Total Differential

**coobe** · April 11th 2010, 07:05 AM

Hello ! Im just diving into multi-variable Calculus and having some problems. Currently im trying to solve this:

Approximate by using the total differential how the surface of a cylinder with soil and lid is changing if its radius = 10cm is increased by 5% and its height = 25cm is decreased by 2%. Compare this to the exact result and explain

any ideas ? especially how to get into this in general

thank you

**earboth** · April 11th 2010, 08:00 AM

Originally Posted by coobe

Hello ! Im just diving into multi-variable Calculus and having some problems. Currently im trying to solve this:

Approximate by using the total differential how the surface of a cylinder with soil and lid is changing if its radius = 10cm is increased by 5% and its height = 25cm is decreased by 2%. Compare this to the exact result and explain

any ideas ? especially how to get into this in general

thank you

1. Your function describes the volume of a cylinder with respect to r and h:

$v(r,h)=\pi r^2 \cdot h$

2. I assume that you know that the total differential of a function z(x,y) is calculated by:

$dz = \frac{\partial z}{\partial x}\cdot dx + \frac{\partial z}{\partial y}\cdot dy$

3. In your case:

$dv = \frac{\partial v}{\partial r}\cdot dr + \frac{\partial v}{\partial h}\cdot dh$

4. You know the values of r and dr = 0.5
and h and dh = 0.5

Plug in the known values and calculate dv.

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