MA50: AP CALCULUS (AB)
HOMEWORK
UNIT 2 – EXAM REVIEW #1 (revised, fall 2013)
Name:
Period:
NO CALCULATORS – Do work on separate sheet of paper. Show enough work to justify your answer.
1) f x  and g x are defined for all x. lim f x   8 and lim g  x   2. Find the following limits.
x 3
a) lim  f x   g x 
x 3

x 3

b) lim f x   g 2 x 
x 3
1
f x   g x 
2
d) lim
x 3
g x 
3g x 
c) lim
x 3 f  x 
2) Find the limits.
3x 2  7 x  40
a) lim
x 5
x5
t2  4
x 2 t 3  8
b) lim
3) Match each problem with its limit. (Justify your answer by showing your work.)
a)
1
 sin x
lim 
 sin 
x 
x
 x
i)
1
b)
1
 cos x
lim 
 cos 
x 
x
 x
ii)
0
c)
lim x cos x  e x
iii)

d)
lim
e)
x 0

cos x
x
 1
lim 1  
x  
x

x 0

iv) 1
v)
does not exist
4) Sketch the graph of the derivative of the function f x .
.
5) f  x   
3
x
a) Find the difference between the average rate of change on the interval 1,3 and the
instantaneous rate of change of x  3.
b) Find the equations of the lines tangent and normal to the curve at x  3.
CALCULATORS ALLOWED – Do work on separate sheet of paper. Justify your answer.
6) Find the limits.
2 x3  9 x 2  3x  4
a) lim
x4
x 2  16
b)
lim
c) lim e x sin x
d)
lim
x 0
x 1
x  12
x 1
x  sin x
x   x  cos x
7) Use limits to define the vertical & horizontal asymptotes of each function.
2  3x 2
2y  3
a) f x  
b) T  y  
2
5  x  6x
4  y2
c) g x  
x4  2x2  1
x2  1
8) Explain why f  x  
a) x  0
x2  4
is continuous or not continuous at each point.
2 x
b) x  2
x2  4
9) Redefine f  x  
so that it is continuous for all real values of x.
2 x
 x 2  1,

 2 x,
10) f  x    1,
 2 x  4,

 0
a)
b)
c)
d)
e)
if
x0
if
0  x 1
if
x 1
if
1 x  2
if
x2
Draw a complete graph. Label significant points.
Find the left hand limits as x approaches 0, 1, and 2.
Find the right hand limits as x approaches 0, 1, and 2.
Find the limits of f x  as x approaches 0, 1, and 2.
State why f x  is or is not continuous at x  0,1,2.
11) Sketch a possible graph for a function g where:
lim g  x   1, g x  1,
lim g x   2,
lim g x   0,
x2
x 0
x 0
lim g x   ,
x  
lim g x   3
x 