Exponential Growth and Decay: Population and Radioactive Material, Study notes of Calculus

Download Exponential Growth and Decay: Population and Radioactive Material and more Study notes Calculus in PDF only on Docsity! Population Growth Radioactive Decay Compound Interest Lecture 5 Section 7.6 Exponential Growth and Decay Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu http://math.uh.edu/∼jiwenhe/Math1432 Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 1 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth Exponential Growth of the World Population Over the course of human civilization population was fairly stable, growing only slowly until about 1 AD. From this point on the population growth accelerated more rapidly and soon reached exponential proportions, leading to more than a quadrupling within the last century. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 2 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth Annual Population Growth Rate 1980-98 (UNESCO) Annual population growth rate is the increase in a country’s population during one year, expressed as a percentage of the population at the start of that period. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 3 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth Annual Population Growth Rate 1980-98 (UNESCO) Let P(t) the size of the population P at time t. Then the growth rate k = P(t + 1)− P(t) P(t) ≈ 1 P(t) lim h→0 P(t + h)− P(t) h = P ′(t) P(t) . Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 3 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth Annual Population Growth Rate 1980-98 (UNESCO) Countries with the most rapid population growth rates tend to be located in Africa and the Middle East. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 3 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth Exponential Growth Theorem If f ′(t) = kf (t) for all t in some interval then f is an exponential function f (t) = Cekt where C is arbitrary constant. If the initial value of f at t = 0 is known, then C = f (0), f (t) = f (0)ekt Proof. f ′(t) f (t) = k ⇒ d dt ln f (t) = k ln f (t) = kt + c ⇒ f (t) = ekt+c = Cekt f (0) = Ce0 = C ⇒ f (t) = f (0)ekt . Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 4 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth Exponential Growth Theorem If f ′(t) = kf (t) for all t in some interval then f is an exponential function f (t) = Cekt where C is arbitrary constant. If the initial value of f at t = 0 is known, then C = f (0), f (t) = f (0)ekt Proof. f ′(t) f (t) = k ⇒ d dt ln f (t) = k ln f (t) = kt + c ⇒ f (t) = ekt+c = Cekt f (0) = Ce0 = C ⇒ f (t) = f (0)ekt . Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 4 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth Exponential Growth Theorem If f ′(t) = kf (t) for all t in some interval then f is an exponential function f (t) = Cekt where C is arbitrary constant. If the initial value of f at t = 0 is known, then C = f (0), f (t) = f (0)ekt Proof. f ′(t) f (t) = k ⇒ d dt ln f (t) = k ln f (t) = kt + c ⇒ f (t) = ekt+c = Cekt f (0) = Ce0 = C ⇒ f (t) = f (0)ekt . Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 4 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth When Will the World Population Double? The population was 4.5 billion in 1980, and 6 billion in 2000. Let P(t) be the population (in billion) t years after 1980 and k the annual population growth rate. P ′(t) = kP(t) with P(0) = 4.5 gives P(t) = 4.5ekt . P(20) = 6 ⇒ 4.5ek20 = 6, 20k = ln(6/4.5), k ≈ 1.43%. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 5 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth When Will the World Population Double? The population was 4.5 billion in 1980, and 6 billion in 2000. Let P(t) be the population (in billion) t years after 1980 and k the annual population growth rate. P ′(t) = kP(t) with P(0) = 4.5 gives P(t) = 4.5ekt . P(20) = 6 ⇒ 4.5ek20 = 6, 20k = ln(6/4.5), k ≈ 1.43%. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 5 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth When Will the World Population Double? The population was 4.5 billion in 1980, and 6 billion in 2000. Let P(t) be the population (in billion) t years after 1980 and k the annual population growth rate. P ′(t) = kP(t) with P(0) = 4.5 gives P(t) = 4.5ekt . P(20) = 6 ⇒ 4.5ek20 = 6, 20k = ln(6/4.5), k ≈ 1.43%. