0. Functions

Changing bases Convert the following expressions to the indicated base.

$\ln \mid x\mid$ using base 5

Convert the following expressions to the indicated base.

$a^{\frac{1}{\ln a}}$ using basa e, for $a>0$ and $a\ne 1$

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>

{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.

b. How long does it take for the population to reach 5000 fish? How long does it take for the population to reach 90% of the carrying capacity?

Computer Explorations

Use a CAS to perform the following steps in Exercises 55–62.

a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.

2y² + (xy)¹/³ = x² + 2, P(1,1)

Suppose a quantity described by the function y(t) = y₀eᵏᵗ, where t is measured in years, has a doubling time of 20 years. Find the rate constant k.

Geometric means A quantity grows exponentially according to y(t) = y₀eᵏᵗ. What is the relationship among m, n, and p such that y(p) = √(y(m)y(n))?

"General relative growth rates Define the relative growth rate of the function f over the time interval T to be the relative change in f over an interval of length T:

R_T = [f(t + T) − f(t)] / f(t)

Show that for the exponential function y(t) = y₀ e^{kt}, the relative growth rate R_T, for fixed T, is constant for all t."

Atmospheric pressure The pressure of Earth’s atmosphere at sea level is approximately 1000 millibars and decreases exponentially with elevation. At an elevation of 30,000 ft (approximately the altitude of Mt. Everest), the pressure is one-third the sea-level pressure. At what elevation is the pressure half the sea-level pressure? At what elevation is it 1% of the sea-level pressure?

Uranium dating Uranium-238 (U-238) has a half-life of 4.5 billion years. Geologists find a rock containing a mixture of U-238 and lead, and they determine that 85% of the original U-238 remains; the other 15% has decayed into lead. How old is the rock?

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.

An individual consumes two cups of coffee, each containing 90 mg of caffeine, two hours apart. Assume the half-life of caffeine for this individual is 5.7 hours.

b. Determine the amount of caffeine in the bloodstream 1 hour after drinking the second cup of coffee.

39–40. LED lighting LED (light-emitting diode) bulbs are rapidly decreasing in cost, and they are more energy-efficient than standard incandescent light bulbs and CFL (compact fluorescent light) bulbs. By some estimates, LED bulbs last more than 40 times longer than incandescent bulbs and more than 8 times longer than CFL bulbs. Haitz’s law, which is explored in the following two exercises, predicts that over time, LED bulbs will exponentially increase in efficiency and exponentially decrease in cost.

Haitz’s law predicts that the cost per lumen of an LED bulb decreases by a factor of 10 every 10 years. This means that 10 years from now, the cost of an LED bulb will be 1/10 of its current cost. Predict the cost of a particular LED bulb in 2021 if it costs 4 dollars in 2018.

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?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. If the rate constant of an exponential growth function is increased, its doubling time is decreased.

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?

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.