6.1: Antiderivatives and Slope Fields

First, a little review:

Consider:2 3y x

then: 2y x 2y x

2 5y x or

It doesn’t 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 don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.

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.

Initial value problems and differential equations can be illustrated with a slope field.

A differentiable function--and the solutions to differentialequations better be differentiable--has tangent lines at every point. Let's draw small pieces of some of these tangent lines of the function: y = sin(0.2x2)

In other words, we're seeking a function whose slope at anypoint in the (x,y)-plane is equal to the value of x2 at that point.

Let's examine a few selected points:At the point (1,2) the slope would be 12=1. At the point (5,3) the slope would be 52 = 25. At the point (-3,11) the slope would be (-3)2 = 9.

Slope fields graphically represent each of the slopes that we find at points all over the plane by a short line segment that isactually as steep as the slope says it should be at that point. We can think of these little line segments as tangent lines to the function y that we've been looking for all this time.

Consider the slope field for f’(x) = x2.

For f’(x) = x2, the following slope field was computergenerated.

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.

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 00 00 0

23

1 0 21 1 2

2 0 4

-1 0 -2

-2 0 -4

2y x

If you know an initial condition, 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.

Construct the slope field (a.k.a. direction field) for

yy21

'x y y’

0 0 00 1 0.50 2 10 3 1.50 -1 -0.50 -2 -10 -3 -1.5…

Construct the slope field (a.k.a. direction field) for

yy21

'

It is left as an exerciseto produce the generalsolution

CAeyx

21

Go to: and enter the equation as:Y=

For more challenging differential equations, we will use the calculator to draw the slope field.

2

2

1

dy xy

dx x

1 2 1/ 1 ^ 2y t y t (Notice that we have to replace x with t , and y with y1.)

(Leave yi1 blank.)

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

1 2 1/ 1 ^ 2y t y t

Set the viewingwindow:

Then draw the graph:

WINDOW

GRAPH

Be sure to change the Graph type back to FUNCTION when you are done graphing slope fields.

Integrals such as are called definite integrals

because we can find a definite value for the answer.

4 2

1x dx

4 2

1x dx

43

1

1

3x C

3 31 14 1

3 3C C

64 1

3 3C C

63

3 21

The constant always cancels when finding a definite integral, so we leave it out!

Integrals such as are called indefinite integrals

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

2x dx

2x dx31

3x C

When finding indefinite integrals, we always include the “plus C”.

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, in example 5, the book shows a technique to graph the integral of a function using the numerical integration function of the calculator (fnINT).

This is extremely slow and usually not worth the trouble.

A better way is to find the indefinite integral and plot the resulting expression on the calculator.

),0,,sin(1 xxxxfnINTy x

tdtty0

sin or

To find the indefinite integral, use:

= sinx – x cos x

Enter the new expression in the Y= screen, and then plot theGraph. You will notice the difference immediately!!

dxxx sin