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

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

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

Problem 7.2.23a

Chapter 7, Problem 7.2.23a

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.
a. Based on these figures, find the doubling time and the projected population in 2050. Assume the growth rate remains constant.

Verified step by step guidance

Identify the type of growth model: Since the population grows at a constant percentage rate, this is an example of exponential growth, which can be modeled by the formula $P(t) = P_0 e^{rt}$, where $P_0$ is the initial population, $r$ is the growth rate (as a decimal), and $t$ is time in years.

Convert the given growth rate from a percentage to a decimal: $0.79\% = 0.0079$ per year. This will be used as the value of $r$ in the exponential growth formula.

Calculate the doubling time using the formula for exponential growth doubling time: $T = \frac{\ln(2)}{r}$. This formula comes from setting $P(t) = 2P_0$ and solving for $t$.

Calculate the projected population in 2050 by first determining the time elapsed from 2020 to 2050, which is $t = 30$ years. Then use the exponential growth formula: $P(30) = 334.4 \times e^{0.0079 \times 30}$, where 334.4 million is the initial population in 2020.

Evaluate the expression for $P(30)$ to find the projected population in 2050. This will give the population assuming the growth rate remains constant over the 30 years.

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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 a process where the quantity increases at a rate proportional to its current value, often modeled by P(t) = P_0 * e^(rt). Here, P_0 is the initial population, r is the growth rate, and t is time. This model is essential for projecting population changes over time assuming a constant growth rate.

Doubling Time

Doubling time is the period required for a quantity growing exponentially to double in size. It can be calculated using the formula T_d = ln(2)/r, where r is the growth rate expressed as a decimal. This concept helps determine how quickly the population will double under a constant growth rate.

Natural Logarithm and Its Application

The natural logarithm (ln) is the inverse of the exponential function and is used to solve for time or growth rate in exponential equations. In population projections, ln helps isolate variables when calculating doubling time or future population values, making it a crucial tool for interpreting growth models.

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Derivative of the Natural Logarithmic Function

Related Practice

Textbook Question

61–62. Points of intersection and area

a. Sketch the graphs of the functions f and g and find the x-coordinate of the points at which they intersect.

f(x) = sech x, g(x) = tanh x; the region bounded by the graphs of f, g, and the y-axis

Textbook Question

Chemotherapy In an experimental study at Dartmouth College, mice with tumors were treated with the chemotherapeutic drug Cisplatin. Before treatment, the tumors consisted entirely of clonogenic cells that divide rapidly, causing the tumors to double in size every 2.9 days. Immediately after treatment, 99% of the cells in the tumor became quiescent cells which do not divide and lose 50% of their volume every 5.7 days. For a particular mouse, assume the tumor size is 0.5 cm³ at the time of treatment.

a. Find an exponential decay function V₁(t) that equals the total volume of the quiescent cells in the tumor t days after treatment.

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.

a. What is the value of the machine after 10 years?

Textbook Question

ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.

a. Make a sketch of the function f(x) = 1/x on the interval [1, 2]. Explain why the area of the region bounded by y = f(x) and the x-axis on [1, 2] is ln 2.

Textbook Question

Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.

a. coth 4

Textbook Question

Terminal velocity Refer to Exercises 95 and 96.

a. Compute a jumper’s terminal velocity, which is defined as lim t → ∞ v(t) = lim t → ∞ √(mg/k) tanh (√(kg/m) t).