Have you every used a picture to describe something? Or have you ever seen something that made you connect it with something else you've seen before? Well, slope fields are like that. They are the slope at every point with x and y inserted into the function. They show how the slopes take their shapes and you can look at the big picture to figure out what function it would be.
For example, this is a slope field of the sine curve
You can tell that that is the sine curve because it looks very similar to sine.
Another big part of chapter 6 is u-substitution, it was very nice because we already went over this concept. BUT, we added integrals with bounds and finding new bounds and inserting it into the anti-derivative. I think that this chapter is so much fun, I finally feel kind of caught up to what we are learning.
This is the substitution rule for definite integrals.
This is the substitution rule for definite integrals.
Here is an example of changing the bounds. You can see that you have to insert the original bound numbers 0, and 1, in order to find the new bounds 7 and 8.
Here is link that has more examples of definite integration with substitution