Math 160 Exam II Practice Problems
1. Find the
2. (5 points each)
a.
Find the exact value of
b.
Find the exact value of
3. Verify
4. Expand
as much as possible.
5. Verify that
6. Solve for x:
7. Use transformations to sketch the graph
the range.
. In addition find the domain and
8. Solve
9. Solve
10.
a.
Find the exact value of
b.
Find
c.
Find
d.
Find
11. Graph
12.
by transforming the graph of
and find its domain/range.
.
a> Explain why
b> lFind
exists
and verify your answer.
13. Use a double angle formula for
to derive a formula for
.
14. Derive a formula that expresses
formula for
as a function of
and
. You may use the
without proof.
15. (5 points each)
a.
Find the exact value of
b. Find the exact value of
16. Verify
17. Expand
as much as possible.
18. Verify that
19. Simplify each of the following:
a.
b.
c.
20. Graph at least two periods of y  4 cot( x 

2
) . Identify the amplitude, period, and phase shift
21. Sketch the graph of f ( x)  2 ln( x  3)  1 . You m6ust find its domain and the range.
22. (5 points each) a) Expand
log b
e x  ex
3
as much as possible. b) Solve for x:
2
w
xy 2
z3
23. A hard-boiled egg at 98 degrees is put in a large sink of 18 degrees of water. After 5
minutes, the egg’s temperature is 38 degrees. How much longer will it take for the egg to
reach 20 degrees?
24. Use transformations to sketch the graph of y  ln( x  4)  3 . Find the domain, range, and
vertical asymptote.
25. It is given that f ( x ) 
f (x ) and f
1
x3
is 1-1. Compute f
2x  1
1
( x) and determine its domain and range for
( x) .
26. Use transformations to graph at least one period of 2 cos(3 x   ) . Also find the amplitude,
period, and phase shift
27. State and prove the formulas for sin
2
 and sin 2 .
28. (5 points each) Find the exact value for each of the following: a) sin
1
1 b)
1
5

3
1
sin(sin 1  cos 1 ) c) cos( x  ) if sin x  and x is in QII . d) sin( 2 cos 1 )
2
13
6
5
2
29. An object is heated to 100 degrees (in Celsius) and is then allowed to cool in a room whose
air temperature is 30 degrees. If the temperature of the object is 80 degrees after five
minutes, when will its temperature be 50 degrees?
30. A certain cell culture has a doubling time of 5 hours. Initially, there were 3000 cells
present. Find the time it takes for the culture to triple.
31. Given
f ( x) 
x 1
x2  1
, g ( x) 
, find the domain of g f ( x)
x2
x2
sSolutions:
1.
Find the
Solution:
2.
a.
Find the exact value of
b.
Find the exact value of
a.
b.
3. Verify
Solution:
4. Expand
as much as possible.
Solution:
5. Verify that
Solution:
6. Solve for x:
Solution:
First combine the logs:
Next raise both sides by the power 2 to cancel the log
However,
when
. Therefore,
7. Use transformations to sketch the graph
the range.
Solution:
The domain is all real numbers:
The range is
. In addition find the domain and
8.
Solve
Solution: Take ln of both sides:
exponents:
. Then bring down the
Next distribute and isolate the terms containing x:
. Isolate the terms containing
x:
9. Solve
10.
a.
Find the exact value of
b.
Find
c.
Find
d.
Find
Solutions
a.
b.
c.
d.
: since the tangent is negative, the angle must be in QIV.
Use a double angle formula for cosine involving sine:
11. Graph
Solution:
by transforming the graph of
and find its domain/range.
1.
2.
3.
4.
Reflection about the y-axis
Shift 3 to the left
Vertical stretch
Shift 1 down
Domain: all real numbers.
Range:
12.
. Explain why the inverse of f exists. Then Find
and verify your answer.
Solution:
f ( x)  3 x  1 is an ont-to-one function by the horizontal line test. Thus the inverse of f exists.
To check:
And
13. Use a double angle formula for
to derive a formula for
.
Pf:
14. Derive a formula that expresses
formula for
as a function of
without proof.
Pf:
15. (5 points each)
a.
Find the exact value of
Solution: recall that
. Using
and
. You may use the
b. Find the exact value of
Recall that
. Using
,
16. Verify
Pf:
17. Expand
as much as possible.
Solution:
18. Verify that
Pf:
Solution: First combine the logs. Then exponentiate both sides.
. Raise both sides by the base 2.
Next check to see if
If
positive.
are valid.
, which makes sense since all the arguments are
If
arguments are negative.
Therefore, If
.
12) Use transformations to
d
Solution: Use transformations:
(1 up)
, which does not make sense since some
and find the domain and the range.
(shift 2 to the right)
(reflection about the y-axis)
Domain: all real numbers
Range:
19. Simplify each of the following:
a.
b.
c.
Solution:
a.
b.
c.
20) Graph at least two periods of y  4 cot( x 

