Sec. 1.2: Finding Limits Graphically and Numerically
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Transcript Sec. 1.2: Finding Limits Graphically and Numerically
AP Calculus BC
Tuesday, 19 November 2013
• OBJECTIVE TSW solve differential equations using
slope fields, Euler’s Method, and separation of
variables.
• ASSIGNMENTS
– Sec. 4.5 is due on Friday (TEST day).
Solving Differential Equations
Slope Fields
Solving DE: Slope Fields
Slope Fields allow you to approximate the
solutions to differential equations graphically.
Solving DE: Slope Fields
Draw a slope field in the grid for the given differential equation. Then,
draw in a solution curve for the initial condition.
y
Example 1
Initial condition
y 1 1
dy
x 1
dx
x
Solving DE: Slope Fields
Draw a slope field in the grid for the given differential equation. Then,
draw in a solution curve for the initial condition.
y
Example 1
Initial condition
dy
x 1
dx
y 1 1
x
Solving DE: Slope Fields
Draw a slope field in the grid for the given differential equation. Then,
draw in a solution curve for the initial condition.
y
Example 1
Initial condition
dy
x 1
dx
y 1 1
x
Solving DE: Slope Fields
Draw a slope field in the grid for the given differential equation. Then,
draw in solution curves for the initial conditions.
y
Example 2
Initial condition
a) y 1 1
dy
2y
dx
x
b) y 2 2
Solving DE: Slope Fields
Draw a slope field in the grid for the given differential equation. Then,
draw in solution curves for the initial conditions.
y
Example 2
Initial condition
dy
2y
dx
a) y 1 1
x
b) y 2 2
Solving DE: Slope Fields
Draw a slope field in the grid for the given differential equation. Then,
draw in solution curves for the initial conditions.
y
Example 2
Initial condition
dy
2y
dx
a) y 1 1
x
b) y 2 2
Solving DE: Slope Fields
Draw a slope field in the grid for the given differential equation. Then,
draw in solution curves for the initial conditions.
y
Example 2
Initial condition
dy
2y
dx
a) y 1 1
x
b) y 2 2
Solving DE: Slope Fields
Draw a slope field in the grid for the given differential equation. Then,
draw in solution curves for the initial conditions.
y
Example 3
Initial condition
a) y 0 1
dy
x y
dx
b) y 1 1
x
Solving DE: Slope Fields
Draw a slope field in the grid for the given differential equation. Then,
draw in solution curves for the initial conditions.
y
Example 3
Initial condition
dy
x y
dx
a) y 0 1
b) y 1 1
x
Solving DE: Slope Fields
Draw a slope field in the grid for the given differential equation. Then,
draw in solution curves for the initial conditions.
y
Example 3
Initial condition
dy
x y
dx
a) y 0 1
b) y 1 1
x
Solving DE: Slope Fields
Draw a slope field in the grid for the given differential equation. Then,
draw in solution curves for the initial conditions.
y
Example 3
Initial condition
dy
x y
dx
a) y 0 1
b) y 1 1
x
Solving DE: Slope Fields
• Assignment
WS Solving Differential Equations – Slope
Fields
Due Friday, 22 November 2013.
WS Slope Fields
Due Friday, 22 November 2013.
Solving DE: Euler’s Method
Illustration of Euler’s
Method; the curve is
being “linearized” to
approximate a
desired solution,
given an initial value.
A better
approximation is
attained with a
smaller step size h.
Solving DE: Euler’s Method
Euler’s Method
To approximate the solution of the initial-value
problem
y ′ = f (x, y), y (x0) = y0,
proceed as follows:
Step 1: Choose a nonzero number h to serve as
an increment or step size along the x-axis,
NOTE: h will
and let
be given.
x1 = x0 + h, x2 = x1 + h, x3 = x2 + h, . . .
Solving DE: Euler’s Method
Euler’s Method
Step 2: Compute successively
y1 = y0 + f (x0, y0)h
y2 = y1 + f (x1, y1)h
y3 = y2 + f (x2, y2)h
...
The numbers y1, y2, y3, . . . in these equations are
the approximation of y(x1), y(x2), y(x3), . . .
