Accumulation of Change (College Board AP® Calculus AB)

Accumulation of change

What is an accumulation of change?

• If you know the rate of change of a quantity,

  • then an accumulation of change is the actual change that occurs to the quantity over a given interval

  • E.g. 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 the accumulation of change over 6 meters is kilograms of strawberries

• Note that the accumulation of change does not tell you the total value for the quantity in question

  • It only tells you the amount that quantity changes by

  • E.g. after 6 meters we know that the amount of strawberries harvested has increased by 3 kilograms

    • But not the total amount of strawberries harvested so far

• To find a total value for the quantity you need to know a boundary value

  • This is a value for the quantity at a particular point

  • E.g. if 23 kilograms of strawberries have been harvested so far

    • then after another 6 meters of harvesting

    • there will be a total of 23+6=29 kilograms of strawberries harvested

What is the connection between accumulation of change and graphs?

• If you have a graph of the rate of change function for a quantity

  • then the area between the rate of change function and the x-axis over a given interval

  • is equal to the accumulation of change over that interval

• There are different methods available to calculate such areas

  • Sometimes simple geometry can be used

    • Areas of rectangles, triangles, trapezoids, semicircles, etc.

    • See the Worked Example

  • A definite integral can also be used

    • This may be necessary if the area is not a simple geometric shape

    • E.g. can be used to calculate the area between the x-axis and the graph of between and

      • See the 'Properties of Definite Integrals' study guide

How is the sign of the rate of change connected to the accumulation of change?

• If the rate of change function is positive over a given interval, then :

  • the graph of the rate of change function will be above the x-axis

  • the accumulation of change over the interval will be positive

  • the quantity will be increasing over the interval

• If the rate of change function is negative over a given interval, then :

  • the graph of the rate of change function will be below the x-axis

  • the accumulation of change over the interval will be negative

  • the quantity will be decreasing over the interval

What are the units for an accumulation of change?

• The units for an accumulation of change will be equal to

  • the units of the rate of change function

  • times the units of the independent variable

• For example

  • If the rate of change is in kilograms per meter, and the independent variable is in meters

    • Then the units of accumulation of change are

      • 

  • If the rate of change is in meters per second per second (i.e. meters per second squared), and the independent variable is in seconds

    • Then the units of accumulation of change are

      •