Summary: Big Picture Calculus – Lecture 4: The Exponential Function

Source: MIT OpenCourseWare, Professor Gilbert Strang

Introduction

The exponential function y = eˣ is the function that only calculus could create. Unlike algebraic functions, reaching eˣ always requires a limiting process—something going to zero or infinity. This makes it fundamentally a calculus concept.

The Defining Property

The most remarkable property of eˣ is that its slope equals itself: dy/dx = y. This is the first and most important differential equation—an equation connecting a function to its slope. Differential equations describe nature, and understanding this one unlocks many others.

While any multiple like 2eˣ or 10eˣ would also satisfy dy/dx = y, specifying a starting point y = 1 at x = 0 pins down the unique solution eˣ.

Exponential Growth Explained

Starting at 1 with slope 1, the function climbs. As y increases, the slope increases equally, causing faster climbing. This accelerates without bound—eˣ grows faster than any polynomial, even x¹⁰⁰.

The Multiplication Property

A key property for any exponential: eˣ × eˣ = eˣ⁺ˣ. This must be verified from the construction to confirm eˣ deserves to be called "a number to the x power."

Constructing the Function

Building y(x) from the requirement dy/dx = y and starting at y = 1:

  • Start with y = 1
  • The derivative must equal 1, so add x (slope of x is 1)
  • Now need derivative to include x, so add ½x²
  • Continue: add ⅙x³, then 1/24·x⁴, and so on

The pattern emerges: each term is xⁿ/n! where n! (n factorial) is n × (n-1) × ... × 1.

The complete series: eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + ...

The series converges because n! grows faster than xⁿ for any fixed x.

The Number e

Setting x = 1 gives e itself: e = 1 + 1 + 1/2 + 1/6 + 1/24 + ...

This equals approximately 2.718281828... It's irrational—not expressible as any fraction or terminating decimal.

The Graph of eˣ

  • At x = 0: y = 1 (starting point)
  • At x = 1: y = e ≈ 2.718
  • The curve rises with ever-increasing steepness for positive x
  • For negative x: e⁻ˣ = 1/eˣ
  • At x = -1: y = 1/e ≈ 0.37
  • The curve decays toward zero but never reaches it

Compound Interest Application

Consider 100% annual interest on $1:

  • Yearly compounding: After one year: $2 (powers of 2 thereafter)
  • Monthly compounding: (1 + 1/12)¹² — better than $2
  • Daily compounding: (1 + 1/365)³⁶⁵ — even better
  • Continuous compounding: As n → ∞, (1 + 1/n)ⁿ → e

This provides another construction of e: the limit of (1 + 1/n)ⁿ as n approaches infinity.

Generalization

The general differential equation dy/dx = cy (where c is the growth or decay rate) has solution y = eᶜˣ. When c is negative, the function decays. This single family of solutions covers exponential growth and decay at any rate.

Production: MIT OpenCourseWare and Gilbert Strang. Funded by the Lord Foundation.