Change with time

Question

I was going through a silly doubt, but aren't able to find its answer. If we say something is changing with time, what do we mean by that? Should we multiply the quantity with time or divide it? Like if I say that the velocity is changing with time, what should I calculate? acceleration or distance? Similarly in definite integral and differentiation. If I say that something is changing with time, should I take differentiation or do definite integral in interval $t$ to $t+\Delta t$?

Answer 1

If $A$ is changing with time over a time interval $\Delta t$, it means that that $A(t+\Delta t)-A(t)\neq 0$. That's, by definition, what change means. Now, one can ask how fast something is changing. One way to quantify this measure of how fast something is changing is simply by defining the rate of change which is given by the ratio of how much something changed and how much time it took, i.e., the rate of change of $A$ over a time interval $\Delta t$ is given by $R=\frac{A(t+\Delta t)-A(t)}{\Delta t}$. Now, if you want to ask how fast something is changing at a point in time, you see what the rate $R$ is when you make the $\Delta t$ infinitesimaly small. This is where the differentiation/derivatives come in. The derivative of $A$ with respect to $t$ at some time $t$ is defined as the limit of the rate of change $R$ as $\Delta t\to 0$. Thus, we have $\frac{dA}{dt}=\lim_{\Delta t\to 0}R=\lim_{\Delta t\to 0}\frac{A(t+\Delta t)-A(t)}{\Delta t}$.

So, in conclusion:

• The rate of change of $A$ over a time interval $\Delta t$ is given by $R=\frac{A(t+\Delta t)-A(t)}{\Delta t}$.
• The rate of change of $A$ at a time $t$ is given by the limit $\lim_{\Delta t\to 0}R=\frac{df}{dt}$.

Now, if you know the rate of change of something and you want to calculate how much the change occurred during a given period of time then you need to multiply the rate of change by the time interval during which the rate of change was calculated. So, $\Delta A=R\cdot\Delta t$. However, if you don't know the rate of change over a time and rather simply know the rate of change at each instant in time during the interval, you add up all small changes in $A$ during that period (which you get by multiplying the small time-interval $dt$ to the rate of change $\frac{dA}{dt}$ at a given instant of time). So, you can write $\Delta A = \int \frac{dA}{dt} \ dt$.

So, in conclusion:

• The change in $A$ over a time interval $\Delta t$, if you know the rate $R$ of change in $A$ over the time interval $\Delta t$, is given by $\Delta A=R\cdot{\Delta t}$.
• The change in $A$ over a time interval $\Delta t$, if you know the rate of change in $A$ at each point in time (i.e., $dA/dt$) over the time interval $\Delta t$, is given by $\Delta A=\int\frac{dA}{dt}\ dt$.

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In other words, to get the rate of change, you differentiate. To get the total amount of change given the rate of change, you integrate. Of course, it should be noted that this is a practical description from a physicist's perspective. A mathematician would be appalled by the sloppiness of every sentence I wrote.