Rates of Change in the Natural and Social Sciences

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Presentation transcript:

Rates of Change in the Natural and Social Sciences CHAPTER 2 2.4 Continuity Rates of Change in the Natural and Social Sciences

CHAPTER 2 2.4 Continuity The difference quotient dy/dx = lim x  0  y/ x is the instantaneous rate of change y with respect to x. CHAPTER 2 2.4 Continuity The difference quotient y / x = (f (x2) - f (x1) ) / (x2 - x1) is the average rate of change of y with respect to x over the interval [x1, x2].

CHAPTER 2 2.4 Continuity Physics If s = f(t) is the position function of a particle that is moving in a straight line, then s /t represents the average velocity over a time period t and v = ds / dt represents the instantaneous velocity (the rate of change of displacement with respect to time). 2.4 Continuity Example A particle moves along the x-axis, its position at time t given by x(t) = t / (1+ t2), t >=0, where t is measured in seconds and x in meters. a) Find the velocity at time t. b) When is the particle moving right or moving left?

CHAPTER 2 2.4 Continuity Chemistry If A + B  C is a chemical reaction, then the instantaneous rate of reaction is: Rate of reaction = lim t  0 [C] / t = d[C] / dt Chemistry 2.4 Continuity Example The data in the table gives the concentration C(t) of hydroxyvaleric acid in moles per liter after t minutes. Find the average rate of reaction for 2 <= t <= 6. t 2 4 6 8 C(t) 0.0800 0.0570 0.0408 0.0295 0.0210

CHAPTER 2 2.4 Continuity Biology The instantaneous rate of growth is obtained from the average rate of growth by letting the time period t approach 0: growth rate = lim t  0 n /t =dn/dt. 2.4 Continuity Example Suppose that a bacteria population starts with 500 bacteria and triples every hour. a) What is the population after 3 hours? b) What is the population after t hours?

CHAPTER 2 2.4 Continuity Economics Suppose that C(x) is the total cost that a company incurs in producing x units of a certain commodity. The function C is called a cost function. The average rate of change of the cost is: C /x = [C(x2) - C(x1)] / (x2 - x1). 2.4 Continuity The cost function for a commodity is: C(x)=84+ 0.16x – 0.0006x2+0.000003x3 Find C’(100). b)Calculate the value of x for which C has an inflection point.

Computer Science CHAPTER 2 2.4 Continuity