(BQ) Part 2 book "3,000 solved problems in calculus" has contents: Exponential growth and decay, inverse trigonometric functions, integration by parts, trigonometric integrands and substitutions, improper integrals, planar vectors, polar coordinates, infinite sequences, partial derivatives,.and other contents. | CHAPTER 26 Exponential Growth and Decay A quantity y is said to grow or decay exponentially in time if D,y = Ky for some constant K. (K is called the growth constant or decay constant, depending on whether it is positive or negative.) Show that y = y0eKt, where ya is the value of y at time t = 0. Hence, ( = 0, y0=Ce° = C. Thus, yleKl is a constant C, y= CeKl. When Kl y = y0e . A bacteria culture grows exponentially so that the initial number has doubled in 3 hours. initial number will be present after 9 hours? How many times the Let y be the number of bacteria. Then y = y0eKl. By the given information, 2y0 = y0e*K, 2 = e}K, In 2 = In e3* = 3K, K = (ln2)/3. When f = 9, y = y0e9K = y0e^"2 = y0(ela2)3 = ya -2 3 = 8y0. Thus, the initial number has been multiplied by 8. A certain chemical decomposes exponentially. will remain after 3 hours? Assume that 200 grams becomes 50 grams in 1 hour. How much Let y be the number of grams present at time t. Then y = yneKl. The given information tells us that When r = 3, y = 200e3* = 200(eA:)3 50 = 200eK, grams. Show that, when a quantity grows or decays exponentially, the rate of increase over a fixed time interval is constant (that is, it depends only on the time interval, not on the time at which the interval begins). The formula for the quantity is y = y0eKl. Let A be a fixed time interval, and let t be any time. y(t + A)/y(0 = yaeK('+^/y0eK> = e*A, which does not depend on t. If the world population in 1980 was billion and if it is growing exponentially with a growth constant K = In 2, find the population in the year 2030. Let y be the population in billions in year t, with t = 0 in 1980. , = 50, y = ° 04. When /=!, y = (1 + r/100)y n . A" = In (! + /•/100) and r = 100(e" - 1). Then Hence, (1 + r/100)yn = yneK, l + r/100 = e* So If a population is increasing exponentially at the rate of 2 percent per year, what will be the percentage increase over a .
