Mathematical Modeling with Differential Equations

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 Eulers Method
• 9.3 Modeling with First-Order Differential
  Equations
• Quiz

3
Overview

• 9.1 First-Order Differential Equations and
  Applications
• 9.2 Direction Fields Eulers 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
Newtons Second Law
10
Overview

• 9.1 First-Order Differential Equations and
  Applications
• 9.2 Direction Fields Eulers 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
• Eulers 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 Newtons
  or Eulers

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 cs
• Dont forget the points on the axes

13
Eulers 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
Eulers 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
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
• 9.2 Direction Fields Eulers 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 Eulers 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, Eulers
  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 Eulers 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

• Barrons How to Prepare for the Advanced
  Placement Exam Calculus
• Anton, Bivens, Davis Calculus
• http//exploration.grc.nasa.gov/education/rocket/I
  mages/newton2r.gif
• http//www.usna.edu/Users/math/meh/euler.html