Solving DE: Euler’s Method
Ex:
Use Euler's method with a step size of 0.5
to solve the initial-value problem
y' = y – x, y (0) = 2
over the interval 0 ≤ x ≤ 1.
x0, y 0 0, 2
y1 y 0 f x0 , y 0 h y1 2 f 0, 2 0.5
y 0 2
y1 2 2 0 0.5 3
x1, y1 0.5, 3
Solving DE: Euler’s Method
Ex:
Use Euler's method with a step size of 0.5
to solve the initial-value problem
y' = y – x, y (0) = 2
over the interval 0 ≤ x ≤ 1.
x1, y1 0.5, 3
y 2 y1 f x1, y1 h y 2 3 f 0.5, 3 0.5
y 2 3 3 0.5 0.5 4.25
y 1 4.25
x2, y 2 1, 4.25
Solving DE: Euler’s Method
Ex:
Use Euler's method with a step size of 0.25
to solve the initial-value problem
y' = y – x, y (0) = 2
over the interval 0 ≤ x ≤ 1.
x0, y 0 0, 2
y1 y 0 f x0 , y 0 h y1 2 f 0, 2 0.25
y 0 2
y1 2 2 0 0.25 2.5
x1, y1 0.25, 2.5
Solving DE: Euler’s Method
Ex:
Use Euler's method with a step size of 0.5
to solve the initial-value problem
y' = y – x, y (0) = 2
over the interval 0 ≤ x ≤ 1.
x1, y1 0.25, 2.5
x2, y 2 0.5, 3.0625
y 2 y1 f x1, y1 h
y 3 y 2 f x2 , y 2 h
y 2 2.5 f 0.25, 2.5 0.25
y 3 3.0625 f 0.5, 3.0625 0.25
y 2 2.5 2.5 0.25 0.25
y 3 3.0625 3.0625 0.5 0.25
y 2 3.0625
y 3 3.703125
x2, y 2 0.5, 3.0625
x3 , y 3 0.75, 3.703125
Solving DE: Euler’s Method
Ex:
Use Euler's method with a step size of 0.5
to solve the initial-value problem
y' = y – x, y (0) = 2
over the interval 0 ≤ x ≤ 1.
x3 , y 3 0.75, 3.703125
y 4 y 3 f x3 , y 3 h
y 4 3.703125 f 0.75, 3.703125 0.25
y 4 3.703125 3.703125 0.75 0.25
y 4 4.44140625
x4 , y 4 1, 4.44140625
When using Euler’s
Method, use all decimals !
y 1 4.44140625
Solving DE: Euler’s Method
WS Euler’s Method
Due on Friday, 22 November 2013.
AP Calculus BC
Wednesday, 20 November 2013
• OBJECTIVE TSW (1) solve differential equations
using the technique of separation of variables, and (2)
review for the test covering indefinite integration.
• ASSIGNMENTS DUE FRIDAY
–
–
–
–
–
(11/22/13)
Sec. 4.5
WS Slope Fields
WS Solving Differential Equations – Slope Fields
WS Euler's Method
WS Separation of Variables
Solving Differential Equations
Separation of Variables
Solving Differential Equations:
Separation of Variables
We have seen differential equations that are
explicitly in terms of x:
dy
Ex :
4x 1
dx
Ex : y 3 x 2
dy
x
Ex :
dx
16 x 2
Solving Differential Equations:
Separation of Variables
We solved these by multiplying both sides
by dx and integrating. For example,
dy
4x 1
dx
dy 4 x 1 dx
dy 4x 1 dx
y 2x 2 x C
Solving Differential Equations:
Separation of Variables
Sometimes, though, the differential
equations are not as straightforward.
dy
Ex : Solve the DE:
4 xy 2.
dx
Get terms with "y" on
1
one side, all other terms
dy 4 xdx
2
on the other side.
y
1
NOTE: The solution is NOT
dy
4
xdx
2
1
y
y 2 C.
2x
1
2x 2 C
y
1
This is called the general solution
y 2
because it has "C".
2x C
Solving Differential Equations:
Separation of Variables
Let's add an initial condition: y(0) = 1
1
1
y 2
1
2
2x C
2 0 C
1
1
C
1 C
y
1
2x 2 1
This is called the
particular solution.
Solving Differential Equations:
Separation of Variables
Solve
dy
4y cos y 3 x 2 0, y 0 0
dx
dy
4y cos y 3 x 2
dx
4y cos y dy 3x 2dx
2
4
y
cos
y
dy
3
x
dx
2y 2 sin y x 3 C
The equation cannot be
simplified for y.
Solving Differential Equations:
Separation of Variables
For the particular solution, y(0) = 0, so
2y 2 sin y x 3 C
2 0 sin 0 0 C
2
3
0 C
2y 2 sin y x 3
Solving Differential Equations:
Separation of Variables
WS Separation of Variables
Due on Friday, 22 November 2013 (TEST day).