Question

If a person's temperature after x hours of strenuous exercise is t(x) = x3(8 − x2) + 98.6 degrees fahrenheit (for 0 ≤ x ≤ 2), find the rate of change of the temperature after 1 hour.

Answer 1

Given function : $t(x) = x^3(8-x^2) + 98.6$

Soltution: Distributing x^3 over (8-x^2), we get 8x^3 -x^5.

Therefore,

$t(x) = x^3(8-x^2) + 98.6=-x^5 + 8x^3 + 98.6$

Finding first derivative of the above function, because first derivative represents the rate of change.

$t'(x) = -5x^4 +8*3x^2$

$t'(x) = -5x^4 +24x^2$

We need to find rate of change of the temperature after 1 hour.

So, we need to plug x=1 in first derivarive of the function.

$t'(1) = -5(1)^4 +24(1)^2$

$= -5 +24 = 19.$

Therefore, after 1 hour, the person's temprature is increasing by 19° per hour.