The Number e and Continuous Compounding
Master Euler's number, the natural exponential function, and continuous compound interest applications.
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Questions Covered in This Set
10 cards to master
What is the approximate value of Euler's number e?
e ≈ 2.71828... It's the limit of (1 + 1/n)^n as n approaches infinity.
What is the continuous compound interest formula?
A = Pe^(rt), where A is final amount, P is principal, r is annual rate (decimal), and t is time in years.
What is the domain and range of f(x) = e^x?
Domain: all real numbers (-∞, ∞). Range: all positive real numbers (0, ∞).
What is the y-intercept of the natural exponential function?
(0, 1) because e^0 = 1.
What is the Rule of 69.3?
With continuous compounding, divide 69.3 by the interest rate percentage to approximate the time it takes for an investment to double.
How does e arise from compound interest?
As compounding frequency increases toward infinity (continuous), (1 + 1/n)^n approaches e ≈ 2.718.
What unique property does e^x have in calculus?
The derivative of e^x is e^x itself—it equals its own rate of change.
What is the horizontal asymptote of f(x) = e^x?
y = 0 (the x-axis) as x approaches negative infinity.
Give two real-world applications of e^x beyond finance.
Population growth (bacteria, viruses), radioactive decay, Newton's Law of Cooling, and drug concentration in bloodstream.
If $5,000 is invested at 6% continuously for 10 years, what is the final amount?
A = 5000·e^(0.6) ≈ $9,110.59
