Download MATH-154-FINAL-EXAM

Math 154 - College Algebra
Name: _____________________________
Final Comprehensive Exam Review
Date: ______________________________
Instructor: __________________________
College Center: ______________________
For Instructor Use Only
Final Exam Grade: _________
Final Course Grade: ________
Read Each Problem Carefully.
Show All Your Work Neatly In The Space Provided.
No cell phones allowed on this exam.
No graphing calculators allowed on this exam.
Scientific calculators are permitted.
Exponential Decay Model: y  aebx , b  0
y  aebx , b  0
a
Logistic growth Model y 
1  be  rx
Logarithmic Models: y  a  b ln x, or y  a  b log10 x
Exponential Growth Model:
log a x 
log b x
log b a
log a a x  x
d
 x2  x1 2   y2  y1 2
4
V   r3
3
A  Pe
x
rt
b  b2  4ac
2a
A = πr2
a log a x  x
 n

S    i (n  2)   )r 2 
 i 1

 x1  x2 y1  y2 
,


2 
 2
A  4 r 2
 r
A  P 1  
 n
1n
 A
i 
P
C = 2πr
1
nt
Evaluate the following expression. No Decimal Answers. Show your work.
1/ 2
1. (2 pts) 8
2 3
2. (2 pts)
 4 x6 


 9 


Simplify the following and give the result with positive exponents. exponents positive. Show your work
Show your work. Give the result in exact form.
3
3. (3 pts) 5 125x 4
4. (3 pts) 3
81
64x3
Rewrite the following expression by rationalizing the denominator.
Simplify and give the result in exact form.
5. (4 pts)
2
6. (4 pts)
5 3
2
6 2
Perform the indicated operation and simplify the following expressions:
7. (4 pts)
2a  2
3a  3a
2

3a 2  a  2
2
9a  4
8. (4 pts)
10 x
2
x 9

5
x 3
Perform the indicated operation and simplify the following expressions:
9. (4 pts)
 2 3
 

 3m 2 
 3
1 
 4m  2 
3m 

Application Problem.
10.
The approximate annual interest rate r of a monthly installment loan can be modeled by the formula
 24  NM  P  


N


r
NM 

P

12 

Where N is the total number of payments, M is the monthly payment, and P is the amount financed.
a) (3pt)
Simplify the expression for the annual interest rate.
b) (3pt)
Use the simplified expression to approximate the annual interest rate for a 4-year loan of
$15,000 that has a monthly payment of $400.
Solve the following equations using the quadratic formula.
11. 5 x 2  8 x  2  0
a) (2pt) Give the result in simplified exact form.
b) (2pt) Give the result rounded to 3 decimal places.
Application Problem.
12.
The cost to produce bottled spring water is given by the cost equation
C  16x  63 , where x is the number of bottles in thousands. The total revenue from the sale of these
bottles is given by the equation R   x 2  326 x  18463 .
a) (3pt)
Determine the profit equation ( profit = revenue-cost ).
b) (3pt)
After a bad flood contaminates the drinking water of a nearby community, the owners
decide to bottle and donate as many bottles of water as they can, without taking a loss.
How many bottles will they produce for the flood victims?
13. Consider the following equation.
x2  y 2  2 x  4 y  4  0
a) (3pt) Write the equation of the circle in standard form.
b) (2pt) Sketch the circle. Label the axes.
Consider the graph of f ( x)  x .
Use your knowledge of transformations.
14. g ( x)  
x 4 3
a) (2pt) Clearly state and indicate the transformations.
15. Given the functions: f ( x)  x  4 and g  x   3x  2 ,
a) (2pt)
b) (2pt)
c) (2pt)
d) (2pt)
Find f g .
What is the domain of f g ?
Find g f .
What is the domain of g f ?
16. (3 pts) Given the functions f ( x) 
2x
and
x3
g ( x) 
5
, find  g f   3 .
x
17. (3 pts) Given the functions f ( x)  x 2  3x and
g ( x)  x  2 , find  f g   2 .
Application Problem.
18.
The number of bookcases b that a factory can produce per day is a function of the number of hours h it
operates
b(h)  40h ,
0  h  12
The daily cost c to manufacture b bookcases is given by the function
c(b)  0.1b2  90b  800 ,
0  h  12
a) (3pt) Evaluate b  5 and interpret your answer.
b) (3pt) Evaluate c  5 and interpret your answer.
c) (3pt) Evaluate  c b  h  and interpret your answer.
d) (3pt) Evaluate  c b 10 and interpret your answer.
19. (3pt) Given f ( x)  x 2  3, x  0 , find f 1 ( x) if the inverse exists.
20. (3pt) Given f ( x)  3 x  5, , find f 1 ( x) if the inverse exists.
21. Rewrite in exponential form.
a) (3pt) logb 27  3
b) (3pt) ln x  2
22. Rewrite in logarithmic form.
a) (3pt) 22 
1
4
b) (3pt) e2.8  x
Evaluate the following logarithm by changing the base to 10.
Round the result to 3 decimal places,
23. (3 pts) log 1  36
 
3
Application Problem.
24. (5 pts)
You decide to invest $8000 for 3 years at an amount rate of 8%. How much more is the return
if the interest is compounded continuously than monthly. Round your results to the nearest
dollar. Show your work.
Application Problem.
25. (5 pt) Find the half-live, of radioactive iodine (131I) if, after 20 days, only 0.53 kg of the
initial 3 kg remains.
Condense the following expressions to a logarithm of a single quantity and simplify if possible.
26. (3 pts)
ln  x  2  ln 3x
28. (3 pts)
1
 log5 x  log5 y   2 log5  x  1
2
27. (3 pts)
3ln  x  7   ln x
Fully expand to a linear expression.
29. (3 pts)
log6 3 x 2 y5
30. (3 pts)
ln e19 x 20
Solve the following equations algebraically. Approximate the result to 3 decimal places.
31. (3 pts)
32 x1  15
33. (3 pts)
40e0.6 x  3  237
34. (3 pts)
e2 x  4e x  3  0 Circle your solution(s)
35. (3 pts)
log4  x  2   2 Circle your solution(s)
36. (3 pts)
3ln 2x  12 Circle your solution(s)
37. (3 pts)
38. (3 pts)
32. (3 pts)
4 x  15
log2 x  log2  x  7   3 Circle your solution(s
1
ln  x  2   ln  4 x  3  ln
Circle your solution(s)
x
Application Problem.
39. (5 pt) You are planning for an early retirement. How long, to the nearest tenth of a year, will it take
$25,000 to grow to half a million dollars at 9% interest? Show your work.