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

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

Briggs 3rd Edition

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

문제 7.2.42

7장, 문제 7.2.42

Tripling time A quantity increases according to the exponential function y(t) = y₀eᵏᵗ. What is the tripling time for the quantity? What is the time required for the quantity to increase p-fold?

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Start with the given exponential growth function: \(y(t) = y_0 e^{k t}\), where \(y_0\) is the initial quantity, \(k\) is the growth rate, and \(t\) is time.

To find the tripling time, denote this time as \(T_3\), where the quantity becomes three times the initial amount: \(y(T_3) = 3 y_0\).

Substitute into the equation: \(3 y_0 = y_0 e^{k T_3}\). Divide both sides by \(y_0\) to simplify: \(3 = e^{k T_3}\).

Take the natural logarithm of both sides to solve for \(T_3\): \(\ln(3) = k T_3\), which gives \(T_3 = \frac{\ln(3)}{k}\).

For the general \(p\)-fold increase time \(T_p\), set \(y(T_p) = p y_0\) and follow the same steps: \(p = e^{k T_p}\), so \(T_p = \frac{\ln(p)}{k}\).

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주요 개념

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Exponential Growth Function

An exponential growth function is expressed as y(t) = y₀e^(kt), where y₀ is the initial amount, k is the growth rate, and t is time. The quantity grows continuously at a rate proportional to its current value, leading to rapid increases over time.

Tripling Time

Tripling time is the time required for a quantity to become three times its initial value in an exponential growth process. It is found by solving y(t) = 3y₀, which leads to t = (ln 3)/k, using natural logarithms to isolate time.

General p-Fold Increase Time

The time for a quantity to increase p-fold in exponential growth is found by setting y(t) = p y₀ and solving for t. This gives t = (ln p)/k, showing that the time depends logarithmically on the factor p and inversely on the growth rate k.

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