# Is differentiation a possible approach?

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1. Apr 22, 2015

### 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.

Best regards,
wirefree

2. Apr 22, 2015

### HallsofIvy

Staff Emeritus
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".

3. Apr 23, 2015

### 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

4. Apr 24, 2015

### wirefree

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

Regards,
wirefree

5. Apr 24, 2015

### HallsofIvy

Staff Emeritus
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.

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