0:00

[music] There's a visual way to gain some insight into these anti-differentiation

problems. A visual method goes by the name of slope

fields. So, what's a slope field?

The idea is that I start with some function.

Like, here's an example of function x squared minus x.

But instead of just graphing the function, like I normally do here, I'm graphing the

function's values. I make a graph.

But the thing that I'm graphing is little tiny line segments of that given slope,

right? So, instead of plotting a value at some

height, I draw little tiny line segments with that slope.

Now, you can see in this example that if you plug in 0 or 1 into this function, the

output 0. And in this slope field, if you look at a

point whose x value is 0 or whose x value is 1, there I'm plotting just horizontal

lines, lines of slope zero. Over here, I've got very high slope, and

over here I've got very high slope. And that just reflects the fact that this

function's value is very high when x is negative and when x is positive.

And in between here, I've got just a little tiny region where the lines are

pointing down, right? Where the slope of these little line

segments is negative. And of course, that's because there's a

little tiny region here where this function's value is negative.

Well, let's look at a specific example. With this specific example, function F of

x equals x times cosine x. Now, I can draw the slope field for this

example. Well, in that case, here's the slope field

that I get. Imagine that I have an anti-derivative, a

function f, whose derivative was F. That is, suppose I had some function, f,

whose derivative was F. The slope field is drawn so that all these

little line segments have slope equal to the value of F, so I could use this slope

field to try to draw a picture, the graph of my function f, right?

Let's suppose that my function maybe starts here.

And then, I could try to follow the slope field along and try to use the information

in the, in the slope field to try to draw a picture of what my function, f, must

look like. So, these slope fields let you see the

anti-derivative, right? If you follow the slope field, you're

tracing out a function whose derivative is given by the slope field.

Now, we can find this anti-derivative, but we're trying to anti-differentiate x

cosine x. Well, our first guess might be x sine x.

I mean, that's a reasonable guess because differentiating sine will give me the

cosine and it, but what happens if I actually differentiate this product?

Well, that's the sort of thing that I do with the product rule which is x times the

derivative, which is cosine. So, that's looking good.

But it's plus the derivative of the first, which is 1 times the second, which is sine

x. So, the anti-derivative of x cosine x

isn't just x sine x because I've got this pesky additional sine x term.

But I also know that the derivative of cosine is minus sine.

So, if I add these things together, right? If I add them together, what's the

derivative of x sine x plus cosine x? Well, it's these two things added

together. But that's x cosine x, which is good, plus

sine x minus sine x. So, it's just x cosine x.

So that means the general anti-derivative of x cosine x is x sine x plus cosine x

plus some constant c. And yeah, we can see this in the graph.

So here, I've graphed x sine x plus cosine x.

And you can see that it's really moving in the direction of the slope field, which

makes sense. Because the slope field is coming from x

cosine x, which is the derivative of this thing.

And, you know, I drew the slope field by drawing little tiny line segments whose

slope is x cosine x. So, since this thing differentiates to x

cosine x, this thing is really moving along the slope field.

The slope field picture also gives you some insight into how the plus c really

works. What is the plus c really accomplishing,

right? It's moving the graph up and down.

And if you start with, you know, say x sine x plus cosine x, and then you slide

that graph up and down, it's still following the slope field, right?

It's still an anti-derivative for x cosine x.