Calculus I
Sample Exam #01
1. Sketch the graph of the function and define the domain and range.
1
3
x2
b) g ( x) 
x 1  2
x2 1
x2  5x  6
d) j ( x) 
 x2  x  6
x 3
a) f ( x)  
c) h( x) 
2. Evaluate the following.
 5 
a) sin   
 6 
e) sin 1
2
2
 7 
b) tan  

 6 
f)
e  4ln 3

3
d) cos 1  

 2 
 4 
c) sec  

 3 

g) ln 2e  e ln e1

1

4
h) log8 
3. Solve for x.
a) ln  7 x 2  8  ln10 x
b)   e12 x  3
4. Sketch the graph of csc and cot  on the interval    ,   . Draw each graph on a
separate set of axis. Clearly label the location of any vertical asymptotes.
5. Express the function f  x   2 x  3  4 x  1 in piecewise form without the absolute
values.
6. Define the Domain and Range of f 1  x  given f  x   ln  x  3  2 .
7. For the function f (x) graphed below, answer the following:
4
3
2
1
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-1
-2
-3
-4
x 0
b) lim f ( x) 
c) lim f ( x) 
d)
f (0) 
f)
f (4) 
g) lim f ( x) 
h)
a) lim f ( x) 
e)
x 0
x 
x 4
For parts i) through m) answer True or False.
i) f (x) is continuous from the right at x   6 .
j) f (x) is not continuous at x   6 .
k) f (x) is continuous from the left at 0.
l) f (x) is not continuous from the right at x  2 .
lim f ( x) 
x  2
lim f ( x) 
x  6
7
8. Sketch a possible graph for a function f (x) with the specified properties:
lim f  x     and lim f  x   0 ;
f  1  f  3  1 ;
x 3 
x 3
lim f x     and lim f x    ;
x  1
lim f  x   2 and lim f  x    
x  
x  1
x 
4
3
2
1
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-1
-2
-3
-4
9. Find a value for the constant k that will make the function continuous at x  2
for part a) and at x  3 for part b).
a)
2x  3

f ( x)  
2
  kx  2
x  2
x  2
 x2
 4k
b) g ( x)  
x  3
x  3
7
10. Find the following limits. Rationalize the denominator if necessary.
 3x 4  x 
a) lim  2

x   

 x 8 
b) lim

x 6  5 x3  x3


x2
c) lim 

2
x  
 x 4 2
d) lim

x2  7 x 
x  
x  

x2  4 x

11. Find the following limits:
 x 
a) lim sin 

x  
 2  3x 
c) lim
 0
tan 7

b) lim
t 0
d) lim
x 0
tan 3x
sin 6 x
t2
1  cos 2 t
Math3A
Practice Exam Fall 2015
1. Sketch the graph of the function and define the domain and range.
a) f ( x ) = −
3x 4
6 ( x + 4 ) ( x − 5)
2
2
b) h ( x ) =
2 x5
( x + 3) ( x − 6 )
2
3
2. Express the function f ( x ) =
−3 x + 2 + 2 x − 1 in piecewise form without the
absolute values.
3. Evaluate the following.
 5π 
a) sin  −

 6 

3
b) cos −1  −

 2 
4. A robot moves in the positive direction along a straight line so after t minutes its distance
is s ( t ) = 2t 2 feet from the origin. Find the average velocity of the robot over the time
interval [ 2, 4] . Then find a formula for the instantaneous velocity at to using the definition
s ( to + h ) − s ( to )
.
h →0
h
vinst = lim
5.
For the function f (x) graphed below, answer the following:
4
3
2
1
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
-1
-2
-3
-4
x→ 0
b) lim− f ( x) =
c) lim f ( x) =
d)
f (−4) =
f)
f (0) =
g) lim f ( x) =
h)
a) lim f ( x) =
e)
x→ 0
x →+∞
x→ 2
For parts i) through l) answer True or False.
i) f (x) is continuous from the left at x = −4 .
j) f (x) has a removable discontinuity at x = −6 .
k) f (x) is not continuous from the left at 0.
l) f (x) is continuous from the right at x = 4 .
lim f ( x) =
x→ − 4
lim f ( x) =
x→ − 6
6. Sketch a possible graph for a function f (x) with the specified properties:
f=
( 2) 1 ;
( 4 ) f=
lim f ( x ) = 0 and lim+ f ( x ) = − ∞ ;
x →−3 −
lim f ( x ) = − ∞ and lim+ f ( x ) = +∞ ;
x → − 1−
x→−1
x →−3
lim f ( x ) = + ∞ and lim f ( x ) = 2
x →+ ∞
x →−∞
4
3
2
1
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-1
-2
-3
-4
7. Find a value for the constant k that will make the functions continuous at x = 0 .
a)
 sin 2 x
x≠0

