Math 136 Calculus 2 - Lab 4 Slope Fields and Antiderivatives

M136 - Calculus II - Lab 4. Slope Fields and Antiderivatives

Over the course of this semester, the function [Maple Math] has arisen on several occasions. Aside from the fact that this function plays an important role in several branches of science, this function has also played an important role in our studies. In our case, its importance has stemmed primarily from the fact that it possesses no elementary antiderivative. It is valuable to be able to find an analytic expression for an antiderivative when one exists, but quite ofter the real world is not so nice. It may be necessary to understand the behavior of the anitderivative of a function even when there is no easy way to define that antiderivative.

In the study of definite integrals, functions such as f ( x ) = [Maple Math] force us to consider numerical methods for calculating integrals. The fundamental theorem of calculus allows us to define a function via the definite integral. In the case of f ( x ), we could determine pointwise values for the anitderivative via the formula

[Maple OLE 2.0 Object]

where C is taken to be our initial condition, i.e., [Maple Math] . We now have another way of defining F ( x ), we can represent F ( x ) as a power series:

[Maple OLE 2.0 Object]

At this point then, we have developed numerical and analytic (although not closed form) representations of the antiderivative. These representations have their usefulness, but in order to gain an intuitive understanding of the behavior of the function, we would like to have a graphical representation. We can get such a representation by using a tool that is also very handy in the study of differential equations; namely, we will use what is termed a slope or direction field to graph the function.

The idea behind the slope field is very simple. In the case of a first order linear differential equation, we are given a definition for the derivative of the function we hope to find. The equation

[Maple OLE 2.0 Object]

can easily be rewritten as such an equation by simply differentiating both sides of this equation:

[Maple OLE 2.0 Object]

Now a solution to this equation will have the property that the slope of the tangent line to the curve will be equal to [Maple Math] for any given value of x . To draw the slope field, we simply plot line segments for several points in the xy -plane. At each point, the slope of the line segment would be given by [Maple Math] .

[Maple Plot]

This slope field is essentially tracing out the graph of y ( x ) for us. Notice that there are several curves on this plot. However, each curve has a different y -intercept. In other words we are getting the approximate plot of the family of antiderivatives for f ( x ) = [Maple Math] - each y -intercept corresponding to a different initial condition. Suppose we wanted the graph of the antiderivative satisfying the initial condition y (0)=0. We can use the slope field to sketch its' graph:

[Maple Plot]

Of course plotting the slope field by hand could be very tedious. Fortunately, we don't have to. Maple has a function that will plot the slope field for us. It is called "dfieldplot" and is contained in the package "DEtools". We must remember to load this package before we can use the command:

> with(DEtools);

[Maple Math]
[Maple Math]

The slope field in Figure 1 was generated with the command

> dfieldplot(diff(y(x),x)=exp(-x^2),[y(x)],x=-2..2,y=-2..2):

Exercises:

1. On the slope field given below, graph the solution y ( t ) satisfying the initial conditions

[Maple OLE 2.0 Object]

[Maple Plot]

2. Consider the slope field given below. Is there a solution satsifying the initial condition y (0) = 0 ?

Explain.

[Maple Plot]

[Maple OLE 2.0 Object]

a) Find the fourth degree Taylor polynomial that approximates F ( x ).

b) Graph the function. Sketch the solution on a printout of the slope field. Note: you have been given an initial condition. What is it?