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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions문제 7.2.23b
7장, 문제 7.2.23b

Projection sensitivity
According to the 2014 national population projections published by the U.S. Census Bureau, the U.S. population is projected to be 334.4 million in 2020 with an estimated growth rate of 0.79%/yr.
b. Suppose the actual growth rate is instead 0.7%. What are the resulting doubling time and projected 2050 population?

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Identify the formula for exponential growth of a population: \(P(t) = P_0 \times e^{rt}\), where \(P_0\) is the initial population, \(r\) is the growth rate (expressed as a decimal), and \(t\) is the time in years from the initial time.
Calculate the doubling time using the formula for exponential growth doubling time: \(T_d = \frac{\ln(2)}{r}\). Here, \(r\) is the actual growth rate of 0.7%, so convert it to decimal form as \(r = 0.007\).
Determine the time interval from the base year to 2050. Since the base year is 2020, the time \(t\) is \(2050 - 2020 = 30\) years.
Use the exponential growth formula to find the projected population in 2050: \(P(30) = 334.4 \times e^{0.007 \times 30}\). Here, 334.4 million is the population in 2020, and \(r = 0.007\) is the growth rate.
Evaluate the expressions for doubling time and projected population after substituting the values, but do not calculate the final numerical results as per instructions.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Growth Model

Exponential growth describes populations increasing at a rate proportional to their current size, modeled by P(t) = P_0 * e^(rt), where P_0 is the initial population, r is the growth rate, and t is time. This model helps predict future population sizes based on continuous growth rates.
추천 영상:
09:29
Exponential Growth & Decay

Doubling Time

Doubling time is the period required for a quantity growing exponentially to double in size. It is calculated using the formula T_d = ln(2)/r, where r is the growth rate. This concept helps understand how quickly a population grows under a given rate.
추천 영상:
02:32
Verifying Solutions of Differential Equations Example 3

Applying Growth Rate to Population Projections

To project future population, the exponential growth formula uses the actual growth rate and time elapsed. Adjusting the growth rate changes the projected population, illustrating sensitivity in predictions and the importance of accurate growth estimates.
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가이드 코스
04:16
Intro To Related Rates
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Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:


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37–38. Caffeine After an individual drinks a beverage containing caffeine, the amount of caffeine in the bloodstream can be modeled by an exponential decay function, with a half-life that depends on several factors, including age and body weight. For the sake of simplicity, assume the caffeine in the following drinks immediately enters the bloodstream upon consumption.


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Overtaking City A has a current population of 500,000 people and grows at a rate of 3%/yr. City B has a current population of 300,000 and grows at a rate of 5%/yr.

b. Suppose City C has a current population of y₀ < 500,000 and a growth rate of p > 3%/yr. What is the relationship between y₀ and p such that Cities A and C have the same population in 10 years?

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A running model A model for the startup of a runner in a short race results in the velocity function v(t) = a(1 - e⁻ᵗ/ᶜ), where a and c are positive constants, t is measured in seconds, and v has units of m/s. (Source: Joe Keller, A Theory of Competitive Running, Physics Today, 26, Sep 1973)


b. Using the velocity in part (a) and assuming s(0) = 0, find the position function s(t), for t ≥ 0.

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교과서 질문

Many formulas There are several ways to express the indefinite integral of sech x.


b. Show that ∫ sech x dx = sin⁻¹ (tanh x) + C. (Hint: Show that sech x = sech² x / √(1 − tanh² x) and then make a change of variables.)

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교과서 질문

"Integral formula Carry out the following steps to derive the formula ∫ csch x dx = ln |tanh(x / 2)| + C (Theorem 7.6).


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