If 1 dollars is invested over a 10-year period, then the...

If 1 dollars is invested over a 10-year period, then the balance $A$, where $t$ represents the time in years, is given by $A=1+0.06\|t\|$ or $A=[1+(0.055 / 365)]^{[365 t]}$ depending on whether the interest is simple interest at $6 \%$ or compound interest at $5 \frac{1}{2} \%$ compounded daily. Use a graphing utility to graph each function in the same viewing window. Which grows at a greater rate?

Key Concepts

Simple Interest

Simple interest is a method of calculating interest where the interest charge is computed solely on the original principal amount, without considering any interest accrued in previous periods. This results in linear growth over time, meaning that the increase in the account balance is constant for each period, and the formula typically takes the form A = P + rt, where P is the principal, r is the interest rate, and t is the time elapsed.

Compound Interest

Compound interest calculates interest on both the initial principal and the accumulated interest from previous periods. The effect of compounding results in an exponential growth pattern, as the interest earned in one period contributes to a larger base for calculating interest in future periods. The formula A = P(1 + r/n)^(nt) reflects how frequent compounding (with n representing the number of compounding periods per year) impacts the overall growth of the investment.

Exponential Growth

Exponential growth describes a process where the rate of change of a quantity is proportional to the current amount, leading to accelerated increases over time. In the context of compound interest, even with a lower nominal rate, the investment grows more rapidly due to the recurring application of interest to an increasingly larger base, illustrating a key difference from simple, linear growth.

Graphical Comparison

Graphical comparison involves plotting multiple functions on the same axes to visually analyze and compare their behavior over time. In financial contexts, graphing simple versus compound interest functions helps to illustrate how the linear progression of simple interest contrasts with the accelerating, exponential growth of compound interest, thereby providing a clear visual representation of which growth method yields a higher return over a given period.

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