3x
f ( x) = 
 2
0
3 x − 2k + 1 x =
1 − cos x

x≠0
 2 + 5x
b) g ( x) = 

0
 − 3k + x x =
7
8. Find the equation of the tangent line on f ( x ) = 12 at xo = −2 . Use the definition
x
mtan = lim
x → xo
f ( x ) − f ( xo )
to find the slope of the tangent line.
x − xo
9. The figures below show the position versus time curves of four different particles
moving on a straight line. For each graph determine whether its instantaneous
velocity is increasing or decreasing with time.
s (t )
s (t )
t
s (t )
s (t )
t
t
t
10. Find the following limits:


a) lim  x − 5 
x →5
x
−
5


 1

− 1 

b) lim  x + h
x
h →0


h


 cot x − 1 
c) lim 
π
x→
 sin x − cos x 
4
 ( x + h )2 − x 2 
d) lim 

h →0
h


11. Use the following definition mtan = lim
h →0
f ( x + h) − f ( x)
h
to find the slope of a
tangent line on f ( x ) = sin x at an arbitrary x value.
( )
(
)
Hint: Remember that lim sin h = 1 and lim 1 − cos h = 0 .
h →0
h →0
h
h
Math3A
Practice Exam Fall 2015
Math3A
Exam #01
Spring 2016 Solutions
1. Sketch the graph of the function and define the domain and range. Give the equations of any
vertical or horizontal asymptotes if they exist.
a) f  x   
6 x5
2  x  4   x  5
3
2
b) h  x   x 2  4
2. Express the function f  x   2 x  4  3 x  2 in piecewise form without the absolute values.
3. Given f  x   ax 2  bx  c answer the following questions.
a) Given a  1, c  1, and 0  b  2 , define the domain and range of the function.
b) Given a  1, c  1, and 0  b  2 , how many y-intercepts does the function have?
c) Given a  1, c  1, and 0  b  2 , how many x-intercepts does the function have?
4. A robot moves in the positive direction along a straight line so after t minutes its distance is
s  t   3t 2 feet from the origin. Find the average velocity of the robot over the time interval 1,5 .
Then find a formula for the instantaneous velocity at to using the definition
vinst  lim
h 0
s  to  h   s  to 
.
h
5. Circle the limits below that represent the slope of the tangent line on the function g  x  at
x b.
 g  a   g b  
lim 

a b
a b


 g  a   g b  
lim 

a b
ba


 g  a   g b  
lim 

b a
a b


 g b  g  x  
lim 

b x
bx


 g  x   g b 
lim 

x b
x b


6. For the function f (x) graphed below, answer the following:
4
3
2
1
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
-1
-2
-3
-4
x 0
b) lim f ( x) 
c) lim f ( x) 
d)
f (3) 
f)
f (0) 
g) lim f ( x) 
h)
a) lim f ( x) 
e)
x 0
x 
x 4
For parts i) through l) answer True or False.
i) f (x) is continuous at x  1 .
j) f (x) has a removable discontinuity at x  6 .
k) f (x) is not continuous at 0.
l) f (x) has a removable discontinuity at x  4 .
lim f ( x) 
x  3
lim f ( x) 
x  6
7. Find a value for the constant k that will make the functions continuous at x  0 .
a)
 sec  x  2 
f ( x)  
2
  kx  2
x  2
x  2
 sin 2 x

b) g ( x)  
3x
 4k
8. Answer True or False.
a) If lim
f  x   f 1
f 1  h   f 1
 3 then lim
 3.
h

