Calculus Explanation: Compound vs. Simple Interest Over 25 Years
To explain the difference between the two graphs over 25 years using calculus, we look at both the rate of change (derivatives) that creates the gap and the area between the curves (definite integrals) that measures the cumulative difference over time.
At Year 25, the vertical dollar gap between annual compounding ($54,274.33) and simple interest ($27,500.00) reaches $26,774.33. Integrating the difference between these two curves from \(t = 0\) to \(t = 25\) yields a total shaded area of 185,627.93 dollar-years.
1. The Equations for Each Graph
Let \(t\) represent time in years, with a starting principal \(P = 10,000\) and interest rate \(r = 0.07\):
- Compound Interest Function (Curved): \(C(t) = 10,000(1.07)^t\)
- Simple Interest Function (Linear): \(S(t) = 10,000 + 700t\)
Note: In pure calculus, continuous compounding \(C(t) = 10,000e^{0.07t}\) is often used because \(e\) simplifies differentiation, whereas banks and annuities use annual compounding \((1.07)^t\).
2. Derivatives: Why the Gap Accelerates
Taking the first derivative \(\left(\frac{d}{dt}\right)\) shows the instantaneous speed (slope) at which your money is growing at any exact moment \(t\):
- Simple Interest Slope: \(S'(t) = 700\). The derivative is a constant, meaning your wealth grows at a flat speed of $700 per year forever. The second derivative is \(S''(t) = 0\) (zero acceleration).
- Compound Interest Slope: \(C'(t) = 10,000(1.07)^t \ln(1.07) \approx 676.59(1.07)^t\). Because \(t\) remains in the exponent of the derivative, your growth speed constantly increases. At \(t = 25\), your balance is growing at an instantaneous rate of $3,672.14 per year—more than five times faster than the simple interest line.
3. Definite Integral: Measuring the Area Under & Between the Graphs
In calculus, "the difference under the two graphs" is found by subtracting the area under the simple interest line from the area under the compound interest curve using a definite integral from \(t = 0\) to \(t = 25\):
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Find the Antiderivative:
\[\left[ \frac{10,000(1.07)^t}{\ln(1.07)} - 10,000t - 350t^2 \right]_{0}^{25}\]
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Evaluate from 0 to 25:
- Area under Compound Curve: \(\frac{10,000(1.07)^{25}}{\ln(1.07)} - \frac{10,000}{\ln(1.07)} = 802,175.43 - 147,800.00 = \mathbf{654,375.43}\)
- Area under Simple Line: \(10,000(25) + 350(25)^2 = 250,000 + 218,750 = \mathbf{468,750.00}\)
- Net Difference Between Curves: \(654,375.43 - 468,750.00 = \mathbf{185,627.93}\)
What That Number Actually Means
While the vertical distance at \(t = 25\) tells you how much extra cash you have in your pocket at the finish line ($26,774.33), the area between the curves (185,627.93 dollar-years) measures your cumulative "time-weighted wealth advantage."
If you divide that total area by the 25-year timeframe \(\left(\frac{185,627.93}{25}\right)\), it tells you that your average balance over the entire 25-year period was $7,425.12 higher than it would have been without compounding.
