3.7 Rates of Change In the Natural and Social Sciences

 

Application problems from "the sciences"

 

Average Rate of Change

 

Slop of secant line from to  

Know 2 values of dependent and independent variable.

 

                  

 

 

Instantaneous Rate of Change

 

         

 

 

 

Distance Function

 

         

 

a)  Find the velocity at t

 

 

 

b)  Find the position, s, at t = 1, 2, and 3

 

 

 

 

c)  Find the velocity at t = 1 and 3

 

 

 

d)  When is the particle at rest?

 

e)  When is the particle moving forward?

 

 

 

 

f)  When is the particle moving backward?

 

 

 

 

g)  What is the total distance traveled on [0,1], [0,2], [0,3]

 

 

 

 

 

#14  A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s.  Find the rate at which the area within the circle is increasing

a)  as r changes from 2 sec to 3 sec.

 

 

 

 

b)  after 1 sec, after 3 sec and after 5 sec.

 

 

 

 

Any conclusions??

 

 

 

 

#28  The cost function for production of a commodity is 

a)  Find and interpret C ‘(100)

 

 

 

 

b)  Compare C ‘(100) with the cost of producing the 101st item.

 

 

 

 

 

 

 

The interest rate on U.S. treasury bills in a function of time.  The following table gives midyear values of this function f(t) over a nine-yeaqr period (as a percent per year)

 

t

I(t)

1983

8.62

1984

9.57

1985

7.49

1986

5.97

1987

5.83

1988

6.67

1989

8.11

1990

7.51

1991

5.41

1992

3.46

a)  Estimate the rate of change in the interest rate in 1985 and in 1990 by averaging the slopes of two secant lines.

 

 

 

 

 

 

 

 

 

 

b)  Use a graphing calculator to model these data by a fourth-degree polynomial.  Let t = 0 be 1980.

 

 

c)  Estimate the rate of change of interest rates in 19985 and 1990 using the polynomial obtained in b)

 

 

 

 

 

Assignment:  3.7 pg 179;  1 a-e) 3 a-e), 7, 9, 13, 15, 20 [Ans: a) b) , 21, 27