should rate of change be negative
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?
$\endgroup$ 12 Answers
$\begingroup$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).
$\endgroup$ 3 $\begingroup$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.
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