Is differentiation a possible approach?

[wirefree]

Question:
I have a function of time. Its expression has a constant 'b' in it. I am asked to ascertain how changing 'b' affects the function.

Specifically, I have velocity as a function of time which accounts for drag forces; 'b' is the drag coefficient. I am asked to ascertain how changing 'b' affects how quickly terminal velocity is attained.

Attempt 1:
I understand that Calculus is the study of change. I am tempted to employ it. Hence, I obtained the derivative of the velocity function $$\frac{dv}{db}$$

Trouble:
My limited understanding also tells me 'derivatives' help me determine how quickly a function changes given a change in one of its variables. I doubt if that's what my velocity-b context is demanding.

Attempt 2:
I have also labored through studying $$v(t)$$ graphs for different values of 'b'.

Trouble:
I am not in favor of the method from Attempt 2 since it's labor-intensive and, perhaps, crude (do you agree?).Any guidance as to how to address this question would be greatly appreciated.wirefree

 

[HallsofIvy]

The derivative of v with respect to b ($\partial v/\partial b$) while holding other variables or parameters constant is the "rate of change of v as b changes".

 

[wirefree]

Appreciate the response, HallsofIvy. It has prompted me to consider the situation.

The derivative of v with respect to b while holding other variables or parameters constant, or the rate of change of v as b changes, if positive, will indicate that v increases as b increases. But that's not what the question concerns itself with. To restate it: for different values of b, does v change over time faster?

How do I address such a situation, please?

Regards,
wirefree

 

[wirefree]

Would appreciate some guidance on how to interpret the above expression.

Regards,
wirefree

 

[HallsofIvy]

Appreciate the response, HallsofIvy. It has prompted me to consider the situation.

The derivative of v with respect to b while holding other variables or parameters constant, or the rate of change of v as b changes, if positive, will indicate that v increases as b increases. But that's not what the question concerns itself with. To restate it: for different values of b, does v change over time faster?

Faster than what? Or do you mean that the rate of change itself is increasing? That will be true when the second derivative is positive.

How do I address such a situation, please?

Regards,
wirefree