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 5 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth When Will the World Population Double? How long will it take for the population to double from 1980? To find the “double time”, we solve 2P(0) = P(0)ekt for t. ekt = 2, kt = ln 2, t = ln 2k ≈ 0.69 k% = 69 1.43 ≈ 48.5 The population will double in 48.5 years (from 1980); that is, the population will reach 9 billion midyear in the year 2028. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 5 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth When Will the World Population Double? How long will it take for the population to double from 1980? To find the “double time”, we solve 2P(0) = P(0)ekt for t. ekt = 2, kt = ln 2, t = ln 2k ≈ 0.69 k% = 69 1.43 ≈ 48.5 The population will double in 48.5 years (from 1980); that is, the population will reach 9 billion midyear in the year 2028. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 5 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth When Will the World Population Double? Future population projections are notoriously inaccurate: d dk (P(48.5)) = P(0) d dk ( e48.5k ) = 48.5 P(48.5). A difference of just 0.1% between predicted and actual growth rates translates into hundreds of millions of lives 48.5× 9× 0.1% ≈ 0.43 billion: Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 5 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth When Will the World Population Double? As of January 1, 2002, demographers were predicting that the world population will peak at 9 billion in the year 2070 and will begin to decline thereafter. A sustainable growth of the world population can not be an exponential growth. But what is it?! Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 5 / 18 Population Growth Radioactive Decay Compound Interest Human Population Growth Quiz Quiz 1. lim x→−∞ ex =: (a) 0, (b) −∞, (c) ∞. 2. lim x→∞ ex =: (a) 0, (b) −∞, (c) ∞. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 6 / 18 Population Growth Radioactive Decay Compound Interest Radioactive Decay Radioactive Waste Workers at a nuclear power plant standing near a storage pond filled with spent fuel. Radioactive waste (or nuclear waste) is a material deemed no longer useful that has been contaminated by or contains radionuclides. Radionuclides are unstable atoms of an element that decay, or disintegrate spontaneously, emitting energy in the form of radiation. Radioactive waste has been created by humans as a by-product of various endeavors since the discovery of radioactivity in 1896 by Antoine Henri Becquerel. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 7 / 18 Population Growth Radioactive Decay Compound Interest Radioactive Decay Radioactive Waste Workers at a nuclear power plant standing near a storage pond filled with spent fuel. Radioactive waste (or nuclear waste) is a material deemed no longer useful that has been contaminated by or contains radionuclides. Radionuclides are unstable atoms of an element that decay, or disintegrate spontaneously, emitting energy in the form of radiation. Radioactive waste has been created by humans as a by-product of various endeavors since the discovery of radioactivity in 1896 by Antoine Henri Becquerel. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 7 / 18 Population Growth Radioactive Decay Compound Interest Radioactive Decay Half-Life of Radioactive Material A half-life is a measure of time required for an amount of radioactive material to decrease by one-half of its initial amount. The half-life of a radionuclide can vary from fractions of a second to millions of years: sodium-26 (1.07 seconds), hydrogen-3 (12.3 years), carbon-14 (5,730 years), uranium-238 (4.47 billion years). Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 8 / 18 Population Growth Radioactive Decay Compound Interest Radioactive Decay Half-Life of Radioactive Material A half-life is a measure of time required for an amount of radioactive material to decrease by one-half of its initial amount. The half-life of a radionuclide can vary from fractions of a second to millions of years: sodium-26 (1.07 seconds), hydrogen-3 (12.3 years), carbon-14 (5,730 years), uranium-238 (4.47 billion years). Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 8 / 18 Population Growth Radioactive Decay Compound Interest Radioactive Decay Cobalt-60 Gamma Knife Radiosurgery Unit (www1.wfubmc.edu) Cobalt-60, used extensively in medical radiology, has a half-life of 5.3 years. The decay constant k is given by k = − ln 2 Thalf ≈ −0.69 5.3 ≈ −0.131. If the initial sample of cobalt-60 has a mass of 100 grams, then the amount of the sample that will remain t years after is A(t) = A(0)ekt = 100e−0.131t . Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 9 / 18 Population Growth Radioactive Decay Compound Interest Radioactive Decay Cobalt-60 Gamma Knife Radiosurgery Unit (www1.wfubmc.edu) Cobalt-60, used extensively in medical radiology, has a half-life of 5.3 years. The decay constant k is given by k = − ln 2 Thalf ≈ −0.69 5.3 ≈ −0.131. If the initial sample of cobalt-60 has a mass of 100 grams, then the amount of the sample that will remain t years after is A(t) = A(0)ekt = 100e−0.131t . Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 9 / 18 Population Growth Radioactive Decay Compound Interest Radioactive Decay Radiocarbon Dating: Carbon-14 The mathematics of radioactive decay is useful for many branches of science far removed from nuclear physics. One reason is that, in the late 1940’s, Willard F. Libby discovered natural carbon-14 (radiocarbon), a radioactive isotope of carbon with a half-life of 5730 years. W. F. Libby was awarded the Nobel Prize in 1960 for his work on carbon-14 and its use in dating archaeological artifacts, and natural tritium, and its use in hydrology and geophysics. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 10 / 18 Population Growth Radioactive Decay Compound Interest Radioactive Decay Radiocarbon Dating: Carbon-14 All living organisms take in carbon through their food supply. While living, the ratio of radiocarbon to nonradioactive carbon that makes up the organism stays constant, since the organism takes in a constant supply of both in its food. After it dies, however, it no longer takes in either form of carbon. The ratio of radiocarbon to nonradioactive carbon then decreases with time as the radiocarbon decays away. The ratio decreases exponentially with time, so a 5600-year-old organic object has about half the radiocarbon/carbon ratio as a living organic object of the same type today. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 11 / 18 Population Growth Radioactive Decay Compound Interest Radioactive Decay Radiocarbon Dating: Carbon-14 All living organisms take in carbon through their food supply. While living, the ratio of radiocarbon to nonradioactive carbon that makes up the organism stays constant, since the organism takes in a constant supply of both in its food. After it dies, however, it no longer takes in either form of carbon. The ratio of radiocarbon to nonradioactive carbon then decreases with time as the radiocarbon decays away. The ratio decreases exponentially with time, so a 5600-year-old organic object has about half the radiocarbon/carbon ratio as a living organic object of the same type today. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 11 / 18 Population Growth Radioactive Decay Compound Interest Radioactive Decay Radiocarbon Dating The Ancient footprints of Acahualinca preserved in volcanic mud near the lake in Managua, Nicaragua: 5,945 +/- 145 years by radiocarbon dating. The Chauvet Cave in southern France contains the oldest known cave paintings, based on radiocarbon dating (30,000 to 32,000 years ago). The technique of radiocarbon dating has been used to date objects as old as 50,000 years and has therefore been of enormous significance to archaeologists and anthropologists. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 12 / 18 Population Growth Radioactive Decay Compound Interest Radioactive Decay Quiz (cont.) Quiz (cont.) 3. d dx e−x =: (a) −e−x , (b) ex , (c) e−x . 4. d2 dx2 e−x =: (a) −e−x , (b) ex , (c) e−x . Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 13 / 18 Population Growth Radioactive Decay Compound Interest Compound Interest Compounding The value, at the end of the year, of a principle of $1000 invested at 6% compounded: annually (once per year): A(1) = 1000(1 + 0.06) = $1060. quarterly (4 times per year): A(1) = 1000(1 + (0.06/4))4 ≈ $1061.36. monthly (12 times per year): A(1) = 1000(1+(0.06/12))12 ≈ $1061.67. continuously: A(1) = 1000 lim n→∞ ( 1 + 0.06 n )n . Let x = n0.06 . A(1) = 1000 [ lim x→∞ ( 1 + 1 x )x]0.06 = 1000e0.06 ≈ $1061.84. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 14 / 18 Population Growth Radioactive Decay Compound Interest Compound Interest Compounding The value, at the end of the year, of a principle of $1000 invested at 6% compounded: annually (once per year): A(1) = 1000(1 + 0.06) = $1060. quarterly (4 times per year): A(1) = 1000(1 + (0.06/4))4 ≈ $1061.36. monthly (12 times per year): A(1) = 1000(1+(0.06/12))12 ≈ $1061.67. continuously: A(1) = 1000 lim n→∞ ( 1 + 0.06 n )n . Let x = n0.06 . A(1) = 1000 [ lim x→∞ ( 1 + 1 x )x]0.06 = 1000e0.06 ≈ $1061.84. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 14 / 18 Population Growth Radioactive Decay Compound Interest Compound Interest Compounding The value, at the end of the year, of a principle of $1000 invested at 6% compounded: annually (once per year): A(1) = 1000(1 + 0.06) = $1060. quarterly (4 times per year): A(1) = 1000(1 + (0.06/4))4 ≈ $1061.36. monthly (12 times per year): A(1) = 1000(1+(0.06/12))12 ≈ $1061.67. continuously: A(1) = 1000 lim n→∞ ( 1 + 0.06 n )n . Let x = n0.06 . A(1) = 1000 [ lim x→∞ ( 1 + 1 x )x]0.06 = 1000e0.06 ≈ $1061.84. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 14 / 18 Population Growth Radioactive Decay Compound Interest Compound Interest Number e as limx→∞ ( 1 + 1x )x Theorem e = lim x→∞ ( 1 + 1 x )x Proof. Define g(h) = { 1 h ln(1 + h), h ∈ (−1, 0) ∪ (0,∞), 1 h = 0 At x = 1, the logarithm function has derivative (ln x)′|x=1 = lim h→0 ln(1 + h)− ln 1 h = 1 x ∣∣∣∣ x=1 thus g is continuous at 0. The composition eg(h) is continuous at 0; lim h→0 (1 + h) 1 h = e. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 15 / 18 Population Growth Radioactive Decay Compound Interest Compound Interest Number e as limx→∞ ( 1 + 1x )x Theorem e = lim x→∞ ( 1 + 1 x )x Proof. Define g(h) = { 1 h ln(1 + h), h ∈ (−1, 0) ∪ (0,∞), 1 h = 0 At x = 1, the logarithm function has derivative (ln x)′|x=1 = lim h→0 ln(1 + h)− ln 1 h = 1 x ∣∣∣∣ x=1 thus g is continuous at 0. The composition eg(h) is continuous at 0; lim h→0 (1 + h) 1 h = e. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 15 / 18 Population Growth Radioactive Decay Compound Interest Compound Interest Continuous Compound Interest The economists’ formula for continuous compounding is A(t) = A(0)ert t is measured in years, A(t) = the principle in dollars at time t, A(0) = the initial investment, r = the annual interest rate. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 16 / 18 Population Growth Radioactive Decay Compound Interest Compound Interest Continuous Compound Interest The economists’ formula for continuous compounding is A(t) = A(0)ert t is measured in years, A(t) = the principle in dollars at time t, A(0) = the initial investment, r = the annual interest rate. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 16 / 18 Population Growth Radioactive Decay Compound Interest Compound Interest Continuous Compound Interest The economists’ formula for continuous compounding is A(t) = A(0)ert t is measured in years, A(t) = the principle in dollars at time t, A(0) = the initial investment, r = the annual interest rate. Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 16 / 18 Population Growth Radioactive Decay Compound Interest Compound Interest Outline Population Growth Human Population Growth Radioactive Decay Radioactive Decay Compound Interest Compound Interest Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 17 / 18 Population Growth Radioactive Decay Compound Interest Compound Interest Online Resources www.unesco.org/education www.populationinstitute.org www.globalchange.umich.edu www.geohive.com www.nrc.gov www.pollutionissues.com www1.wfubmc.edu/neurosurgery/Radiosurgery mathdl.maa.org www.britannica.com en.wikipedia.org Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 5 January 29, 2008 18 / 18