Calculus (6th edition) by James Stewart
Section 2.1- The Tangent and Velocity Problems
5.
If a ball is thrown into the air with a velocity of 40 ft/s, its height in
feet after t seconds is given by      
a) Find the average velocity for the time period beginning when
   and lasting
i) 0.5 second
The average velocity is equal to
Change in height
ab  ab


Change in time

ab  ab  a      b
   ft/s

ii) 0.1 second
The average velocity is equal to
ab  ab


ab  ab  a      b
   ft/s

iii) 0.05 second
ab  ab


ab  ab  a      b
   ft/s

iv) 0.01 second
ab  ab


ab  ab  a      b
   ft/s

b) To get the instantaneous velocity at    s, find the average
velocity for the time period beginning with 2 and lasting for
h seconds. Then see what happens as h approaches 0.
The average velocity from    to      seconds is
a  b  ab


a  b  a  b  a  b 
    a      b 
               
ab       
a  b  ab
      
   





 a  b
  a  b 

lim a  ba  b    


Therefore the instantaneous velocity when    s is   ft/s.
7.
The table shows the position of a cyclist.
 ab  



      


a) Find the average velocity for each time period:
(i) [1,3]
ab  ab
  


The average velocity is


 m/sec.
(iii) [3,5
ab  ab
  
The average velocity is




 m/sec.
b) Use the graph of  as a function of  to estimate the instantaneous
velocity when   
We can just get a rough approximation. Shown below is a
graph of  as a function of  Also shown is the tangent
line at the point where    The velocity is equal to the slope
of the tangent line.
Note we can just get a rough estimate of the slope of the
tangent line. Look at the points on the line where    and
   The change in t is 1 and a rough estimate of the change
in  is 6. Therefore an estimate of the instantaneous velocity

 6 m/sec.
is
