Partial Differentials of two functions with 2 variables each

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Homework Help Overview

The discussion revolves around finding the partial derivatives ∂s/∂V and ∂h/∂V for two functions involving two variables each, specifically in the context of volume (V) and surface area (S) of a cylinder defined by the equations V = π*r^2*h and S = 2π*r*h + 2*π*r^2. Participants express uncertainty about how to start the problem.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss expressing S as a function of V and h, with one suggesting a relationship between the components of S and V. There is also a consideration of how to take partial derivatives with respect to V.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the relationships between the variables and questioning the correct expressions to use. Some guidance has been offered regarding the formulation of S in terms of V and h, but there is no explicit consensus on the approach yet.

Contextual Notes

Participants are navigating the challenge of expressing the surface area in terms of volume and height, which may involve assumptions about the relationships between the variables that are not fully clarified in the initial posts.

CorvusCorax
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From the two equations given below, find ∂s/∂V (holding h constant) and ∂h/∂V (holding r constant
V = π*r^2*h, S = 2π*r*h + 2*π*r^2
Not entirely sure where to start...
 
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Welcome to PF!

Hi CorvusCorax! Welcome to PF! :smile:
CorvusCorax said:
… find ∂s/∂V (holding h constant)
…
V = π*r^2*h, S = 2π*r*h + 2*π*r^2

What is S as a function of V and h ? :wink:
 
tiny-tim said:
Hi CorvusCorax! Welcome to PF! :smile:


What is S as a function of V and h ? :wink:

Thanks. Been a lurker for years on several sites, decided it is probably time to join up for my own questions.

I'd say the first part of S, 2*pi*r*h, is just V'. I guess I could say 2*pi*r^2 is equivalent to 2*(V/h). So at that point just take the partial w/respect to V?
 
CorvusCorax said:
I guess I could say 2*pi*r^2 is equivalent to 2*(V/h).

Yes. :smile:
I'd say the first part of S, 2*pi*r*h, is just V'.

No, that's not an expression in V and h. :redface:

(and now I'm off to bed :zzz:)
 

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