Say I have a spherical snowball. I want it's average rate of change of surface area as radius goes from 25cm to 20cm. I did the calculation.
$f(r)=4*\pi*r^2$.
That's the formula of surface area of sphere
So I did $\dfrac{f(25)-f(20)}{25-20}$
Which gave me a positive number but I am confuse. Snowballs are melting so should not their rates of change be negative?
The average rate of change you're thinking of is $$\frac{\text{new }f-\text{old }f}{|\text{new }r - \text{old }r|}$$ which is the negative of what you've calculated. The ordering is important. You've calculated the average rate of change going from $r=20$ to $r=25$, which I'm sure you would agree should be positive and then makes sense.
Regarding the modulus: Without it, we're calculating the gradient of the line segment connecting the two points. The ordering then doesn't matter. This result would be positive (just consider the graph).
When looking at the average rate of change. We are always positive. When looking at the average change in the quantities. We will look at:
$$\frac{ \text{ new } f - \text { old } f}{\text{ magnitude of change in time/size/etc }}$$
However, we will always refer to $\frac{f(b)-f(a)}{b-a}$ when finding average rate of change, this is a positive quantity, since we are referring to how much it has changed, not how it has changed.
Yes it is common to get the wrong sign in a problem, usually if you exchange the two components you will see the correct output. Later on you might find this occurring in integration(not sure of your current level).
– Display Name Sep 04 '14 at 22:24