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

2. 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].

3. 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?

4. 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

5. 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?

6. 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)= x – x x3
Find C’(100).
b)Calculate the value of x for which C has an inflection point.

7. Computer Science
CHAPTER 2
2.4 Continuity