2
) . Identify the amplitude, period, and phase
shift
21) Sketch the graph of f ( x)  2 ln( x  3)  1 . You m6ust find its domain and the range.
Solution:
ln x  ln( x  3)  2 ln( x  3)  2 ln( x  3)  1
1) 3 to the right
2) Vertical stretch
3) 1 up
Domain:
x3
Range: all real numbers.
22) (5 points each) a) Expand
Solutions
log b
xy 2
z3 w
as much as possible. b) Solve for x:
e x  ex
3
2
a.
log b
xy 2
3
1
 log b x  2 log b y  3 log b z  log b w
2
w
z
e  ex
b.
 3  e x  e  x  6  e x  6  e  x  0 . Next multiply each term by the LCD e x
2
x
x
x
e (e  6  e )  0  e 2 x  6e x  1  0 . Let y  e x . Then e 2 x  (e x ) 2  y 2 .
x
e 2 x  6e x  1  0  y 2  6 y ,1  0  y  3  5  e x  3  5, e x  3  5 . Since 3  5 and 3  5
are both positive, they are both possible. By taking ln of both sides, we get
e x  3  5,  x  ln( 3  5 ), e x  3  5  x  ln( 3  5 )
23) A hard-boiled egg at 98 degrees is put in a large sink of 18 degrees of water. After 5
minutes, the egg’s temperature is 38 degrees. How much longer will it take for the egg
to reach 20 degrees?
Solution: Use Newton’s Law of Cooling:
T  TM  Ae  kt , where T is the temperature of the egg.
We know that
Tm  18 , the water temperature.
t  0  T  98
. So 98  18  Ae  A  80
0
t  5  T  38 , so 38  18  80e 5 k  e 5 k 
Thus k 
Find
1
1
 5k   ln 4 ( ln  ln 1  ln 4   ln 4 )
4
4
ln 4
5
t when T  20 . Thus 20  18  80e
(
ln 4
)t
5
. But
e
(
ln 4
)t
5
e
t
ln 4 (  )
5
 (e
ln 4
)
t
( )
5
4

t
5
24) Use transformations to sketch the graph of y  ln( x  4)  3 . Find the domain, range, and
vertical asymptote.
Solution: First note that for y  ln x , the domain is (0, ) , the range is (, ) , the vertical
asymptote is x=0.
The horizontal transformation ln x  ln( x  4) is to move the graph 4 units
to the left. This moves changes the domain to (4, ) and the vertical asymptote to x  4
The vertical transformation is Shift the graph 3 to the right. But this move does not affect the
range since it is already all real numbers.
25) It is given that f ( x ) 
for f (x ) and
f
1
x3
is 1-1. Compute f
2x  1
1
( x) and determine its domain and range
( x) .
Solution:
Let y 
y3
x3
. Switching x and y, we get x 
. To solve for y, multiply by the LCD and
2x  1
2y 1
isolate the terms containing y:
x
y3
 x(2 y  1)  y  3  2 xy  x  y  3  2 xy  y  x  3 . Now
2y 1
factor out y and solve for y by dividing by its coefficient:
 2 xy  y  x  3  y (2 x  1)  x  3  y 
x3
 f
2x  1
The domain of f is all real numbers except x 
x
1
. Thus the range of f and f
2
1
1
( x) 
x3
2x  1
1
. The domain of f
2
1
is all real numbers except
are both all real numbers except y 
1
2
26) Use transformations to graph at least one period of 2 cos(3 x   ) . Also find the amplitude,
period, and phase shift
28) State and prove the formulas for sin
2
 and sin 2 .
29) 5 points each) Find the exact value for each of the following: a) sin
1
1 b)
1
5

3
1
sin(sin 1  cos 1 ) c) cos( x  ) if sin x  and x is in QII . d) sin( 2 cos 1 )
2
13
6
5
2
Solutions:
a) sin
1
1

2
b) Use sin(A+B)=sinAcosB+cosAsinB
1
5
1
5
1
5
 cos 1 )  sin(sin 1 ) cos(cos 1 )  cos(sin 1 ) sin 1 (cos 1 )
2
13
2
13
2
13
1 5
3 12
5  12 3
 ( )( )  ( )( ) 
2 13
2 13
26
sin(sin
1
c) Use cos(A+B)=cosAcosB-sinAsinB

4
3
3 1
4 3 3


)  cos x cos( )  sin( x ) sin( )  ( )( )  ( )( ) 
5 2
5 2
10
6
6
6
cos( x 
d) use sin(2A)-2sinAcosA
1
1
1
3 1
3
sin( 2 cos 1 )  2 sin(cos 1 ) cos(cos 1 )  2( )( ) 
2
2
2
2 2
2
29) An object is heated to 100 degrees (in Celsius) and is then allowed to cool in a room whose
air temperature is 30 degrees. If the temperature of the object is 80 degrees after five
minutes, when will its temperature be 50 degrees?
T  TM  Ae  kt . Tm  30 since the room temp is 30. It is also given that, when t = 0 T  100
1 5
. Find t when
5 7
 100  30  A  A  70 .When t = 5, T is 80  80  30  70e k (5)  k   ln  
1
T = 50. 50  30  70e 5
5
ln( ) t
7
2
5
 t  5 ln( ) / ln
7
7
30) A certain cell culture has a doubling time of 5 hours. Initially, there were 3000 cells
present. Find the time it takes for the culture to triple.
Use
P  Aekt . When t = 5, P=2A . Thus 2 A  Ae 5 k  k 
ln 2
3 A  Ae
5
t

32. Given f ( x) 
ln 2
5
t  ln 3  t 
ln 2
. Let P= 3A and solve for t:
5
5 ln 3
ln 2
x 1
x2  1
, g ( x) 
, find the domain of g f ( x)
x2
x2
First find the domain of f: all real numbers except 2. Next find the domain of g: all real numbers
except -2. But this is not the problem: we need to solve f(x)=-2:
f ( x) 
x 1
 2  2 x  4  x  1  x  1 . Thus the domain is all real numbers except 2 and 1.
x2