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Mathematical Modeling with Differential Equations

Mathematical Modeling with Differential Equations. Chapter 9: By, Will Alisberg Edited By Emily Moon. Overview. 9.1 First-Order Differential Equations and Applications 9.2 Direction Fields; Euler’s Method 9.3 Modeling with First-Order Differential Equations Quiz. Overview.

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Mathematical Modeling with Differential Equations

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  1. Mathematical Modeling with Differential Equations Chapter 9: By, Will Alisberg Edited By Emily Moon

  2. Overview • 9.1 First-Order Differential Equations and Applications • 9.2 Direction Fields; Euler’s Method • 9.3 Modeling with First-Order Differential Equations • Quiz

  3. Overview • 9.1 First-Order Differential Equations and Applications • 9.2 Direction Fields; Euler’s Method • 9.3 Modeling with First-Order Differential Equations • Quiz

  4. Key Definitions • Differential Equation- Any equation in which the derivative affects the f(x)… e.g. f(x)=f’(x)/(2x) • Order- the highest degree of differentiation in a differential equation • Integral Curve- Graph of a solution of a differential equation

  5. First Order Initial Value Problems • Find a general formula for y(x) and use initial condition to solve for C. • Replace variables to solve

  6. General Solution • Start by Converting to: • Calculate x) • Use General Solution:

  7. My Turn! So… Set up the integral for the given differential equation

  8. Your Turn! Set up the integral to solve for y Wonhee Lee

  9. Newton’s Second Law

  10. Overview • 9.1 First-Order Differential Equations and Applications • 9.2 Direction Fields; Euler’s Method • 9.3 Modeling with First-Order Differential Equations • Quiz

  11. Key Definitions • Direction Field- A graph showing the slope of a function at each point • Euler’s Method- A technique for obtaining approximations of f(x) • Absolute Error- Difference between approximated value of f(x) and actual value • Percentage error- Absolute Error divided by the Exact value of f(x), Multiply the decimal by 100 to obtain a percentage • Iteration- One cycle of a method such as Newton’s or Euler’s

  12. Direction Field • Show Slopes at Various Points on a Graph • Follow the trail of lines • Different arrows with the same value of x represent different c’s • Don’t forget the points on the axes

  13. Euler’s Method: Theory • Approximates values of f(x) through small changes in x and its derivative • The algebraic idea behind slope fields • More make a more accurate approximation

  14. Euler’s Method: Calculation • Starting with a known point on a function, knowing the equation for the function. • Use • Repeat • Note: with very small values of we will get

  15. With a step size of approximate Knowing Your Turn! Wonhee Lee Just kidding- Go ahead Anna

  16. Overview • 9.1 First-Order Differential Equations and Applications • 9.2 Direction Fields; Euler’s Method • 9.3 Modeling with First-Order Differential Equations • Quiz

  17. Key Defintions • Uninhibited growth model- y(x) will not have a point at which it will not be defined • Carrying Capacity- The magnitude of a population an environment can support • Exponential growth- No matter how large y is, it will grow by a% in the same amount of time • Exponential decay- No matter how large y is, it will decrease by b% in the same amount of time • Half-Life- The time it takes a population to reduce itself to half its original size

  18. Exponential Growth and Decay Where k is a constant, if k is negative, y will decrase, if k is positive, y will increase

  19. My Turn! • The bacteria in a certain culture continuously increases so that the population triples every six hours, how many will there be 12 hours after the population reaches 64000?

  20. Your Turn! • The concentration of Drug Z in a bloodstream has a half life of 2 hours and 12 minutes. Drug Z is effective when 10% or more of one tablet is in a bloodstream. How long after 2 tablets of Drug Z are taken will the drug become inaffective? Jiwoo, from Maryland

  21. Answer

  22. Overview • 9.1 First-Order Differential Equations and Applications • 9.2 Direction Fields; Euler’s Method • 9.3 Modeling with First-Order Differential Equations • Quiz

  23. Quiz! • If a substance decomposes at a rate proportional to the substance present, and the amount decreases from 40 g to 10 g in 2 hrs, then the constant of proportionality (k) is A. -ln2 B. -.5 C -.25 D. ln (.25) E. ln (.125) 2. The solution curve of that passes through the point (2,3) is A. B. C. D. E.

  24. More Quiz Questions • True or False? If the second derivative of a function is a constant positive number, Euler’s Method will approximate a number smaller than the true value of y? • A stone is thrown at a target so that its velocity after t seconds is (100-20t) ft/sec. If the stone hits the target in 1 sec, then the distance from the sling to the target is: A. 80 ft B. 90 ft C. 100 ft D. 110 ft E. 120 ft

  25. Last Quiz Question • If you use Euler’s method with = .1 for the differential equation y’(x)=x with the initial value y(1)=5, then, when x= 1.2, y is approximately: A. 5.10 B. 5.20 C. 5.21 D. 6.05 E. 7.10

  26. Quiz Answers • 1A • 2C • 3True • 4B • 5C

  27. Bibliography • Barron’s “How to Prepare for the Advanced Placement Exam: Calculus • Anton, Bivens, Davis “Calculus” • http://exploration.grc.nasa.gov/education/rocket/Images/newton2r.gif • http://www.usna.edu/Users/math/meh/euler.html

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