Questions: The temperature T of a person during an iftiness is given by T(t)=10t/t^2+1+98.6, where T is the temperature, in degrees Fahrenheit, at time t, in hours. (a) Find the rate of change of the temperature with respect to time. (b) Find the temperature at t=1 hr. (c) Find the rate of change of the temperature at t=1 hr.

Solution Steps

To find the rate of change of the temperature with respect to time, we need to find the derivative of T(t) with respect to t. T(t) = 10t/(t² + 1) + 98.6 Using the quotient rule for differentiation, the derivative of 10t/(t² + 1) is (10(t² + 1) - 10t(2t))/(t² + 1)² = (10t² + 10 - 20t²)/(t² + 1)² = (10 - 10t²)/(t² + 1)² = 10(1 - t²)/(t² + 1)² Thus, T'(t) = 10(1 - t²)/(t² + 1)²

Substitute t = 1 into the equation for T(t): T(1) = 10(1)/(1² + 1) + 98.6 T(1) = 10/2 + 98.6 T(1) = 5 + 98.6 T(1) = 103.6

Substitute t = 1 into the equation for T'(t): T'(1) = 10(1 - 1²)/(1² + 1)² T'(1) = 10(0)/2² T'(1) = 0/4 T'(1) = 0

Final Answer