Question

The number of tree species s in a given area a in the pasoh forest reserve in malaysia has been modeled by the power function s(a) = 0.882a0.842 where a is measured in square meters.† find s'(107). (round your answer to five decimal places.)

Answer 1

$s(a) = 0.882 a^{0.842}$

s'(107) means the value of derivative of s(a) at a = 107. So, first step is to find the derivative of s(a).

$s(a) = 0.882 a^{0.842}$
Taking derivative of both sides, we get:
$\frac{d}{da}(s(a))= \frac{d}{da}( 0.882 a^{0.842})$

Taking the common out from right hand side of the equation:
$s'(a)= 0.882\frac{d}{da}(a^{0.842})$

Applying the power rule of derivative:

$s'(a) = 0.882 ( 0.842 a^{(0.842-1)}) \\ s'(a) = 0.742644 a^{-0.158}$

Now we can substitute 107 in place of a to get s'(107):
$s'(107) = 0.742644( 107)^{-0.158} \\ s'(107)= 0.35494$

Thus the value of s'(107) rounded to 5 decimal places is 0.35494