Page 1 College Algebra : Computing Lab 1 /5 Name Due 1/18/18 ________________________ Using your calculator evaluate the following for and . Bœ Cœ# % 1. ( œ # ___________________ 2. Ð (Ñ œ # ___________________ 3. $ÐBCÑ#C &B$C B # # # œ ___________________ 4. C B B C Š ‹ " " œ ___________________ Winplot is a free general-purpose plotting utility which can be downloaded from http://faculty.madisoncollege.edu/alehnen/winptut/Install_Winplot.html . There is an online tutorial at . http://faculty.madisoncollege.edu/alehnen/winptut/winplot_intro_spr2012.ppt 5. In use the command sequence to make a 2-dim Window with and Winplot View/view/set corners &ŸBŸ& #! Ÿ C Ÿ #! . From the menu use the check radio button to indicate that you want a grid with values shown on the a View/Grid rectangular scale xes. Use a horizontal scale of 0.5 with a of 2 . This labels every other horizontal "tic mark" on the axis. Use a ver interval freq B tical interval of 2 with a of 1. In the menu enter the following two expressions Only type what is displayed in the "box" freq Equa/1. Explicit Þ . C œ 0ÐBÑ œ "#B "$B "% # C œ 0ÐBÑ œ Ð$B #ÑÐ%B (Ñ Use different colors for each graph. Print and attach your plot with the lab. To print, first enlarge the print size through the command sequence with the set to 16 cm, then use . File/Format width File/Print How do the graphs of the two expressions compare? Explain why this is so.
Transcript
Page 1 College Algebra : Computing Lab 1 /5
Name Due 1/18/18________________________
Using your calculator evaluate the following for and .B œ C œ � #%
1. � ( œ# ___________________
2. Ð � (Ñ œ# ___________________
3. $ÐB�CÑ�#C&B�$C �B
#
# # œ ___________________
4. CB B
C � Š ‹
�"�"
œ ___________________
Winplot is a free general-purpose plotting utility which can be downloaded from
There is an online tutorial at .http://faculty.madisoncollege.edu/alehnen/winptut/winplot_intro_spr2012.ppt
5. In use the command sequence to make a 2-dim Window with and Winplot View/view/set corners � & Ÿ B Ÿ & � #! Ÿ C Ÿ #! .
From the menu use the check radio button to indicate that you want a grid with values shown on the a View/Grid rectangular scale xes.
Use a horizontal scale of 0.5 with a of 2 . This labels every other horizontal "tic mark" on the axis. Use a verinterval freq B tical interval
of 2 with a of 1. In the menu enter the following two expressions Only type what is displayed in the "box"freq Equa/1. Explicit Þ .
C œ 0ÐBÑ œ "#B � "$B � "% #
C œ 0ÐBÑ œ Ð$B � #ÑÐ%B � (Ñ
Use different colors for each graph. Print and attach your plot with the lab. To print, first enlarge the print size through the command
sequence with the set to 16 cm, then use .File/Format width File/Print
How do the graphs of the two expressions compare? Explain why this is so.
Page 2 College Algebra : Project 1 /40
Name Due 1/25/18________________________
Problems 27 through 39 and 54 through 60 are each worth 1 point. All other problems are each worth point."#
True/False
1. is not a rational number.____________ !
2. .____________ šB B R U ± §# − ›3. .____________ š B ± B Á g N Q # − ›+4. There are some terminating or repeating decimals which are also irrational numbers.____________
5. is an irrational number.____________ È*'
6. For any real number must always equal .____________ B ± B ± B,
7. For any real number must always equal .____________ B ± � B ± B,
8. For any real number must always equal .____________ B ± � B ± B ±, ±
Solve and graph the solutions of the following inequalitiesÞ
49. & � #B & ""
50. #ÐB � $Ñ � #ÐB � "Ñ ' � #ÐB � #Ñ
51. ± $B � ' ± *
52. ± %B � ' ± Ÿ "!
