Is there an easy way to relate f, f', and f''?

[in the rye]

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.

 

[Mark44]

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.

Start with something simple. You know what the graph of y = f(x) = x² looks like, right? What does the graph of y = f'(x) look like? What does the graph of y = f''(x) look like?

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.

No. if the graph of f is concave up, f'' is positive. Think about the example I gave previously, with f(x) = x².

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.

See above.

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.

In this context, "increasing" means "becoming less negative." To see this, hold a pencil at an angle that represents a negative slope. What will the pencil be doing as the slope gets less negative (i.e., the slope increases) as you move from left to right on the graph?

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

It is very bad, IMO, to use the word "it" in mathematics, especially when "it" changes meaning in different parts of the sentence.
"We looked at concavity, and when it (what does "it" refer to here?) was concave up we would say that it (what does "it" refer to here?) is increasing..."
Unless it is intuitively obvious to the most casual observer what the antecedent of "it" is, you should never use this word.

, 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.

 

[FactChecker]

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.

You are very close to the truth. The only problem is that you think that a function must be always increasing to be concave up. Not true. If it is turning up, it would be concave up even if it is decreasing. If f' is negative, that just means that f is sloped down. It will still be concave up if it is turning up (f'' > 0 so f' is increasing).

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,

But f doesn't have to start out increasing. It can be decreasing, but less and less till it might begin increasing. That would also be concave up

.or decreasing at a decreasing rate, whereas concave down was increasing at a decreasing rate, or decreasing at an increasing rate.

Same here. f doesn't have to start out decreasing. It can start out increasing and turn downward till it is decreasing. That would be concave down.

Which has helped me a little bit conceptually. And relate this idea of acceleration.

Acceleration is the first derivative of velocity and the second derivative of position. To put my earlier comments in these terms: Suppose your car was facing up hill coasting backward. Let's call that a decreasing f = altitude and negative velocity. Then giving it gas would slow the backward drift, increase the velocity toward positive (climbing the hill). Even though it is slowing the backward drift, it is positive acceleration. Eventually the velocity would go positive and f=altitude would slope up.

 

[in the rye]

Ahh, I think it makes sense, now. I wasn't really thinking of f as a position. But, it makes sense. If you have f(x) = x² it's position from the left is approaching 0, so, in terms of velocity, it is negative because it is going back to the origin. Which means that the derivative would be f(x) = x, having a negative velocity from (-inf, 0), and a positive velocity from (0, inf). This means that since the slope of x is a constant, it is neither accelerating or deceleration, it is remaining a constant. In this case, a constant of 1. So the double derivative would just be a f(x) = 1. It makes sense, it's just when Stewart throws complicated graphs it throws me off. For some reason I want to draw the derivative as being fairly similar to the function itself. Coincidentally, I do better on questions where they describe the graph, and ask to draw it. For example, if they say f''(x) > 0 from [0,4]. I know that the graph is concave up from that interval. I think maybe analyzing the graphs and describing what they're doing on a separate sheet, and THEN drawing it might be the way to go for me.

Thanks.

 

[mathwonk]

f = height, f' = change in height (slope), f'' = (f')' = change in slope (convexity).

I.e. if slope is positive then height is increasing, if convexity is positive then slope is increasing.