The Natural Base e
There is a number so special in calculus that it gets its own letter. We define and learn how it describes continuous growth better than any other base.
Introduction
You are familiar with the number , which relates a circle's circumference to its diameter. In the world of calculus and growth, there is another irrational constant that reigns supreme: Euler's number, denoted as .
Prerequisite Connection
Recall that grows faster as gets larger. We will find where fits on this spectrum (between 2 and 3).
Today's Increment
We define and learn to manipulate expressions involving , treating it exactly like any other base.
Why This Matters
In calculus, is the "perfect" function because it is immune to the operation of differentiation—it is its own derivative. This unique property simplifies thousands of complex calculations in engineering and physics.
Key Concepts
Definition of
The number is defined as the value that approaches as gets infinitely large:
It is an irrational number, meaning its decimal expansion never repeats or terminates.
The Natural Exponential Function
The function is called the natural exponential function. Since , it represents exponential growth.
Passes through
Passes through
Horizontal Asymptote:
Why is it "Natural"?
In calculus, is the only function that is its own derivative. That is, the slope of the graph at any point equals the y-value at that point.
Worked Examples
Example 1: Simplifying with Base e
BasicSimplify the expression:
Multiply the Numerator
Combine coefficients and add exponents: and .
Numerator:
Divide by the Denominator
Subtract exponents: .
Answer
Example 2: Evaluating e
IntermediateUse a calculator to approximate to four decimal places, and interpret this value graphically.
Calculation
Find the button on your calculator. Input .
Result:
Graphical Interpretation
Since the exponent is negative, the value is small (between 0 and 1). This point lies on the left tail of the graph, close to the asymptote.
Answer
Example 3: Transformations of Rate
AdvancedGraph . Determine the asymptote, domain, and range.
Horizontal Shift
The in the exponent shifts the graph **right 2** units. The reference point moves to an x-coordinate of 2.
Vertical Shift
The at the end shifts the graph **up 1** unit. This moves the horizontal asymptote from to .
New Reference Point
Original generic point moves right 2, up 1 to . The range becomes .
Answer
Asymptote: . Domain: . Range: .
Common Pitfalls
Treating "e" as a Variable
is not like or . It is a specific number, just like . Do not try to solve for "e".
Rounding Errors
When calculating, use the button on your calculator rather than typing 2.718. Using the approximation too early can lead to significant errors in the final answer.
Real-World Application
Continuous Compound Interest
In finance, banks pay interest on your deposit. If they calculate that interest once a year, you get a certain amount. If they calculate it every month, you get slightly more (interest on interest).
If they calculate it every second of every day—continuously—the formula switches from standard algebra to using : . This is the theoretical maximum limit of interest growth.
Practice Quiz
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