Why does [f(b)-f(a)]/(b-a) make sense?

[egio]

Hello,

As I progress into advanced mathematics I have noticed that [f(b) - f(a)] / (b - a) appears all the time, representing different things. For instance, it can represent slope, and if seen as Δposition/Δtime, it represents average velocity.

How can I read this expression, more so the numerator and denominator separately, intuitively? For instance, the formula for a mean is (∑x)/N, which makes sense to me. However, I'm not quite there yet with [f(b) - f(a)] / (b - a), and as I encounter different versions of it throughout calculus, I think it would help me to truly understand and read it in a general way, and know why it applies to so many different applications.

Thanks!

 

[micromass]

That's the entire usefulness of calculus. If there was only one interpretation of this quantity, then calculus wouldn't be this useful. Instead, there are plenty, plenty of different interpretation available. It's useless to search for one interpretation that will contain them all. It is far more useful to keep all the distinct interpretations in the back of your mind and use all of them simultaneously. So whenever you encounter $\frac{f(b)-f(a)}{b-a}$, you should try to give it all the different interpretations you know about - slope, velocity, acceleration, etc. - and see what different concepts you get.

 

[Mark44]

Hello,

As I progress into advanced mathematics I have noticed that [f(b) - f(a)] / (b - a) appears all the time, representing different things. For instance, it can represent slope, and if seen as Δposition/Δtime, it represents average velocity.

The quotient above represents the change in function values divided by the change in the independent variable. If you have a curve y = f(x), two points on the curve are (a, f(a)) and (b, f(b)). $\frac{f(b) - f(a)}{b - a}$ gives the slope of the line segment joining the two points; that is, the slope of the secant line.

If the function represents distance travelled, then the quotient can be interpreted as average velocity. If the function represents the population of, say, bacteria, at time t, then the quotient represents the average rate of growth between two times.

How can I read this expression, more so the numerator and denominator separately, intuitively? For instance, the formula for a mean is (∑x)/N, which makes sense to me. However, I'm not quite there yet with [f(b) - f(a)] / (b - a), and as I encounter different versions of it throughout calculus, I think it would help me to truly understand and read it in a general way, and know why it applies to so many different applications.

Thanks!

 

[Ssnow]

Basically it is a ratio used to compare two quantities. The idea is simpler but much powerful if you think the contribute in the calculus ... You can interpret as a velocity where the function is the space and the variable the time. If you have a population, it represents the change in the time of the number of organism. Also in chemistry you can represent the change of the concentration of a substance respect the time ...

 

[robphy]

Here's how average velocity is connected to $\frac{\Delta x}{\Delta t}$.
Let's suppose the velocity is piecewise-constant during the entire time-interval.
$\begin{align*} v_{avg} &\equiv \frac{\int_a^b v\ dt}{\int_a^b \ dt}\\ &=\frac{v_1\Delta t_1+v_2\Delta t_2+v_3\Delta t_3}{\Delta t_1+\Delta t_2+\Delta t_3}\\ &=\frac{\Delta x_1+\Delta x_2+\Delta x_3}{\Delta t_1+\Delta t_2+\Delta t_3}\\ &=\frac{\Delta x_{total}}{\Delta t_{total}}\\ &=\frac{x(b)-x(a)}{b-a} \end{align*}$
since $x(b)=x(a)+\Delta x_{total}$ and $b=a+\Delta t$.