The Exponential Function: y = eˣ
Lecture Digest — Big Picture Calculus, Lecture 4
Lecturer: Professor Gilbert Strang
Source: MIT OpenCourseWare
Topic: The exponential function and its unique properties
Executive Summary
This lecture establishes the exponential function y = eˣ as the function that "only calculus could create." Unlike algebraic functions, eˣ requires a limiting process to define—whether through infinite series or continuous compounding. Its defining property is extraordinary: the slope equals the function itself (dy/dx = y). Professor Strang constructs the function from first principles, proves its multiplicative property, determines the value of e ≈ 2.718, and connects the mathematics to real-world compound interest.
Why the Exponential Function Matters
The exponential function holds a unique position in mathematics because it is the solution to the most fundamental differential equation: dy/dx = y. This makes it indispensable for modeling any phenomenon where the rate of change is proportional to the current value—compound interest, population growth, radioactive decay, and countless other natural processes.
The function cannot be constructed through pure algebra. Every approach requires a limiting process: something going to zero or infinity. This is why eˣ belongs distinctly to calculus.
The Defining Property: Slope Equals Function
The exponential function satisfies the differential equation:
dy/dx = y
with initial condition y(0) = 1
This means at any point on the curve, the slope equals the height. The consequences are remarkable: starting at y = 1 with slope = 1, the function climbs. As it climbs higher, the slope increases (since slope = y), causing faster climbing. This self-reinforcing growth means eˣ eventually outpaces any polynomial—even x¹⁰⁰.
The initial condition y(0) = 1 is essential; without it, any constant multiple like 2eˣ or 10eˣ would also satisfy dy/dx = y.
Constructing eˣ: The Infinite Series
Professor Strang builds the function by requiring y = dy/dx at every stage:
Starting with y = 1 (so dy/dx = 1), we need to add x to make the derivative equal 1. But then y = 1 + x requires dy/dx = 1 + x, so we add x²/2. Continuing this pattern yields:
eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + ... + xⁿ/n! + ...
The factorial denominators (n! = n × (n-1) × ... × 1) grow faster than any power of x, ensuring the series converges for all x values. This infinite series is the rigorous definition of eˣ.
The Multiplicative Property
A true exponential function must satisfy:
eˣ · eˣ = eˣ⁺ˣ
This property—essential to the concept of exponentiation—emerges naturally from the series construction. The proof involves multiplying the two series expansions and recognizing the binomial pattern in the resulting coefficients.
An important consequence: e⁻ˣ = 1/eˣ. Since eˣ · e⁻ˣ = e⁰ = 1, the function for negative x is simply the reciprocal. This explains exponential decay: as x → -∞, eˣ → 0.
The Number e
Setting x = 1 in the series gives:
e = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + ... e ≈ 2.71828...
The number e is irrational—it cannot be expressed as a fraction or terminating decimal. Professor Strang notes you can verify e lies between 2 and 3: the first three terms already sum to 2.5, but the rapidly shrinking remaining terms never push the total past 3.
The Graph of eˣ
Key features of the curve:
- Passes through (0, 1) with slope 1
- Passes through (1, e) ≈ (1, 2.718)
- For x > 0: exponential growth—the curve climbs ever more steeply
- For x < 0: exponential decay—the curve approaches but never reaches zero
- At x = -1: y = 1/e ≈ 0.37
The curve is always positive, always increasing, and always concave up.
Application: Compound Interest and the Emergence of e
The number e arises naturally from compound interest. Consider $1 at 100% annual interest:
| Compounding Frequency | Formula | Year-End Value |
|---|---|---|
| Annually | 1 × 2 | $2.00 |
| Monthly | (1 + 1/12)¹² | ~$2.61 |
| Daily | (1 + 1/365)³⁶⁵ | ~$2.71 |
| Continuously | lim (1 + 1/n)ⁿ | e ≈ $2.718 |
As compounding becomes continuous, the limit approaches e exactly. This is a second, independent construction of the number e—and it matches the series definition perfectly.
Important caveat: The expression (1 + 1/n)ⁿ approaches e as n → ∞. This is not "1^∞ = e" (which is meaningless), but rather a delicate limit where both the base approaches 1 and the exponent approaches infinity in a specific coordinated way.
Generalization: dy/dx = cy
The general first-order linear differential equation with constant coefficient:
dy/dx = cy, with y(0) = 1
has solution:
y = eᶜˣ
When c > 0, this models growth (interest, population). When c < 0, it models decay (radioactive decay, cooling). The constant c represents the rate—100% interest corresponds to c = 1; a 5% decay rate corresponds to c = -0.05.
This single function family solves an enormous class of differential equations central to physics, biology, economics, and engineering.
Key Takeaways for Study
Memorize the defining property: dy/dx = y, y(0) = 1 → y = eˣ
Know the series: eˣ = Σ (xⁿ/n!) from n = 0 to ∞
Understand why it converges: n! grows faster than xⁿ for any x
Remember key values: e⁰ = 1, e¹ ≈ 2.718, e⁻¹ ≈ 0.368
Recognize the compound interest connection: e = lim (1 + 1/n)ⁿ
Internalize the multiplicative property: eᵃ · eᵇ = eᵃ⁺ᵇ and e⁻ˣ = 1/eˣ
Generalize to dy/dx = cy: solution is y = eᶜˣ
Connections to Other Topics
- Differential equations: This lecture introduces the concept; later lectures will explore more complex equations
- Limits: The rigorous theory of limits underlies both the series definition and the compound interest construction
- Logarithms: The natural logarithm ln(x) is the inverse of eˣ (likely covered in subsequent lectures)
- Complex numbers: Euler's formula e^(ix) = cos(x) + i·sin(x) extends these ideas spectacularly
This digest reorganizes and synthesizes Professor Strang's Lecture 4 from MIT OpenCourseWare's Big Picture Calculus series.