Understanding Euler's Method and Logistic Growth in Calculus
Learn about Euler's Method, a recursive process used in calculus to approximate solutions to differential equations, and explore Logistic Growth models for population dynamics. Discover how these concepts are applied in real-world scenarios and improve your understanding of advanced calculus topics.
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mastermathmentor.com presents CALCULUS ON THE WALL Helping students learn and teachers teach 17. Euler s Method & Logistic Growth Created by: Stu Schwartz Sample Version Un-narrated Graphics: Apple Grapher: Version 2.3 Math Type: Version 6.7 Intaglio: 2.9.5a Fathom: Version 2.11
The Recursive Process Euler s method is a recursive process. By that we mean that we start with a point and apply the process to get another point. We use that new point, apply the process, and get another point, continuing the process for a finite number of points. Euler s method uses the formula: dy = (dy/dx) x. Let y = f(x) be the solution to the differential equation dy/dx = x2 - 2y with f(2) = -3. Approximate f(3) using Euler s method with x = 0.5. Start with a DEQ, a point (x0, y0), and a given x. x x x x ynew=yold + dy ynew=yold + dy ynew=yold + dy ynew=yold + dy 2 2 2 2 -3 -3 -3 -3 2.5 2.5 2.5 2.5 3 3 3 3 Calculuate the slope dy/dx using the DEQ at the point. -3 + 10(0.5) = 2 = 2 2.52 2(2) = 2.25 -3 + 10(0.5) 2+2.25(0.5) = 3.125 dy/dx dy/dx dy/dx dy/dx 22 - 2(-3) = 10 = 10 = 10 22 - 2(-3) 22 - 2(-3) Find the value of dy = (dy/dx) x. 1. Create the chart and plug in your initial point. 2. Calculate dy/dx using the DEQ. Find new values of x and y: xnew = xold+ x ynew = yold + dy 3. Calculate the new value of y using dy (dy/dx) x. 4. Repeat until you get to x = 3. f(3) 3.125 www.mastermathmentor.com
Approximate and Actual Solutions Find the difference in values of the approximate solution and actual of dy/dx = (2 cos x)/y on [0, 1] with f(0) = 1 and x= 0.2 using Euler s method. x x ynew =yold + dy 0 0.2 0 0.2 1 1+2(.2) = 1.400 0.4 0.4 1.4+1.4(.2) = 1.680 0.6 0.6 1.68+1.097(.2) = 1.899 0.8 0.8 1.899+.869(0.2) = 2.073 1 1 2.073+.672(0.2) = 2.208 ynew =yold + dy 1 1+2(.2) = 1.400 1.4+1.4(.2) = 1.680 1.68+1.097(.2) = 1.899 1.899+.869(0.2) = 2.073 2.073+.672(0.2) = 2.208 dy/dx dy/dx 2 2 1.400 1.400 1.097 1.097 0.869 0.869 0.672 0.672 1 1.340 1.599 1.805 1.967 2.089 y= 4sinx+1 Difference 0 0.060 0.081 0.094 0.106 0.119 y2 2 y2= 4sinx+C 1= 4sin0+C C =1 y = 4sinx+1 dy dx=2cosx ydy = 2cosxdx = 2sinx+C y = 2cosxdx ydy www.mastermathmentor.com
Logistic Growth We have learned several types of growth models in AP Calculus. In growth models, a population P is a function of time t. These are formed by differentials equations. The first two you know, the last is new. Linear growth: The rate of change of the population P is constant. Exponential growth: The rate of change of the population P is proportional to itself. Logistic growth: The rate of change of the population P is proportional to itself and the room for growth C P where C is the maximum sustainable population and is called the carrying capacity. dP dt dP = kdt P = kt +C = k dP P ln P = kt +C P = Cekt = kdt dP dt dP P = kP = kdt dP dP dt = kdt = kP C - P ( ) www.mastermathmentor.com
Application of Logistic Growth A flu spreads in a school of 1,500 students at a rate proportional both to the population P of students already infected and those unaffected. dP dt = kP 1500-P ( ) a. Write the DEQ that describes that situation. b. If 100 students are infected on January 31, and 150 students infected on February 1, how many will be infected on February 14? 1500 1500 1+deKt c. On what day is the flu spreading the fastest? What is the spread? 1+de-1500kt= t = 0:100 =1500 P = 1+d d =14 1500 1+14eK t =1: 150 = Fastest spread when P = 750 1500 1+14e-0.44183t 14e-0.44183t=1 t ln 1 14 -0.44183= 5.97 February 6 -0.44183 750 ( -1500 14eK= 9 K = ln 9 14 1500 1+14e-0.44183t P 14 ( 750 = ( ) -0.44183 ( ) P = ) 1,458 students ) 1500- 750 ( ) 166students day www.mastermathmentor.com
Purchase the full version - 13 slides Content: Euler s Method & Logistic Growth www.mastermathmentor.com Powerpoint slides to help teach in the classroom Only $10.95 for the full presentation of 13 content slides. (download version) You receive: A narrated version (15 megs) An un-narrated version (2 megs) A PDF of the narration used if you want to provide the commentary Note: Downloading the full presentation may require high-speed internet. Or purchase the entire set of MasterMathMentor.com BC Calculus On the Wall Powerpoint presentations - Only $69.95 - sent to you on a 2-gig flash drive. Topics cover: L Hospital s Rule Integration Techniques Euler s Method & Logistic Growth Parametric and Polar Equations Vector-Valued Functions Taylor Polynomials Infinite Series Power Series www.mastermathmentor.com
