Exponential Growth Wins
Aligned to F-LE.A.3 — Common Core State Standards for Mathematics.
What this lesson teaches
A quantity growing exponentially will eventually overtake any quantity growing linearly, quadratically, or as a polynomial, no matter how large the other one starts. The repeated multiplying compounds quickly.
Worked examples
Example 1
Compare f(x) = 2^x and g(x) = 10x. At x = 4, f = 16 and g = 40, so g leads. But at x = 7, f = 128 and g = 70 — the exponential has passed the linear.
Example 2
Even a slow exponential wins: 1.01^x eventually passes 1000x, just not until x is in the thousands.
Example 3
A quadratic also loses to an exponential. x^100 is beaten by 2^x, far out, which is why 'polynomial time' matters in computing.
Practice questions
- Compare 2^x and 5x at x = 5.
- Which is larger at x = 10: 2^x or x^2?
- Explain in words why doubling beats adding a fixed amount over time.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
El Crecimiento Exponencial Gana
Una cantidad que crece de forma exponencial terminará por superar a cualquier cantidad que crezca de forma lineal, cuadrática o polinomial, sin importar qué tan grande empiece la otra. La multiplicación repetida se acumula rápido.
Ejemplo 1: Compara f(x) = 2^x y g(x) = 10x. En x = 4, f = 16 y g = 40, así que g va adelante. Pero en x = 7, f = 128 y g = 70: la exponencial ya pasó a la lineal.
Ejemplo 2: Hasta una exponencial lenta gana: 1.01^x acaba rebasando a 1000x, solo que no hasta que x llega a los miles.
Ejemplo 3: Una cuadrática también pierde contra una exponencial. x^100 es superada por 2^x, muy lejos, por eso importa el 'tiempo polinomial' en computación.
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