Why is exponential growth often called one of the most powerful forces in nature and finance?
To understand why, compare the two fundamental ways quantities can change:
Linear Growth (Addition): Increases by a constant amount each step:
$$y(t) = y_0 + m \cdot t$$
For example, saving a flat $200 every year.
Exponential Growth (Multiplication): Multiplies by a constant percentage rate each step:
$$y(t) = y_0 \cdot (1 + r)^t$$
For example, an investment earning 8% compound interest annually.
In the early years, linear growth often seems to be winning because its constant addition provides a head start. But because exponential growth multiplies its ever-growing total, it inevitably accelerates, crosses over, and skyrockets past any linear rate!
Using plot2(x, y1, y2, label1, label2, title, xtitle, ytitle), we can plot two curves side-by-side on the same chart to compare their growth trajectories.
plot2(x,y1,y2,label1,label2)
Move the mouse over a dotted box for more information.
Now you try. Try changing the annual compound interest rate from 1.08 to 1.10 (10%) and see how much sooner the crossover happens!
Type your code here:
See your results here:
The code has ???? for the growth factor. Replace ???? with 1.08 and click Run to observe the exponential crossover against simple interest!
Notes:
Notice the dramatic crossover point around Year 23. By Year 40, the compound account ($21,725) is worth more than double the linear account ($9,000)!
This compounding effect applies to biology (bacterial population growth), epidemiology (viral spread), and technology (Moore's law).
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