Accumulated growth is the total change that builds up over time when something is growing at a changing rate. In calculus, this idea is represented by the area under a rate curve, such as r(t), from a starting time to an ending time. It matters in many real situations because populations, bank balances, temperatures, and chemical amounts often do not change at a constant rate.
Instead of multiplying one rate by one time, we add up many tiny contributions.
Key Facts
- Total accumulated growth from t = a to t = b is ∫_a^b r(t) dt.
- If r(t) is a rate of change of Q(t), then ∫_a^b r(t) dt = Q(b) - Q(a).
- Final amount equals initial amount plus accumulated change: Q(b) = Q(a) + ∫_a^b r(t) dt.
- For a constant growth rate r, accumulation is r(b - a), which is the area of a rectangle.
- A rate graph above the time axis gives positive accumulation, while a rate graph below the time axis gives negative accumulation.
- Units of accumulation are rate units times time units, such as people/year × years = people.
Vocabulary
- Accumulation
- Accumulation is the total amount of change gathered over an interval by adding many small changes.
- Rate of growth
- A rate of growth describes how fast a quantity increases or decreases at a particular time.
- Definite integral
- A definite integral gives the signed area under a curve between two input values.
- Initial value
- An initial value is the amount of a quantity at the beginning of the time interval being studied.
- Net change
- Net change is the final value of a quantity minus its initial value over an interval.
Common Mistakes to Avoid
- Using the final rate as if it were the total growth. A rate such as 40 people/year is not the same as 40 people because the time interval must be included.
- Forgetting the initial value when finding a final amount. The integral gives change, not the final quantity unless the starting amount is zero.
- Ignoring negative regions below the time axis. Area below the axis subtracts from the total net change because the rate is negative.
- Mixing up units in the answer. If r(t) is dollars per month and t is measured in months, the accumulated change is in dollars, not dollars per month.
Practice Questions
- 1 A population grows at a rate r(t) = 20 + 5t people per year, where t is in years. How many people are added from t = 0 to t = 4?
- 2 An investment gains money at a rate r(t) = 100e^(0.02t) dollars per year. Approximate the accumulated gain from t = 0 to t = 10 using ∫_0^10 100e^(0.02t) dt.
- 3 A growth-rate graph is above the time axis for the first half of an interval and below the time axis for the second half. Explain how you would decide whether the final amount is greater than, less than, or equal to the initial amount.