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

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

4. Key Definitions Differential Equation- Any equation in which the derivative affects the f(x)… e.g. f(x)=f’(x)/(2x) 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 Order- the highest degree of differentiation in a differential equation Integral Curve- Graph of a solution of 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. Find a general formula for y(x) and use initial condition to solve for C. Replace variables to solve Replace variables to solve

6. General Solution Start by Converting to: Start by Converting to: Calculate  x) Calculate  x) Use General Solution: 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 9.1 First-Order Differential Equations and Applications 9.2 Direction Fields; Euler’s Method 9.2 Direction Fields; Euler’s Method 9.3 Modeling with First-Order Differential Equations 9.3 Modeling with First-Order Differential Equations Quiz Quiz

11. Key Definitions Direction Field- A graph showing the slope of a function at each point Direction Field- A graph showing the slope of a function at each point Euler’s Method- A technique for obtaining approximations of f(x) Euler’s Method- A technique for obtaining approximations of f(x) Absolute Error- Difference between approximated value of f(x) and actual value 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 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 Iteration- One cycle of a method such as Newton’s or Euler’s

12. Direction Field Show Slopes at Various Points on a Graph Show Slopes at Various Points on a Graph Follow the trail of lines Follow the trail of lines Different arrows with the same value of x represent different c’s Different arrows with the same value of x represent different c’s Don’t forget the points on the axes 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 Approximates values of f(x) through small changes in x and its derivative The algebraic idea behind slope fields The algebraic idea behind slope fields More make a more accurate approximation 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. Starting with a known point on a function, knowing the equation for the function. Use Use Repeat Repeat Note: with very small values of we will get Note: with very small values of we will get

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

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

17. Key Defintions Uninhibited growth model- y(x) will not have a point at which it will not be defined 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 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 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 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 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? 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? 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 9.1 First-Order Differential Equations and Applications 9.2 Direction Fields; Euler’s Method 9.2 Direction Fields; Euler’s Method 9.3 Modeling with First-Order Differential Equations 9.3 Modeling with First-Order Differential Equations Quiz Quiz

23. Quiz! 1. 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) 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? 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 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: 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 1A 2C 2C 3True 3True 4B 4B 5C 5C

27. Bibliography Barron’s “How to Prepare for the Advanced Placement Exam: Calculus Barron’s “How to Prepare for the Advanced Placement Exam: Calculus Anton, Bivens, Davis “Calculus” Anton, Bivens, Davis “Calculus” http://exploration.grc.nasa.gov/education/rocket/Images/newto n2r.gif http://exploration.grc.nasa.gov/education/rocket/Images/newto n2r.gif http://exploration.grc.nasa.gov/education/rocket/Images/newto n2r.gif http://exploration.grc.nasa.gov/education/rocket/Images/newto n2r.gif http://www.usna.edu/Users/math/meh/euler.html http://www.usna.edu/Users/math/meh/euler.html