The Exponential Function: y = eˣ

Lecture Summary — Big Picture Calculus, Lecture 4

Lecturer: Professor Gilbert Strang
Source: MIT OpenCourseWare

Introduction

Professor Strang introduces the exponential function y = eˣ as "the function that only calculus could create." Unlike algebraic functions, every approach to eˣ requires a limiting process—something going to zero or infinity. He defers the full theory of limits to a later lecture and proceeds to construct this important function.

The Defining Property

The exponential function has the remarkable property that its slope equals itself: dy/dx = y. This is our first differential equation—an equation connecting a function to its slope. Differential equations are the fundamental language for describing nature.

Since any constant multiple (2eˣ, 10eˣ) would also satisfy dy/dx = y, we need an initial condition: y(0) = 1. Starting at 1 with slope 1, the function climbs. As y increases, so does the slope, causing faster climbing. The function eventually outpaces any polynomial, even x¹⁰⁰.

Professor Strang outlines his plan: construct the function from its property, verify it satisfies eˣ · eˣ = eˣ⁺ˣ, graph it, determine the value of e, and discuss applications like compound interest.

Constructing the Function

Beginning with y = 1 (so dy/dx = 1), we must add x to get a derivative of 1. But then y = 1 + x, so dy/dx must equal 1 + x, requiring x²/2. Continuing: to get a derivative of x², we need x³/6 (since the derivative of x³ is 3x², so we divide by 3·2 = 6).

The pattern that emerges is:

eˣ = 1 + x + x²/2! + x³/3! + ... + xⁿ/n! + ...

The denominator n! (n factorial = n × (n-1) × ... × 1) grows extremely fast—faster than xⁿ—which ensures the series converges and doesn't "blow up."

Verifying the Multiplication Property

For eˣ to deserve the name "exponential," it must satisfy eˣ · eˣ = eˣ⁺ˣ. Professor Strang indicates this can be verified by multiplying the two series together, with the binomial coefficients emerging naturally. The n! terms become essential in making this work.

The Value of e

Setting x = 1 gives:

e = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + ...

Summing just the first few terms: 1 + 1 + 0.5 + 0.167 = 2.667, already close to e. The remaining terms shrink too rapidly to reach 3. The actual value is e ≈ 2.71828..., an irrational number that cannot be expressed as any fraction or finite decimal.

The Graph

At x = 0, the function starts at y = 1 with slope 1. At x = 1, it reaches height e ≈ 2.718. The curve climbs ever more steeply as x increases.

For negative x: since eˣ · e⁻ˣ = e⁰ = 1, we have e⁻ˣ = 1/eˣ. At x = -1, the height is 1/e ≈ 1/3. As x becomes more negative, the curve decays toward zero but never touches the x-axis. To the right, it grows "exponentially."

Application: Compound Interest

Professor Strang illustrates with $1 at 100% annual interest. If compounded yearly, you have $2 after one year, $4 after two, $8 after three—powers of 2.

Compounding monthly: each month you get 1/12 of your balance added. After 12 months: (1 + 1/12)¹² > 2.

Compounding daily: (1 + 1/365)³⁶⁵, even larger.

In general, compounding n times per year gives (1 + 1/n)ⁿ. As n approaches infinity—continuous compounding—this approaches e. This is not "1^∞ = e" (which is meaningless), but rather a specific limit where both base and exponent change together.

Generalization

The lecture concludes by generalizing to dy/dx = cy, where c is the growth or decay rate. If c = 1, we have our original equation. If c is the interest rate (or negative for decay), the solution is:

y = eᶜˣ

Taking the derivative, the c comes down as a factor, giving cy—exactly what the equation requires. This single modification solves a whole family of differential equations with the function calculus created.

Summary of Professor Gilbert Strang's Lecture 4 from MIT OpenCourseWare's Big Picture Calculus series.