Textbook Question
Theorem 7.8
Differentiate sinh⁻¹ x = ln (x + √(x² + 1)) to show that d/dx (sinh⁻¹ x) = 1 / √(x² + 1).
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.2.23b
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.2.23bChapter 7, Problem 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?
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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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Related Practice
Textbook Question
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?
Textbook Question
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.
Textbook Question
Depreciation of equipment A large die-casting machine used to make automobile engine blocks is purchased for \$2.5 million. For tax purposes, the value of the machine can be depreciated by 6.8% of its current value each year.
b. After how many years is the value of the machine 10% of its original value?
Textbook Question
Definitions of hyperbolic sine and cosine Complete the following steps to prove that when the x- and y-coordinates of a point on the hyperbola x² - y² = 1 are defined as cosh t and sinh t, respectively, where t is twice the area of the shaded region in the figure, x and y can be expressed as
x = cosh t = (eᵗ + e⁻ᵗ) / 2 and y = sinh t = (eᵗ - e⁻ᵗ) / 2.
b. In Chapter 8, the formula for the integral in part (a) is derived:
∫ √(z² − 1) dz = (z/2)√(z² − 1) − (1/2) ln|z + √(z² − 1)| + C.
Evaluate this integral on the interval [1, x], explain why the absolute value can be dropped, and combine the result with part (a) to show that:
t = ln(x + √(x² − 1)).
Textbook Question
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.)