Problem

The population of bacteria (in millions) in a certain culture x hours after an experimental nutrient is introduced into the culture is P(x)=20x5+x2. Use the differential to approximate the changes in population for the following changes in x.
a. 1 to 1.5
b. 4 to 4.25
a. Use the differential to approximate the change in population for x=1 to 1.5 .
Between 1 and 1.5 hours, the population of bacteria changes by million. (Round to three decimal places as needed.)

Answer

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Answer

Final Answer: The approximate change in population of bacteria from x=1 to x=1.5 is 1.111 million.

Steps

Step 1 :The problem is asking for the change in population of bacteria from x=1 to x=1.5. This can be approximated using the differential of the function P(x), which represents the rate of change of the population with respect to time. The differential of a function is given by its derivative. So, first we need to find the derivative of P(x), then we can use this to approximate the change in population.

Step 2 :Let's find the derivative of P(x). Given P(x)=20x5+x2, the derivative P(x) is calculated as P(x)=40x2(x2+5)2+20x2+5.

Step 3 :Next, we calculate the change in x, which is Δx=1.51=0.5.

Step 4 :Then, we use the derivative to approximate the change in population, ΔP=P(1)Δx.

Step 5 :By substituting the values into the equation, we get ΔP=1.111 million.

Step 6 :Final Answer: The approximate change in population of bacteria from x=1 to x=1.5 is 1.111 million.

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