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#1 2021-04-18 06:37:55

mathland
Member
Registered: 2021-03-25
Posts: 444

Rate of Change

The velocity v of a liquid flowing through a cylindrical tube is given by the Hagen–Poiseuille equation
v = k(R^2 − r^2), where R is the radius of the tube, k is a constant
that depends on the length of the tube and the velocity of the liquid at its ends, and r is the variable distance of the liquid from the center of the tube.

(a) Find the rate of change of v with respect to r at the center of
the tube.

(b) What is the rate of change halfway from the center to the wall
of the tube?

(c) What is the rate of change at the wall of the tube?

1. I am not seeking the answer but the set up only for all three parts. I will do the math work.

2. The rate of change is the derivative. Right? I need to clearly understand what is meant by RATE OF CHANGE before moving on with my calculus textbook.

Thank you.

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#2 2021-04-18 22:15:50

Bob
Administrator
Registered: 2010-06-20
Posts: 10,052

Re: Rate of Change

hi mathland,

Rate of change is the gradient of the graph so these questions require dv/dr.  (k and big R are constants for differentiation purposes)

Once you have done (a) you'll be able to do (b) and (c) by substituting values of r.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#3 2021-04-19 00:40:48

mathland
Member
Registered: 2021-03-25
Posts: 444

Re: Rate of Change

Bob wrote:

hi mathland,

Rate of change is the gradient of the graph so these questions require dv/dr.  (k and big R are constants for differentiation purposes)

Once you have done (a) you'll be able to do (b) and (c) by substituting values of r.

Bob

Good Monday to you. I hope you had a great weekend.

Here is my work for part (a).

dV/dr = k(R^2 − r^2)

dV/dr = kR^2 kr^2

dV/dr = 2k - 2k

dV/dr = 0


How does 0 help me solve part (b)?

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#4 2021-04-19 04:00:24

Bob
Administrator
Registered: 2010-06-20
Posts: 10,052

Re: Rate of Change

The graph of constants is a horizontal straight line so their gradient function is zero.  You have treated R as a variable.  It isn't.  It is the fixed radius of the tube.  Little r is the only variable.

So dv/dr = -2kr

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#5 2021-04-19 08:52:52

mathland
Member
Registered: 2021-03-25
Posts: 444

Re: Rate of Change

Bob wrote:

The graph of constants is a horizontal straight line so their gradient function is zero.  You have treated R as a variable.  It isn't.  It is the fixed radius of the tube.  Little r is the only variable.

So dv/dr = -2kr

Bob

You are right. This is the correct answer. Can you set up part (c)?

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#6 2021-04-19 14:08:56

Mathegocart
Member
Registered: 2012-04-29
Posts: 2,226

Re: Rate of Change

mathland wrote:
Bob wrote:

The graph of constants is a horizontal straight line so their gradient function is zero.  You have treated R as a variable.  It isn't.  It is the fixed radius of the tube.  Little r is the only variable.

So dv/dr = -2kr

Bob

You are right. This is the correct answer. Can you set up part (c)?

Same deal as part (b). The distance from the center of the tube to the wall is R. Plug that in for r(i.e, r=R).


The integral of hope is reality.
May bobbym have a wonderful time in the pearly gates of heaven.
He will be sorely missed.

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#7 2021-04-19 15:27:09

mathland
Member
Registered: 2021-03-25
Posts: 444

Re: Rate of Change

Mathegocart wrote:
mathland wrote:
Bob wrote:

The graph of constants is a horizontal straight line so their gradient function is zero.  You have treated R as a variable.  It isn't.  It is the fixed radius of the tube.  Little r is the only variable.

So dv/dr = -2kr

Bob

You are right. This is the correct answer. Can you set up part (c)?

Same deal as part (b). The distance from the center of the tube to the wall is R. Plug that in for r(i.e, r=R).

Are you saying that dV/dr = -2kR for part (c)?

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#8 2021-04-20 01:26:01

Mathegocart
Member
Registered: 2012-04-29
Posts: 2,226

Re: Rate of Change

mathland wrote:
Mathegocart wrote:
mathland wrote:

You are right. This is the correct answer. Can you set up part (c)?

Same deal as part (b). The distance from the center of the tube to the wall is R. Plug that in for r(i.e, r=R).

Are you saying that dV/dr = -2kR for part (c)?

Yes.


The integral of hope is reality.
May bobbym have a wonderful time in the pearly gates of heaven.
He will be sorely missed.

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#9 2021-04-20 10:55:10

mathland
Member
Registered: 2021-03-25
Posts: 444

Re: Rate of Change

Mathegocart wrote:
mathland wrote:
Mathegocart wrote:

Same deal as part (b). The distance from the center of the tube to the wall is R. Plug that in for r(i.e, r=R).

Are you saying that dV/dr = -2kR for part (c)?

Yes.

Easier than I thought. Are you a math teacher?

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#10 2021-04-21 14:16:12

Mathegocart
Member
Registered: 2012-04-29
Posts: 2,226

Re: Rate of Change

mathland wrote:
Mathegocart wrote:
mathland wrote:

Are you saying that dV/dr = -2kR for part (c)?

Yes.

Easier than I thought. Are you a math teacher?

I did use to tutor some kids in mathematics. I do some study of higher level mathematics and calculus in my free time - I'm currently programming and my work could use a massive dose of more convoluted mathematics.


The integral of hope is reality.
May bobbym have a wonderful time in the pearly gates of heaven.
He will be sorely missed.

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#11 2021-04-21 19:53:01

mathland
Member
Registered: 2021-03-25
Posts: 444

Re: Rate of Change

Mathegocart wrote:
mathland wrote:
Mathegocart wrote:

Yes.

Easier than I thought. Are you a math teacher?

I did use to tutor some kids in mathematics. I do some study of higher level mathematics and calculus in my free time - I'm currently programming and my work could use a massive dose of more convoluted mathematics.

In that case, perhaps you can help me on my journey through calculus 1, 2, and 3. I will also post geometry, trigonometry, college algebra and precalculus problems from time to time in terms of review.

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