Power Dissipated in a Resistor (really basic, but confused)

[Livethefire]

Homework Statement

I understand the maths... I'm here to ask WHY we have to do it this way.

The question states:
"The power dissipated in a resistor is given by $P= E^2/R$. If $E=200$ and $R=8$, find the change in $P$ resulting in a drop of $5 Volts$ in $E$ and an increase of $0.2 Ohms$ in $R$."

Homework Equations

Above.

The Attempt at a Solution

Physically I was thinking, okay plug in $200$ and $8$ then subtract from that answer the power calculated when $195$ and $8.2$ are input into the equation.

This gives Change in power$\approx362.8W$

My line of thought was, well if I have a resistor of 8 Ohms and a voltage of 200 across it the power will be a certain value. Then if I had a similar resistor of resistance 8.2 Ohms and a Voltage across if of 195 V then the difference when these values are put into the equation will be the change in power.

Why is this NOT the case? Namely the true answer is apparently: 375W,

You get this by doing the partial derivative of the equation with respect to E and R, I've done the math and it checks out to that answer alright, but as stated- What is wrong with what I have done?

What is my fatal assumption?
Is it because the changes are small and thus calculus needs to be involved?

Thanks for any responce.

 

[Andrew Mason]

Your answer is correct. The precise change in power is 362.8 W. If you take the rate of change of power as a function of voltage x change in voltage + the rate of change of power as a function of resistance x change in resistance, you will only get an approximate answer since P is not a linear function of E or R.

AM

 

[Livethefire]

Thanks for the quick reply.

This is kind of ironic though- that question was in a math class. Ussually they try to be the precise ones, and physicists make the approximations :P.

Im not confused at the question anymore, rather why they would do it that way if they have all the information to get a better answer.

Regardless, thanks for clearing that up.