0
x 1
h
b) If lim
f  a   f b 
a b
  2 then lim
1
b a f  a   f  b 
a b
2
x 1
b a
x0
x0
9. Find the following limits:
 3x 4  x 
a) lim  2

x   

x

8


  x3  2 x 2  7 x  3 

b) lim 
9
2
x  
3

3
x

3
x




x2
c) lim 

2
x 
 x 4 2
d) lim
x  

x 2  cx  x 2  dx

10. Use the following definition mtan  lim
h 0
f  x  h  f  x
h
line on f  x   tan x at an arbitrary x value.


Hint: Note that lim tan h  1 .
h 0
h
to find the slope of a tangent
Math3A
Solution Exam #01
Fall 2016
1. Sketch the graph of the function and define the domain and range.
a) f ( x) 
x2  4
x2  x  2
b) h( x) 
x2  x  6
3 x
2. Given f  x  
a)
b)
 x  3
x  2 1
, answer the following:
lim f ( x) 
x 1
lim f ( x) 
x 
c) What is the domain and range of f ( x) ?
3. Decide which of the polynomial functions in the list might have the given
graph. More than one answer may be possible.
2
-4
5
6
-3
2 x5
4 x2
2  x  4  x  5 
a) y 
b) y  
2 x2
 x  4  x  5
b) y 
2x
 x  3 x  6 
c) y  
3x 4
6  x  4  x  5 
c) y 
2 x3
a) y 
2x2
d) y 
4  x  4  x  5 
d) y 
 x  3  x  6 
2
 x  3 x  6 
2
2
2x
 x  3  x  6 
2
2
4. Express the function f  x   x  3  4 x  2 in piecewise form without the
absolute values.
5. Sketch a graph of the function.
 1
 x  2 , x  0
f  x   2
x  x6 , x  0
 x 2  9
6. For the function f (x) graphed below, answer the following:
4
3
2
1
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
-1
-2
-3
-4
x 4
b) lim f ( x) 
f (6) 
f)
a) lim f ( x) 
e)
x  4
f (0) 
c) lim f ( x) 
d)
g) lim f ( x) 
h)
x 
x  3
For parts i) through m) answer True or False.
f ( x)  0 .
i) The lim
x 0
j) f (x) has a removable discontinuity at x  2 .
k) f (x) is not continuous at x  2 .
f ( x)  1 .
l) The xlim
 2
lim f ( x) 
x  3
lim f ( x) 
x  6
7. Sketch a possible graph for a function f (x) with the specified properties with f (x) having
a domain of all real numbers.
lim f  x     and lim f  x   0 ;
f   4  f  2  1 ;
x 3 
x 3
lim f x     and lim f x    ;
x   1
lim f  x     and lim f  x   2
x  
x  1
x 
4
3
2
1
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
-1
-2
-3
-4
8. Find a value for the constant k that will make the function continuous at x  2 for part
a) and at x  0 for part b).
a)
 sec  x  2 
f ( x)  
2
  kx  2
x  2
x  2
 sin 3x

b) g ( x)   4 x
 5k
x0
x0
9. Answer the following questions.
a) Suppose the lim
x 2
must the lim
h 0
f ( x)  f (2)
 4 . If f  x  is an even function, then what
x2
f (2  h)  f (2)
be equal to?
h
b) Suppose f  x  is an odd function. If the slope of the tangent line on f  x 
at x  a is 4, then what must the lim
h 0
f (  a  h)  f (  a )
be equal to?
h
t  2 t  0

g  t   t 2
0t 2
10. Given
, answer the following:
2t
t

2

g (t ) 
a) lim
t 0
g (t ) 
b) lim
t 1
g (t ) 
c) lim
t2
11. Find the following limits. Rationalize the denominator if necessary.
 x3  2x 2  7x  3 

a) lim 
9
2
x 

3
6 x  3x




b) lim sin 1 
t 0



t2

2  2 cos 2 t  
c) lim
x 
x
2
 ax  x 2  bx
