Calculus for Beginners: Basics & Examples

[Frigus]

My question is as it is as you have read in the heading and please tell me in taking mind that I am not a calculus master but just a beginner.

 

[Nugatory]

Post the exact equation that you’re asking about, please.

 

[Frigus]

This is derivation part sir.

This is resnick halliday (principle of physics) tenth edition page no. 210

[Mentor Note -- image processed for better legibility]

 

[PeroK]

If $dM$ is a negative quantity, as Resnick states, then $M + dM$ is correct.

E.g. suppose $M = 100kg$ and $dM = -1 kg$, then the new mass is:

$M' = M + dM = 100kg + (-1kg) = 99kg$

 

[Frigus]

If $dM$ is a negative quantity, as Resnick states, then $M + dM$ is correct.

E.g. suppose $M = 100kg$ and $dM = -1 kg$, then the new mass is:

$M' = M + dM = 100kg + (-1kg) = 99kg$

Sir I understand it but I was watching this mit lecture in which that sir told (at 0:30)that we write it as positive because this is how we define differential but it didn't made any sense to me so to clarify it I posted this thread.
Can you please explain the point that he wanted to explain.

 

[PeroK]

Sir I understand it but I was watching this mit lecture in which that sir told (at 0:30)that we write it as positive because this is how we define differential but it didn't made any sense to me so to clarify it I posted this thread.
Can you please explain the point that he wanted to explain.

You may be confusing "adding" or "subtracting" things with "positive" and "negative" quantities.

For example, we have the well-known equation: $$s = ut + \frac 1 2 a t^2$$
This is correct, regardless of whether $a$ is a positive or negative quantity.

If $a = +1m/s^2$, then that equation is correct; and if $a = -1m/s^2$ that equation is correct.

In the same way $u$, the initial velocity, could be positive ot negative.

Similarly, for a difference in some quantity $Q$ we have $$Q' = Q + \Delta Q$$
Which says that the new quantity $Q'$ is the original quantity $Q$ plus the change $\Delta Q$. That is correct whether $Q$ and/or $\Delta Q$ are positive or negative quantities.

In general, you don't change an equation because you know a quantity is negative. The equation remains the same, it's just the numbers (when you come to plug them in) can be positive or negative.

 

[Cutter Ketch]

I sympathize with your confusion. With a rocket you know how the mass is changing and that biases your expectations regarding the equation.

Let me reiterate what others have said. Let’s say you have an object which is changing mass over time. The mass as a function of time is

M(t)

the change in mass over time is

$\frac {dM} {dt}$

If I told you $\frac {dM} {dt}$ is positive would you say the object is gaining mass or losing mass? Gaining mass, right? If I said it was negative you would say it is losing mass, right? Now if I write an equation using the symbol $\frac {dM} {dt}$ I'm not indicating what the sign of $\frac {dM} {dt}$ is. It could still be positive (gaining mass) or negative (losing mass). In the equation I might write

M(t) = $M_0 + \frac {dM} {dt}$ t

the symbol has a plus sign. It is added to the mass, but it can take on a positive or negative value

 

[Frigus]

Thanks to everyone