Suppose that you won the top prize from a game show and are given two options. The first option is a cash gift of $10,000 and $1,000 is added per day for the next 7 days. The second option is a cash gift of 1 cent (or $0.01) that grows tenfold each day for 7 days. You must wait the entire time and get all of the prize money at the end of the week. Which option would you choose?

In the first option, the amount of money increases by the same amount ($1,000) each day, so we can represent it with a linear function. In the second option, the money grows by multiples of 10, so we can represent it with an exponential function. Let $f$ represent the amount of money $x$ days after winning with the first option, and let $g$ represent the amount of money $x$ days after winning with the second option.

For the first few days, the second option trails far behind the first. Because of the repeated multiplication by 10, however, after 7 days it surges past the amount in the first option.

What if the factor of growth is much smaller than 10? Suppose we have a third option, represented by a function $h$. The starting amount is still $0.01 and it grows by a factor of 1.5 times each day.

If we graph the function $h(x)=(0.01) \boldcdot (1.5)^x$, we see that it takes many, many more days before we see rapid growth. But given time to continue growing, the amount in this exponential option will eventually also outpace that in the linear option. If the prize rules are changed so that both prizes can grow for more than 38 days, this new exponential prize may be worth more than the linear option, but if the prizes can grow for only a shorter amount of time, the linear option is worth more.

Graph of 2 functions on grid, origin O. Horizontal axis, number of days, from .0 to 40, by 2's. Vertical axis, amount in dollars, from 0 to 90,000 by 10,000's. Linear function, f of x equals 10,000 plus 1,000 x represented by red dots. Exponential function, h of x equals 0 point 01 times 1 point 5 to the x, represented by blue dots. Red dots well above blue dots from 0 days to 37 days. Dots intersect around 38 comma 48,000. Blue dot exceeds red dot at 40 days.