antiderivatives anoyrnd slope fields

8/12/2019 Antiderivatives anoyrnd Slope Fields

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6.1 day 1: Antiderivatives

and Slope Fields

Greg Kelly, Hanford High School, Richland, Washington


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First, a little review:

Consider:2

3y x

then: 2y x 2y x

25y x

or

It doesnt matter whether the constant was 3 or -5, since

when we take the derivative the constant disappears.

However, when we try to reverse the operation:

Given: 2y x find y

2y x C

We dont know what the

constant is, so we put C inthe answer to remind us that

there might have been a

constant.


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If we have some more information we can find C.

Given: and when , find the equation for .2y x y4y 1x

2y x C

24 1 C

3 C2

3y x

This is called an initial value

problem. We need the initial

values to find the constant.

An equation containing a derivative is called a differential

equation. It becomes an initial value problem when you

are given the initial condition and asked to find the original

equation.


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Initial value problems and differential equations can be

illustrated with a slope field.

Slope fields are mostly used as a learning tool and are

mostly done on a computer or graphing calculator, but a

recent AP test asked students to draw a simple one by hand.


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Draw a segment

with slope of 2.

Draw a segment

with slope of 0.

Draw a segment

with slope of 4.

2y x

x y y0 0 00 1 0

0 0

0 0

2

3

1 0 2

1 1 2

2 0 4

-1 0 -2

-2 0 -4


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2y x

If you know an initialcondition, such as (1,-2), you

can sketch the curve.

By following the slope field,you get a rough picture of

what the curve looks like.

In this case, it is a parabola.


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Go to: and enter the equation as:Y=

For more challenging differential equations, we will use the

calculator to draw the slope field.

22

1dy xydx x

1 2 1/ 1 ^ 2y t y t (Notice that we have to replacexwith t , andywithy1.)

(Leave yi1blank.)

On the TI-89:

Push MODE and change the Graph type to DIFF EQUATIONS.MODE

Go to: Y=

Press and make sure FIELDS is set to SLPFLD.I


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1 2 1/ 1 ^ 2y t y t

Set the viewing

window:

Then draw the graph:

WINDOW

GRAPH


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Be sure to change the Graph type back to FUNCTION

when you are done graphing slope fields.


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Integrals such as are called definite integrals

because we can find a definite value for the answer.

42

1

x dx

42

1

x dx4

3

1

1

3x C

3 31 14 1

3 3

C C

64 1

3 3C C

63

3 21

The constant always cancels

when finding a definite

integral, so we leave it out!


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Integrals such as are called indefinite integrals

because we can not find a definite value for the answer.

2x dx

2x dx

31

3x C

When finding indefinite

integrals, we always

include the plus C.


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Many of the integral formulas are listed on page 307. The

first ones that we will be using are just the derivative

formulas in reverse.

On page 308, the book shows a technique to graph the

integral of a function using the numerical integration

function of the calculator (NINT).

1 NINT sin , ,0,y x x x xor0

sinx

y t t dt

This is extremely slow and usually not worth the trouble.

A better way is to use the calculator to find the

indefinite integral and plot the resulting expression.


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To find the indefinite integral on the TI-89, use:

sin ,x x xThe calculator will return: sin cosx x x

Notice that it leaves out the +C.

Use and to put this expression

in the screen, and then plot the graph.

COPY PASTE

Y=


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