Calculus (6th edition) by James Stewart
Section 3.7- Rates of Change in the Natural and Social Sciences
7.
The position function of a particle is given by
         
a) When does the particle reach a velocity of 5 m/s?
         
        a    b   
a  ba  b        
The particle has a velocity of 5 when    seconds.
b) When is the acceleration 0? What is the significance of this
value of t?


 s.


If you examine the graph of  below, you see that
when    s  is a minimum.
           
20
20
10
3t
2
9t
12
0
1
2
3
4
5
10
20 20
0
11.
t
5
a) A company makes computer chips from square wafers of silicon.
It wants to keep the side length of a wafer very close to 15 mm and
it wants to know how the area ab of a wafer changes when the
side length changes. Find  ab and explain its meaning in this
situation.
The area of the square wafer is    


  
  mm /mm.
º

 
This means that when    mm, an increase in  by just 1 mm
will result in an increase of the area by approximately 30 mm.
b) Show that the rate of change of the area of a square with
respect to its side length is half its perimeter. Try to explain
geometrically why this is true by drawing a square whose side
length  is increased by an amount . How can you
approximate the resulting change in area  if  is small?

,




 Œ •  



The perimeter   

 




Look at the drawing below which has a square of length
 inside of a larger square of length    I use  rather than 
x
x
h
h
The increase in area is equal to    which consists of
 rectangles of area  and  square of area  


  
Then


    and



as   ,

  

Since       if  is very small like .01 then  will
be very, very small and A can be approximated by
just  or . 
15.
A spherical balloon is being inflated. Find the rate of increase of the
surface area (    with respect to the radius  when  is
a) 1 ft
b) 2 ft
c) 3 ft. What conclusion can you make?
The rate of change of  with respect to  is     .
When        ft /ft,
when       ab   ft /ft, and
when       ab   ft /ft.
This implies that the rate of change in surface area is increasing as
 gets bigger. This will mean that, assuming  is quite large, a small
change in  can result in a huge increase in surface area!
For example, increasing the radius of the earth by 1 inch will result in
a huge increase in the surface area of the earth.
27.
The cost, in dollars, of producing  yards of a certain fabric is
 ab        
a) Find the marginal cost function.
The marginal cost is   ab
  ab      
b) Find   ab and explain its meaning. What does it predict?
  ab    ab  ab  
$32
Actually
. It is the rate of change of cost when   
yard
It predicts how much it will cost to produce the 201st yard
of fabric.
c) Compare   ab with the cost of producing the 201st yard of
fabric.
The actual cost of producing the 201st yard is
 ab   ab 
  ab  ab  ab 


’  ab  ab  ab “ 
$3632.20  $3600.00  $32.20 .