Calculus How To

Exponential Growth: Simple Definition, Step by Step Examples

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Calculus Definitions >

Exponential growth is an increase in some quantity that follows the relationship

N(t) = A e(kt)

where A and k are positive real-valued constants.

Before diving further into the mathematics, let’s look at a graphical example of exponential growth. This plot assumes that A = 3 and k = 1.

exponential growth

The function’s initial value at t=0 is A=3. The variable k is the growth constant. The larger the value of k, the faster the growth will occur.

Differential Equation

The exponential behavior explored above is the solution to the differential equation below:

dN/dt = kN

The differential equation states that exponential change in a population is directly proportional to its size. Initially, the small population (3 in the above graph) is growing at a relatively slow rate. However, as the population grows, the growth rate increases rapidly.

Exponential Growth: Example Problems

Exponential growth can be found in a range of natural phenomena, from the growth of bacterial populations to the speed of computer processors.

Problem 1: A colony of bacteria doubles its population every 4 hours. If the colony originally has ten bacteria, how large will the colony be 24 hours later?

Solution: Since the colony has an original population of 10, then A=10. Knowing that the population will be 20 four hours later, we can solve for the growth constant.

  • N(t) = A e(kt)
  • 20 = 10 e(k*4 hours)
  • ln(2) = (4 hours)*k
  • k = 0.173 /hours

Then, the growth constant can be used to determine the population’s size one day later.

  • N(24 hours) = 10 e(0.173 /hours * 24 hours)
  • N(24 hours) = 635

Amazingly, the original handful of bacteria will blossom into a colony of nearly a thousand in one day’s time. That’s the power of exponential growth.

Problem 2: A client deposits $100 in a savings account at the local bank. The account grows by 1% interest, compounded annually. What will the value of the account be after ten years?

Solution: The initial size of the account is $100, so A=100. The account’s value will be $101 after one year, due to the interest. Knowing this, we can calculate the growth constant.

N(t) = A e(kt)


101 = 100 e(k*1 year)

ln(1.01) = (1 year)*k

k = 0.00995 /years

To find the value of the account at ten years, t=10.

N(10 years) = $100 e(0.00995/years * 10 years)

N(10 years) = $110.46

References

Matthews, John A. “Exponential Growth.” 2014: 387–387. Print.

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Stephanie Glen. "Exponential Growth: Simple Definition, Step by Step Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/exponential-growth/
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