Integration by Partial Fractions 1. If f(x) = P(x) / Q(x) with P(x) and Q(x) polynomials AND Q(x) a higher order than P(x) AND Q(x) factorable in linear factors then we can rewrite f(x) as a sum of rational fractions with linear denominators, which are each integrable. 2. The numerator of each will be an unknown constant that we must solve for by adding the fractions and equating powers of x 3. If P(x) is a higher order, divide P(x) by Q(x). The remainder will be your "partial fraction" (this probably makes very little sense until you see an example!) Dec 8 1:02 AM ex) 2/(x+5) + 3/(x+1) Dec 8 1:02 AM 1
ex) 4/(x+1) + 2/(x 2) Dec 8 1:02 AM ex) 3x + 4 + 1/(x+3) + 2/(x+1) Dec 8 1:02 AM 2
Try these 7/(x+4) + 8/(x 5) 3x^2+4x+1 + 1/(x+2) 2/(x+1) 2/(2x+5) + 1/(x+1) Dec 8 1:02 AM Exponential Growth and Decay The rate of change is proportional to the quantity. Example population growth is proportional to the current population change is mass of an isotope is proportional to the current mass c o is the quantity at time t = 0 k is the growth/decay constant given a growth k and c o, we can predict y(t) given (y 1, t 1 ) and (y 2, t 2 ), we can calculate k and c o y = c o e kt 3
y = c o e kt where c o = initial value k = growth constant t = time y = final amount This is the governing equation for exponential growth (k>0) and decay (k<0) ex) A colony of bacteria grows exponentially. If there are 10000 bacteria at time = 3 hours and 40000 bacteria at time = 5 hours, what is the growth constant? What was the number of bacteria at time t = 4 hours? Half Life Given a quantity that decays exponentially, the half life is the time it takes for the amount to be reduced to 1/2 the current amount The half life is NOT the decay constant, but we can determine k from the half life t 1/2! Given, y 1 = c o e kt What is the amount one half life later? 4
Polonium 210 has a half life of 138 days. What is the weight of a 10 gram sample after 414 days? How long does it take for the sample to decay to 1 mg? The mass of a nuclear isotope sample is 500 grams. 20 days later the mass of the sample is 415 grams. What is the half life of the isotope? What is the mass of the sample after 1 year? 5
Newtonian Cooling where S is the ambient temp, T is temp, t is time Cooling rate of a mass (solid or liquid) is proportional to the difference between the temperature of the mass and the ambient temperature. This is a "rough" model of conductive heat transfer there are multiple modes of heat transfer and many variables that affect the cooling rate!!! Ln(T S) = k t + c T S = c 1 e kt T o = S + c 1 e 0 c 1 = T o S T = S + (T o S) e kt T = S + c 1 e kt Suppose Fred s corpse was discovered in a motel room at midnight and its temperature was 80 o F. The temperature of the room is kept constant at 60 o. Two hours later the temperature of the corpse dropped to 75 o F. Find the time of Fred s death. 6
Exponential Growth and Decay The number of bacteria in a culture, N, grows at a rate of 2/5 y bacteria per hour, where t is hours. At time t = 0 hours, the number of bacteria was 7500. What is the number of bacteria at time t = 5 hours. A population of animals grows according to dy/dt = ky, where y is the number of animals, t is time in years, and k is a constant. If the population doubles every 10 years, what is the value of k? A puppy weighs 2.0 pounds at birth and 3.5 pounds 2 months later. If the weight of the puppy during the first 6 months increases at a rate proportional to its weight, what is the puppy's weight at 3 months? Polonium 210 has a half life of 138 days. What is the weight of a 10 gram sample after 414 days? How long does it take for the sample to decay to 1 mg? 7
The Voyager 1 spacecraft is powered by a Pu 238 radio isotope thermal generator which produced 470 watts of electricity at launch in 1973. Pu 238 has a half life of 87.7 years. a) How much power is Voyager 1 producing now? b) How much electrical energy, in Joules, has the Voyager 1 spacecraft produced? [1 Joule = 1 watt second] A cup of hot chocolate is initially 200 o F, and is set on a kitchen counter where the ambient air is 68 o. If the hot chocolate cools at a rate proportional to the difference of the hot chocolate s temperature and room temperature a) Write a differential equation that describes the cooling, in terms of the constant b) If the hot chocolate cools to 150 o F after 7 minutes, find the value of the constant k to the nearest thousandth c) Use your value of k to find how long it will take the hot chocolate to reach a temperature of 120 of. 8
Logistic Growth Remember exponential growth? dy/dx = ky as y increases, the rate of changes increases with no bound! would this accurately model the growth of a population over a long period of time? would population really grow without bound? in nature there are self limiting mechanism (food supply, increase in predators, disease, etc) logistic growth is a combination of exponential growth and a limiting factor on the rate of increase Logistic Growth Differential Equation exponential growth limiting term: as P > M, (M P) > 0 which limits the growth rate becomes negative if population exceeds M Solution is k is the growth constant M is the carrying capacity A is a constant, A = M/P o 1 9
The solution to the logistic equation comes from integration with partial fractions Extra Credit you do the rest! A population of fish, P, is described by the logistic model a) What is the carrying capacity? where t is in years. b) Write an expression for P if the initial population is 350 individuals. c) What is the population after 3 years? d) When is the population changing the fastest? e) What is the rate when the population changing the fastest? 10
A biologist surveys a remote Canadian island and determines that there are 900 lemmings on the island and are reproducing with a growth factor of 0.00036, and that the carrying capacity of the island is 1500 lemmings. If the number of lemmings follows a logistic model, how many lemmings are expected after 4 years. How long would it take for the population to reach 1450 lemmings? A rumor is spreading like wildfire at AHS (you supply the rumor). The rate at which the rumor spreads is proportional to the product of the number of students that have heard the rumor (more mouths = faster rate) and the number that have not heard the rumor (fewer people to tell = slower rate). a) If there are 2400 students at the high school, write a differential equation describing the rate of the spread of the rumor, in terms of a growth factor k, and the number of days, d, the rumor has been spreading. b) Find k if 10 students initially knew the rumor, and 2 days later 1/3 of the students knew. c) How long will it take for 80% of the students to hear the rumor? d) How long will it take for 95% of the students to hear the rumor? e) Was your model a logistic model? If so, what is the carrying capacity? f) Will a logistic model ever allow the population to exceed the carrying capacity?(explain) 11