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Chapter 3 | Derivatives
20 on the coordinate axis. The path of the particle is shown on a coordinate axis in Figure 3.24 .
Figure 3.24 The path of the particle can be determined by analyzing v(t).
3.22 A particle moves along a coordinate axis. Its position at time t is given by s ( t ) = t 2 −5 t +1. Is the particle moving from right to left or from left to right at time t =3?
Population Change In addition to analyzing velocity, speed, acceleration, and position, we can use derivatives to analyze various types of populations, including those as diverse as bacteria colonies and cities. We can use a current population, together with a growth rate, to estimate the size of a population in the future. The population growth rate is the rate of change of a population and consequently can be represented by the derivative of the size of the population. Definition If P ( t ) is the number of entities present in a population, then the population growth rate of P ( t ) is defined to be P ′( t ).
Example 3.37 Estimating a Population
The population of a city is tripling every 5 years. If its current population is 10,000, what will be its approximate population 2 years from now?
Solution Let P ( t ) be the population (in thousands) t years from now. Thus, we know that P (0) =10 and based on the information, we anticipate P (5) =30. Now estimate P ′(0), the current growth rate, using P ′(0) ≈ P (5)− P (0) 5−0 = 30−10 5 =4. By applying Equation 3.10 to P ( t ), we can estimate the population 2 years from now by writing P (2) ≈ P (0)+(2) P ′ (0) ≈ 10 + 2(4) = 18; thus, in 2 years the population will be 18,000.
3.23 The current population of a mosquito colony is known to be 3,000; that is, P (0) = 3,000. If P ′(0) =100, estimate the size of the population in 3 days, where t is measured in days.
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