Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problem 7.2.48

Chapter 7, Problem 7.2.48

Rule of 70 Bankers use the Rule of 70, which says that if an account increases at a fixed rate of p%/yr, its doubling time is approximately 70/p. Use linear approximation to explain why and when this is true.

Verified step by step guidance

Start by expressing the doubling time problem in terms of exponential growth: if an amount grows at a rate of \(p\%\) per year, the amount after time \(t\) years is given by \(A(t) = A_0 e^{rt}\), where \(r = \frac{p}{100}\) is the growth rate as a decimal.

The doubling time \(T\) satisfies \(A(T) = 2A_0\), so we set up the equation \(2A_0 = A_0 e^{rT}\), which simplifies to \(2 = e^{rT}\).

Take the natural logarithm of both sides to solve for \(T\): \(\ln(2) = rT\), so \(T = \frac{\ln(2)}{r}\).

Use linear approximation by noting that \(\ln(2) \approx 0.693\), which is close to \(0.7\), and rewrite \(T\) as \(T \approx \frac{0.7}{r} = \frac{0.7}{p/100} = \frac{70}{p}\), which is the Rule of 70.

This approximation is valid when \(p\) is small because the exponential function can be closely approximated by its linearization near zero growth rates, making the Rule of 70 a good estimate for doubling time at modest growth rates.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Growth and Doubling Time

Exponential growth describes how quantities increase at a rate proportional to their current value, leading to doubling over consistent intervals. The doubling time is the period it takes for an amount to double, which depends on the growth rate. Understanding this helps explain why the Rule of 70 approximates doubling time for fixed percentage growth.

Linear Approximation (Differentials)

Linear approximation uses the tangent line at a point to estimate function values near that point. By approximating nonlinear functions with linear ones, we can simplify complex calculations. In this context, it helps approximate the natural logarithm function to derive the Rule of 70.

Natural Logarithm and Its Approximation

The natural logarithm function, ln(x), relates to continuous growth and is key in calculating exact doubling times. Near x = 1, ln(1 + r) ≈ r for small r, which allows simplification of the doubling time formula. This approximation underpins why dividing 70 by the percentage rate gives a close estimate.

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