53. "! � #C � #& � #CŸ $C � &
Page 8 7 � �
Express the following complex numbers in standard rectangular form, .+ � ,3
54. È È#& � � "' œ ________________
55. 3"(& È ÈÈ È�%* � #&
% � �*
œ ________________
56. Each year John contributes $ 00 less than twice as much money as Jim to charity. If together they contribute $ ,% *&!how much does each contribute?
57. A square is deformed into a rectangle by increasing one of its sides by m . The resulting rectangle has m more are$Þ! #%Þ! 2 a
than the original square. What were the square's dimensions?
58. Joe Average has scores of 96, 7 , and 83. What is the required range of scores on the fourth exam that will insure Joe's%average to be between 88 and 82 inclusive? (This is a problem of average difficulty.)
59. How many grams of an alloy which is 2 % copper by weight must be mixed with 500 g of an alloy which is % copper by weight& "#to make a final alloy which is % copper by weight?#!
60. Mary has $ in change. She has four less dimes than pennies, but one more dime than twice the number of quarters."Þ('She also has one more nickle than quarters. How many of each kind of coin does Mary have?
Page 9 College Algebra : Computing Lab 2 /5
Name Due 2/01/18________________________
1. Sketch the function Using your calculator or Winplot estimate to the nearest hundredth the two roots of .0ÐBÑ œ #B � B � ( 0ÐBÑ# ÞUse the convention that root #1 root #2 .�
root #1 root #2 œ œ ___________ _____________
Give the 'exact ' answers for the two roots of 0ÐBÑ .
root #1 root #2 œ œ _____________ _____________
Below approximate the exact results as decimal numbers to at least six places.
root #1 root #2 œ œ _____________ _____________
2. From your graph and your knowledge of the roots of , what is the solution set of ?0ÐBÑ 0ÐBÑ � !
3. What is the solution set of ?#B � ( � B#
4. Solve and graph the solution set of the inequality #B � $B# $B � % .
Page 10 College Algebra : Computing Lab 3 /5
Name Due 2/06/18________________________
1. Using a graphing calculator or Winplot enter the following functions :
C œ C œ C œ C œ $B � #B# � " #B � # #B � "; ;
Set Xmin Ymin , Xmax Ymax , then ZOOM to a square grid and graph the four functions.œ œ � & œ œ &
a) Which of the lines are (parallel)?||
b) What is it about the equations of the functions that makes the graphs of the lines ?||
c) Does the constant in the equation have an effect on whether this line is to any other line?, C œ 7B� , ||
2. On your graphing calculator enter the following functions :
C œ C œ � C œ C œ � #B � #B# � " #B � # #B � "; ;
Set Xmin Ymin , Xmax Ymax , then ZOOM to a square grid and graph the four functions.œ œ � & œ œ &a) Which of the lines are (perpendicular)?¼
b) What is it about the equations of the functions that makes the graphs of the lines ?¼
c) Does the constant in the equation have an effect on whether this line is to any other line?, C œ 7B� , ¼
3. Let be the line that passes through and is parallel to the line ._ Ð � $ß &Ñ $B � #C œ � ")
Let be the line that passes through and is perpendicular to the line .` Ð � $ß &Ñ $B � #C œ � ")Fill in the following :
Line Slope intercept interceptC B_`
Equation of in slope-intercept form: Equation of in slope-intercept form: _ ` ________________ ________________
Sketch the graph of the lines and _ `
Page 11 College Algebra : Computing Lab 4 /5
Name Due 2/12/18________________________
1.Using either your graphing calculator or Winplot enter the following functions then sketch the results in a labeled
