If the rate of change of a population is given by dP/dt= 3.1(P-2)(P-10), P0, for what values of P is the population increasing?

Question

If the rate of change of a population is given by dP/dt= 3.1(P-2)(P-10), P>0, for what values of P is the population increasing?

lochana · Answer

hold on

zepdrix · Answer

$$\Large\rm \lim_{x\to3^+}\frac{x^2+4}{(x-3)(x-8)}$$Hmm the factor x-3 doesn't cancel out with anything.
So we don't have a removable discontinuity at x=3, we have an asymptote instead.
We simply need to figure out if it's approach `positive` or `negative` infinity from the right side of 3.

zepdrix · Answer

Numerator is a square and addition, so it will always be positive, yes?$$\Large\rm \lim_{x\to3^+}\frac{positive}{(x-3)(x-8)}$$

zepdrix · Answer

$$\LARGE\rm \frac{positive}{(3^+-3)(3^+-8)}$$How bout the denominator?
Look at each factor separately.
If we're approaching 3 `from the right`, that means our value will always be a little bit `larger than` 3, yes?

zepdrix · Answer

So this first factor is positive,$$\LARGE\rm \frac{positive}{(positive)(3^+-8)}$$A little bit larger than 3, minus 3, equals a positive value.

zepdrix · Answer

How bout that last factor, positive or negative?

anonymous · Answer

Negative