The number e is one of the most important constants in mathematics, with value e ≈ 2.71828. It appears whenever change is proportional to the current amount, such as in compound interest, population growth, radioactive decay, and cooling. In calculus, e is special because exponential functions with base e have the simplest derivative and integral rules.
This makes e the natural base for describing continuous change.
Understanding Calculus: The Number e
One way to see where e comes from is to compare different compounding schedules. Imagine money earning a fixed annual rate. If the bank adds interest once per year, the balance grows by one large step.
If it adds interest twice, each step is smaller, but the second step earns interest on the first. More frequent compounding keeps raising the final balance, though the increases get smaller. In the limiting case, interest is added continuously.
The special number that appears in this limit is e. It is not chosen because it looks convenient. It is forced by the mathematics of endlessly repeated tiny percentage changes.
The key idea is proportional change. A quantity does not always gain the same number of units each second. A growing bacteria culture gains more bacteria when its population is large than when it is small.
A radioactive sample loses more atoms at first because more unstable atoms are present. In both cases, the rate depends on the current amount. Calculus represents this by saying that the rate of change equals a constant times the amount.
Solving that rule produces an exponential function. A positive constant gives growth.
A negative constant gives decay. The constant controls how quickly the change happens, while the starting amount sets the initial size.
The natural logarithm is useful because it reverses exponential growth. It can turn a problem involving repeated multiplication into one involving addition. This is especially helpful when finding time.
For example, a scientist may know a sample's starting mass, its remaining mass, and its decay constant. Taking the natural logarithm lets them isolate the time. Logarithms are defined only for positive inputs in this setting.
That restriction makes sense because an exponential expression with a positive base never becomes zero or negative. Students should keep track of units too. If a rate is measured per day, time must be measured in days for the model to work properly.
Graphs reveal another important feature. An exponential growth curve rises slowly at first, then becomes steep because each increase is based on a larger amount. A decay curve falls quickly early on, then levels toward zero without crossing it in an ideal model.
The graph of the natural logarithm has the opposite behavior. It grows without bound, but very slowly, and it is only drawn to the right of zero. When learning these functions, distinguish a percentage rate from a fixed added amount.
A savings account receiving ten dollars each month follows a linear pattern. An account receiving a fixed percentage follows an exponential pattern. This difference determines which equation, graph, and calculus method make sense.
Key Facts
- e ≈ 2.71828
- e = lim as n approaches infinity of (1 + 1/n)^n
- For continuous growth, A = Pe^(rt)
- The derivative of e^x is d/dx(e^x) = e^x
- The derivative of a^x is d/dx(a^x) = a^x ln(a)
- ln(x) is the inverse of e^x, so ln(e^x) = x and e^(ln x) = x for x > 0
Vocabulary
- e
- The number e is an irrational constant approximately equal to 2.71828 that naturally describes continuous growth and decay.
- Natural base
- The natural base is e, the base for exponential functions whose rate of change matches their current value.
- Continuous compounding
- Continuous compounding is the process of applying interest or growth at every instant, modeled by A = Pe^(rt).
- Natural logarithm
- The natural logarithm ln(x) is the logarithm with base e and is the inverse function of e^x.
- Exponential growth
- Exponential growth occurs when a quantity increases at a rate proportional to its current amount.
Common Mistakes to Avoid
- Treating e as exactly 2.71828 is wrong because e is irrational and the decimal approximation never ends or repeats.
- Using A = P(1 + r)^t for continuous compounding is wrong because that formula assumes compounding once per time period, not continuously.
- Forgetting the chain rule in d/dx(e^(kx)) is wrong because the derivative is k e^(kx), not just e^(kx).
- Thinking ln(x) means 1/x is wrong because ln(x) is a logarithm, while 1/x is its derivative.
Practice Questions
- 1 Compute (1 + 1/100)^100 and compare it with e ≈ 2.71828. Is it less than or greater than e?
- 2 An account has $500 invested at an annual rate of 6% compounded continuously. Use A = Pe^(rt) to find the amount after 4 years.
- 3 Explain why e^x is called the natural exponential function in calculus, using its derivative as part of your reasoning.