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

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

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

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

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

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

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

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

Newton’s Second Law

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

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

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

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

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

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

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

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

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

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?

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

Answer

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

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.

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

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 B C D E. 7.10

Quiz Answers 1A 1A 2C 2C 3True 3True 4B 4B 5C 5C

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” n2r.gif n2r.gif n2r.gif n2r.gif