[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.
.