Difference Quotient vs ARC of a Function

[opus]

These are the two things that I'm going over in my PreCalculus class- the Difference Quotient and the Average Rate of Change of a Function. I'm confused as to what exactly they are, and how they relate to each other.

Average Rate of Change= $\frac {f \left(b\right) - f\left(a\right)} {b - a }$
To my understanding, this is the average rate of change of the function from value b to value a. Getting something like a value of 10 for this would make sense. However, in some of the examples such as:
Find the average rate of change of $f\left(x\right)=2x^2-3$ as x changes from x=c to x=c+h and h cannot equal 0.
$\frac {f \left(c+h\right) - f\left(c\right)} {\left(c+h\right) - c }$,
and yields the result
=4c+2h
What does this even mean?

As for the difference quotient,
Difference Quotient= $\frac {f \left(x+h\right) - f\left(x\right)} {h}$, h cannot equal 0.
Is this equation stating the difference from the value of the function f at x+h to the value of the function f at x? What is the purpose of the h?

I don't have a problem computing these, I just don't know what they're saying or what the purpose is behind them.

 

[mfb]

=4c+2h
What does this even mean?

It means your average rate of change depends on the range you average over. Pick a different range (different c or different h) and you get different average rate of change.

Is this equation stating the difference from the value of the function f at x+h to the value of the function f at x?

Divided by h. Up to this point it is the same as before. Later you'll take the limit for h->0 and get the derivative.

 

[opus]

It means your average rate of change depends on the range you average over. Pick a different range (different c or different h) and you get different average rate of change.Divided by h. Up to this point it is the same as before. Later you'll take the limit for h->0 and get the derivative.

Ok the first part makes sense...taking the average over different ranges.
For the second part, I don't understand what dividing by h does. It makes sense in the average rate of change formula- change in y over change in x. But what does dividing by h do in the difference quotient formula?

 

[mfb]

For the second part, I don't understand what dividing by h does.

The same as above, it gives the average rate of change. Your two fractions are exactly the same, just with c plugged in for x in the first one.

(c+h)−c = h

 

[opus]

Ohhhh okay. I've got it now. Thank you mfb.

 

[fresh_42]

Ok the first part makes sense...taking the average over different ranges.
For the second part, I don't understand what dividing by h does. It makes sense in the average rate of change formula- change in y over change in x. But what does dividing by h do in the difference quotient formula?

A quotient of differences means you draw a secant, which represents your average. Its slope depends on the range $b-a=h$ you define for it, the denominator. Therefore we narrow down this distance and get a tangent as limit. Wikipedia has some nice pictures included: https://en.wikipedia.org/wiki/Tangent.
So you divide anyway, it is simply a difference in notation:
$$
\dfrac{f(b)-f(a)}{b-a}=\dfrac{f((b-a)+a)-f(a)}{b-a}=\dfrac{f(h+a)-f(a)}{h}
$$
where we set $h=b-a$ and if you like $a=x$, in order to get the exact formula of your difference quotient. The limit process is only to get from secants to a tangent, because secants are many, but the tangent is only one, and we want to have a definition which is not many possibilities.

The last substitution $x=a$, however, is very important! It says, that we actually have a point $a$ where the tangent is placed at, not a variable $x$. If we pretend as if this was the same, we identify the tangent $t(a)$ with the function $x \mapsto t(x)\,$, although these are two distinct things: one is a single line $t(a)$, the other is a relation of location $x$ to unique lines $t(x)$. The notation $f'(x)$ often leads to the fact, that this distinction is forgotten.

 

[opus]

Oh wow. That is a great explanation. I'm going to need to put a little time into that to fully understand what you're saying, but relating it like that is so helpful. I'll be back with questions!

 

[mfb]

@fresh_42: I think you are missing -f(a) in the last fraction.

 

[fresh_42]

@fresh_42: I think you are missing -f(a) in the last fraction.

Yep, corrected and thanks! I could have bet that someone would complain about my very sloppy (and wrong) identification of tangents with their slopes, resp. the derivative. I hadn't expected a typo, though.

To explain this comment @opus:

The limit of the quotient of differences is the first derivative. It is a function that relates points $x=a$ to slopes $\dfrac{d}{dx}f =f(x)$ at this point $\left. \dfrac{d}{dx} \right|_{x=a} f(x) = f'(a)$ of the tangent line at $a$, whose full equation is $t(x)=t_a(x)=f'(a)\cdot x + f(a) - f'(a)\cdot a$. When people speak of derivatives as linear approximation to a function $f(x)$, they mean that the straight tangent is a good approximation to the function in a small neighborhood of the point, where the tangent is considered.

So all in all we have the following situation:

• $\dfrac{f(b)-f(a)}{b-a} = \dfrac{f(x+h)-f(x)}{h}$ is the slope of a secant from $x=a$ to $x=b$, resp. from $x$ to $x+h$. As this is a specific point at which we consider secants, it would have been better to write $x=a$ instead. If we narrow down the distance between $x=a$ and $x=b$, that is $h \rightarrow 0$, we will get a tangent line at this point, which is the result of limit process here.

• The limit is the slope of the tangent. We write this slope of the tangent $t_a(x)$ at the point $x=a$ as $f'(a)=\lim_{h \to 0} \dfrac{f(a+h)-f(a)}{h} =\left. \dfrac{d}{dx} \right|_{x=a}f(x)$

• The relation $\text{ location } \rightarrow \text{ slope }$ at a point is a function itself: the (first) derivative $x \longmapsto f'(x)$, sometimes also called differential.

• The tangent $t_a(x)$ at the point $x=a$ itself is of course not only the slope, but also the fact that this line shares exactly one point with the function, namely $(f(a),a) = (t_a(a),a)$. Therefore we get the straight line $t_a(x)=f'(a)\cdot x + f(a) - f'(a)\cdot a$. Now you might object, that a tangent still can intersect the function at another point further away. This is true. It demonstrates another important property of differentiation: Differentiation is a local property - here at the point $x=a$ - and we are only interested in what's going on here. Whether this tangent line crosses the function again somewhere else doesn't bother us, only that it is no secant at $x=a$.

• Last but not least, we can concentrate on the slope alone. That is, we only consider the linear part of the tangent function: $f'(a)\cdot x$, which basically means to parallel translate the tangent $t_a(x)$ along the $x-$axis until it crosses the origin. Thus we have mapped a location, a point $x=a$ to a linear function $\nabla_a f = (\, x \longmapsto f'(a)\cdot x \,)$, i.e. the multiplication by the slope $f'(a)$ at $a$. This is meant, if people speak of derivatives as linear functions. This becomes more important if physics gets involved.

All of the above could be meant if people simply write $f'(x)$. It depends on the context, which view is in focus. In the end, it is simply the limit of your quotient of differences. Very roughly speaking we may say: the entire concept is a linear approximation of the function. The error is negligible not far away from $x=a$ so we can calculate with something linear, which is easy - curved is not as easy.