Suppose a population 𝑃 is given by the formula 𝑃=277×(1.2)^𝑡
Calculate the rate of change of 𝑃 when 𝑡=3.
To find the rate of change of 𝑃 when 𝑡=3, we need to take the derivative of the function 𝑃=277×(1.2)^𝑡 with respect to 𝑡.
Using the chain rule, we get:
𝑑𝑃/𝑑𝑡 = 277(1.2)^𝑡 * ln(1.2)
Now, when 𝑡=3, we can plug in this value and calculate the rate of change of 𝑃:
𝑑𝑃/𝑑𝑡 = 277(1.2)^3 * ln(1.2)
𝑑𝑃/𝑑𝑡 ≈ 118.78
Therefore, the rate of change of 𝑃 when 𝑡=3 is approximately 118.78. Note that this is the instantaneous rate of change at that particular point. The rate of change at other values of 𝑡 will be different.