Chapter 9: By, Will Alisberg
Edited By Emily Moon

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

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

 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

 Find a general formula for y(x) and use initial condition to solve for C.
 Replace variables to solve




Start by Converting to:
Calculate
 x)
 Use General Solution: dy dx
 p ( x ) y
 q ( x )
 ( x )  e P ( x ) y 

1
 ( x )
  q ( x )

dy dx
 x
3  y

4 y dy

5 y
 dx p ( x )
 
5 x
3 q ( x )
 x
3
P ( x )
 
5 x

( x )
 e

5 x y

So…
1 e

5 x
 e

5 x
( x
3
) dx
Set up the integral for the given differential equation

Set up the integral to solve for y x
2 dy dx
 dy dx
 x
2 y
 x

1
 y
Wonhee Lee
( x
2 
1 )( dy dx
 y )
 x

1 dy
 y
 dx p ( x )

1 x
1

1 q ( x )

P ( x )
 x
1

1 x

( x )
 e x y

1 e x
 e x x

1

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




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

 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

 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

 Approximates values of f(x) through small changes in x and its derivative


The algebraic idea behind slope fields
More
 x make a more accurate approximation




Starting with a known point on a function, knowing the equation for the function.
Use y
1
 y
0
 f

( x
0
)(
 x ) x
1
 x
0
  x
Repeat
 Note: with very small values of
 x we will get y
 y
0
 f

( x ) dx

With a step size of

1 approximate
Knowing dy dx

3
: x x

4
Wonhee Lee y ( 1 )

4 Just kidding- Go ahead Anna y

4

3

1 .
5

1

.
75

10 .
25

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

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

Where k is a constant, if k is negative, y will decrase, if k is positive, y will increase y
 y
0 e kt

 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?
y

64000 e kt
3
 e
6 k k
 ln 3
6 y

64000 e
2 ln 3 y

576000

 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

y
 y
0 e kt
.
5
 e
2 .
2 k k
 ln .
5
2 .
2 t
.
1

2 e t (ln .
5 )
2 .
2

9 .
508



9.1 First-Order Differential Equations and
Applications
9.2 Direction Fields; Euler’s Method
 9.3 Modeling with First-Order Differential
Equations
 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)
2. The solution curve of y  ( x )  y that passes through the point (2,3) is
A.
D. y  e y  e x x  3
 ( e 2
B.
 3)
E. y  2 x  5 x y  e
.406
C.


 y  .406
e x


 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

 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

 1A
 2C
 3True
 4B
 5C

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