change are determined by multiplying the units of the rate of change function by the units of the independent variable.

Examples:

• If the rate of change is in kilograms per meter, and the independent variable is in meters: $$\frac{\text{kilograms}}{\text{meters}} \times \text{meters} = \text{kilograms}$$
• If the rate of change is in meters per second squared, and the independent variable is in seconds: $$\frac{\text{meters}}{\text{seconds}^2} \times \text{seconds} = \frac{\text{meters}}{\text{seconds}} = \text{meters per second}$$

#Worked Example

Scenario: The rate of change of the volume of liquid in a tank $r(t)$ is modeled over time $t$ (in minutes), with $r(t)$ in gallons per minute. Initially, the tank contains 7 gallons of liquid.

#(a) Calculate the accumulation of change between $t = 0$ and $t = 5$.

This is equal to the area between the graph of $r$ and the $t$-axis.

• The area consists of a 5 by 2 rectangle and a right triangle with base 3 and height 5: $$5 \cdot 2 + \frac{1}{2}(5 \cdot 3) = 17.5 \text{ gallons}$$

#(b) Calculate the accumulation of change between $t = 5$ and $t = 9$.

The area consists of a semicircle of radius 2, but since $r$ is negative here: $$\frac{1}{2}(\pi \cdot 2^2) = 2\pi$$ So, the accumulation of change is $-2\pi$ gallons.

#(c) What is the volume of liquid in the tank at $t = 9$?

This will be the initial volume plus the total accumulation of change: $$7 + (17.5 - 2\pi) = 24.5 - 2\pi \approx 18.217 \text{ gallons}$$

#Practice Questions

Practice Question

1. If the rate of change of a quantity is 3 units per second over 4 seconds, what is the accumulation of change?

Practice Question

2. Given a rate of change function $f(x) = 2x$ from $x = 1$ to $x = 4$, calculate the accumulation of change using definite integrals.

Practice Question

3. Explain how the sign of the rate of change affects the accumulation of change and provide an example.

#Glossary

• Accumulation of Change: The total change in a quantity over a given interval based on its rate of change.
• Rate of Change: The speed at which a quantity changes over time.
• Boundary Value: A known value of a quantity at a specific point.
• Definite Integral: A mathematical tool to calculate the area under a curve between two points on the x-axis.

#Summary and Key Takeaways

#Summary

• Accumulation of change is the total change in a quantity over an interval.
• It can be graphically represented by the area under the rate of change function.
• The sign of the rate of change determines whether the quantity is increasing or decreasing.
• Units of accumulation of change depend on the rate of change and the independent variable.

#Key Takeaways

• Understand the relationship between rate of change and accumulation of change.
• Be able to calculate areas under curves using geometry and definite integrals.
• Recognize how the sign of the rate of change affects the accumulation of change.
• Always consider the units when dealing with accumulation of change.