Euler's Number Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

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A bank offers continuous compounding at an annual rate of 5%5\%. Using A=PertA = Pe^{rt}, find how much \1000growstoafter grows to after 10$ years.

Solution

  1. 1
    Identify: P=1000P = 1000, r=0.05r = 0.05, t=10t = 10.
  2. 2
    Compute A = 1000 \cdot e^{0.05 \times 10} = 1000 \cdot e^{0.5} \approx 1000 \times 1.6487 = \1648.72$.

Answer

A \approx \1648.72$
Continuous compounding uses the formula A=PertA = Pe^{rt}, where ee appears because the compounding interval shrinks to zero. This is a direct real-world application of Euler's number.

About Euler's Number

Euler's number eβ‰ˆ2.71828e \approx 2.71828 is the unique base for which the exponential function exe^x is its own derivative β€” the natural base for growth and decay.

Learn more about Euler's Number β†’

More Euler's Number Examples

Example 1 easy

Evaluate [formula] and state what this limit defines.

Example 2 hard

Show that the derivative of [formula] is itself, i.e., [formula], using the limit definition of the

Example 3 easy

Using a calculator, compute [formula] to four decimal places. Then determine whether [formula].

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