Math 131 Week in Review Sections 2.1-2.5
J. Lewis
1. The velocity of an object traveling in a straight line is given by v ( t )  20 
t ft/sec.
Find the average acceleration (average rate of change of velocity) of the object from
a) t=1 to t=4 sec.
b) from t=4 to t=7 sec.
v(4  h)  v(4)
Use your calculator to find lim
h 0
h
.
2. Use the calculator to find the limits.
x
a)
5
e 1
lim
x 0
b)
x

 4 sin x

3. f ( x )   ln( x  1)

1

x

lim
( 2  h )  32
h 0
lim

f ( x ),
lim
h 0
h
4h 2
h
x0
0  x  e 1
e 1 x
Find each limit or state DNE for a) c = 0
x c
c)
lim
x c
f ( x ),

and b) c = e-1
lim f ( x )
x c
2
4. f ( x ) 
x 9
2
x x6
Find the limits lim
x c

f ( x) ,
lim
x c

do not exist.
a) c = 3
b) c = -2
5. Evaluate each limit.
a)
lim ln | x |
x 0
b)
lim x ln | x |
x 0
c)
lim e
x

2
sin x
f ( x) ,
lim f ( x ) . State DNE for any that
x c
6. Evaluate each limit.
3
a)
c)
lim
2
3
8 x  5 x  65

lim 
x 
b)
2
x 9
x 
lim
2
8 x  5 x  65
2
x 9
x  
x

x  9x  x

2
d)
lim
5e  7
x  4e x  2
x
e)
lim
5e  7
x   4 e x  2
7. At what x-values if any is f not continuous? Give a graphical reason and a definition of continuity
reason for each discontinuity.
a)
b)
c)
 x2  9
 2
x  x  6
f (x)  

1



1
 sin  
f ( x)  
x
 1

1
 x tan  
f (x)  
x

0
x  3,
x  2
x  3,
x  2
x0
x0
x0
x0
 x 2
x 1
 3 x  b
1 x
8. f ( x )  
a) Find the value of b so that f is continuous at x = 1.
b) For the value of b found in part a, find the average rate of change of f(x) from x=0 to x=1.
c) For the value of b found in part a, find the average rate of change of f(x) from x=1to x=2.
d) Guess lim
h 0
f (1  h )  f (1)

h
and
f (1  h )  f (1)
lim
h 0
9. Find the average rate of change of f ( x ) 

3
h
. What do you see in the graph at x=1?
x from x=0 to x=1. Does lim
h 0
f (h )  f (0)
h
exist?