Difference Quotient vs ARC of a Function

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• opus
In summary, PreCalculus covers two main concepts: the Difference Quotient and the Average Rate of Change of a Function. These concepts are related, as the average rate of change is dependent on the range over which it is averaged. The Difference Quotient formula is essentially the same as the Average Rate of Change formula, but with the addition of the variable h, which is later used to take the limit and find the derivative. It is important to note that in the difference quotient, the variable x represents the point at which the tangent is placed, not the variable for the function.
opus
Gold Member
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.

opus said:
=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.
opus said:
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
mfb said:
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?

opus said:
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.

opus said:
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.

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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!

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

mfb said:
@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.

1. What is the difference between Difference Quotient and ARC of a Function?

The difference quotient is a mathematical concept used to find the slope of a curve at a specific point, while the ARC (average rate of change) of a function is a measure of the overall change of a function over a given interval. In other words, the difference quotient is a local measure of change, while the ARC is a global measure of change.

2. How are Difference Quotient and ARC of a Function calculated?

The difference quotient is calculated by finding the change in the y-values (vertical change) divided by the change in the x-values (horizontal change) between two points on a curve. ARC is calculated by finding the average rate of change of a function over a given interval, which is the total change in the y-values divided by the total change in the x-values within that interval.

3. What information does the Difference Quotient provide about a function?

The difference quotient provides information about the slope of a curve at a specific point. This can be useful in understanding the behavior of a function around that point, such as whether it is increasing or decreasing, and the steepness of the curve at that point.

4. In what situations would you use the Difference Quotient vs ARC of a Function?

The difference quotient is commonly used in calculus to find the derivative of a function at a specific point. It is also used in physics and engineering to calculate instantaneous velocity or acceleration. The ARC of a function is useful in studying the overall change of a function over time, such as in economics or finance.

5. Can you use Difference Quotient and ARC of a Function together?

Yes, the difference quotient and ARC of a function can be used together to gain a better understanding of the behavior of a function. For example, the ARC can provide information about the overall trend of a function, while the difference quotient can provide insight into the rate of change at a specific point on that trend.

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