SECTION 5.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 355 5.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models OBJECTIVES 1 Model Populations That Obey the Law of Uninhibited Growth (p. 355) 2 Model Populations That Obey the Law of Uninhibited Decay (p. 357) 3 Use Newton’s Law of Cooling (p. 359) 4 Use Logistic Models (p. 360) 1 Model Populations That Obey the Law of Uninhibited Growth Many natural phenomena follow the law that an amount A varies with time t according to the function ( ) = A t A ekt 0 (1) Here A0 is the original amount ( ) = t 0 and ≠ k 0 is a constant. If > k 0, then equation (1) states that the amount A is increasing over time; if < k 0, the amount A is decreasing over time. In either case, when an amount A varies over time according to equation (1), it is said to follow the exponential law , or the law of uninhibited growth ( ) > k 0 or decay ( ) < k 0 . See Figure 54. For example, in Section 5.7, continuously compounded interest was shown to follow the law of uninhibited growth. In this section we shall look at some additional phenomena that follow the exponential law. Cell division is the growth process of many living organisms, such as amoebas, plants, and human skin cells. Based on an ideal situation in which no cells die and no by-products are produced, the number of cells present at a given time follows the law of uninhibited growth.Actually, however, after enough time has passed, growth at an exponential rate will cease as a consequence of factors such as lack of living space and dwindling food supply.The law of uninhibited growth accurately models only the early stages of the cell division process. The cell division process begins with a culture containing N0 cells. Each cell in the culture grows for a certain period of time and then divides into two identical cells. Assume that the time needed for each cell to divide in two is constant and does not change as the number of cells increases. These new cells then grow, and eventually each divides in two, and so on. Uninhibited Growth of Cells A model that gives the number N of cells in a culture after a time t has passed (in the early stages of growth) is ( ) = > N t N e k 0 kt 0 (2) where N0 is the initial number of cells and k is a positive constant that represents the growth rate of the cells. Using formula (2) to model the growth of cells employs a function that yields positive real numbers, even though the number of cells being counted must be an integer. This is a common practice in many applications. Exponential growth A (a) A(t ) 5 A0e kt , k . 0 t A0 Exponential decay t A A0 (b) A(t ) 5 A0e kt , k , 0 Figure 54
RkJQdWJsaXNoZXIy NjM5ODQ=