The population of bacteria in one cubic centimeter of the blood of a sick person has been modeled by the function P(t) = 115 t(0.88^t) where t is the time, in days, since the person became ill.
Estimate how fast the population is changing 14 days after the onset of the illness. Round your answer to two decimal places. Rate of Change
This is a difficult derivative.
By the product rule differentiate with respect to t
P' (t) =(115t)(.88^t)(ln .88) + (.88^t)(115)
so
P'(14) = .... you do the button pushing
To estimate the rate at which the population of bacteria is changing 14 days after the onset of the illness, we need to find the derivative of the given function with respect to time (t).
The derivative of P(t) with respect to t represents the instantaneous rate of change of the population at any given time. In this case, we want to evaluate it at t = 14.
Here is the step-by-step process to find the derivative:
Step 1: Differentiate the function P(t) = 115 t(0.88^t) term by term.
The derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.
The derivative of t is 1.
For the term (0.88^t), we need to apply the chain rule. The derivative of (0.88^t) is (0.88^t) times the natural logarithm of the base, which is 0.88.
Step 2: Simplify the derivative expression.
P'(t) = 115 * (1) * (0.88^t) + 115 * t * (0.88^t) * ln(0.88)
Simplifying further:
P'(t) = 115 * (0.88^t) + 115t * (0.88^t) * ln(0.88)
Step 3: Evaluate the derivative at t = 14.
P'(14) = 115 * (0.88^14) + 115*14 * (0.88^14) * ln(0.88)
Calculating the numerical value using a calculator or software, we can find P'(14).
Therefore, the estimated rate at which the population of bacteria is changing 14 days after the onset of the illness will be the value of P'(14) rounded to two decimal places.