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.
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.