We haven't covered the shortcuts to differentiation, yet. That is, we are calculating differentiation by use of limits. Anyways, I am going through Stewarts calculus, and on chapter 2.8 it relates f, f', and f'' together. Conceptually, I have a really difficult time relating them, particularly f''. I know that f' is the slope of f. And I know f' is related to velocity, where f'' is related to acceleration. However, I really have trouble visualizing their graphs. Maybe it's because I'm still new to Calculus. However, it can take me 15-20 minutes just to get a correct graph. I know with practice I'll get it, but aside from memorizing basic rules, I don't see the graphs conceptually. I've essentially had to take to memorizing that when f is concave up, f' is increasing, and therefore f'' is increasing. This, however, seems like a surefire way to make a mistake on a test. To a degree it makes sense because if it is concave up, the slope is getting larger, and thereby making f'' larger, since it is essentially the slope of the slope. Come to think of it, I think my primary confusion is when I am giving a graph of f' and asked to draw f. Whenever f' is negative, but increasing, I can't seem to see this as being concave up. The way I've been somewhat looking at it is from my Pre-Calc book. We looked at concavity, and when it was concave up we would say that it is increasing at an increasing rate, or decreasing at a decreasing rate, whereas concave down was increasing at a decreasing rate, or decreasing at an increasing rate. Which has helped me a little bit conceptually. And relate this idea of acceleration. But, for some reason, it's not sticking, and Stewart doesn't have many examples over it... Thanks.