(indicate which curve is which function) graph. On the calculator Set Xmin ; Xmax and use a square zoom.œ � "! œ "!In Winplot use to with left down , right up , then from select .View/view set corners View/Zoom Squareœ œ � "! œ œ "!From the menu select , then in the dialogue box enter for "name" and for name(x)= , followed by pressiEqua User functions f x^2 " " ng
the "enter" button. Repeat this procedure to define as and then close the dialogue box1ÐBÑ .0ÐB � %Ñ � $
Enter the six curves one at a time by using the dialogue box from the menu.Explicit Equa
Standard Notation TI-84
Curve 1. Y X C œ 0ÐBÑ œ B#1
2œCurve 2. Y Y X 4 3 C œ 1ÐBÑ œ 0ÐB � %Ñ � $ 2 1œ Ð � Ñ �Curve 3. Y Y X C œ 1Ð � BÑ 3 2œ Ð � ÑCurve 4. Y Y X C œ � 1ÐBÑ 4 2œ � Ð ÑCurve 5. Y Y 2X C œ 0Ð#BÑ 5 1œ Ð ÑCurve 6. Y 2Y X C œ #0ÐBÑ 6 1œ Ð Ñ
2. Repeat problem 1 for 0ÐBÑ œ lBl
Graph of Problem 1 Graph of Problem 2
Page 12 College Algebra : Project 2 /40
Name Due 2/15/18________________________
Problems 12, 13, 21, 22, 23, 25, 28, 29, 30 are each worth 2 points. All other problems are worth 1 points.
Solve the following equations.
1. $ � "&BB œ "&B � )(#
2. "#B � %B œ #B$ #
3. ± ± œ � #B � "B � $ B#
4. Solve and graph the solutions of the inequality B Ÿ % � B# � %B .
5. A rectangle is longer than it is wide and has an area of What are the rectangle's dimensions?( "#! m m . #
6. Mary and Steve each travel 845 miles. Mary drove an average of 5 mph faster than Steve and completed the trip in one hour's less time
than Steve. What was each person's average speed?
Page 13 2 � �
Solve the following equations.
7. # & �
�BB �" B B � œ
10 2
8. #%B�# œ #B � '
9. B œ È&B � #%
10. È ÈB � # � B� & œ � "
11. Solve the formula, , for+ œ � . B,�-B,�B .
12. Solve and graph the solution of the inequality $B � B % .
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13. A storage tank has two inlet pipes. The smaller pipe takes 6 minutes longer by itself to fill the tank than does the larger inlet. When both
pipes are open the tank fills in 4 minutes. How long does it take the large inlet pipe acting alone to fill the tank?
14. Write the equation of a circle of radius centered at .% Ð � # $Ñ ,
15. Sketch the circle: B � C � 'B � )C œ !# #
16. Find the slope, distance and midpoint between the points and .Ð � " &Ñ Ð$ � ( Ñ , ,
slope distance œ œ œ mid point _____________ _____________ _____________
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17. Graph the line and fill in the following :� $B � &C œ � "!
slope intercept intercept œ œ œ _____________ _____________ _____________B C
18. Graph the line and fill in the following:� *C œ � #(
slope intercept intercept œ œ œ _____________ _____________ _____________B C
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19. Graph the line and fill in the following :$B œ � "#
slope intercept intercept œ œ œ _____________ _____________ _____________B C
20. Indicate which of the following relations define as a function of or as a function of .B C C B B C
a) b) C œ #B B œ #C# #� ( � (
c) d) B B# # #� C œ "' � C œ "'
21. The function is defined as follows : The dependent variable (output) is three times the cube of the independent variable0ÐBÑ (argument
or input).Use order of operations!
a) 0ÐBÑ œ ________________________
b) 0Ð#BÑ œ ________________________
c) 0ÐB � "Ñ œ ________________________
d) 0ÐCÑ œ ________________________
e) 0ÐB œ#Ñ ________________________
Page 17 6 � �
Evaluate the following :
22. 0ÐBÑ œ $B# � &B � #
a) 0Ð!Ñ œ _______________
b) 0Ð"Ñ œ _______________
c) 0 0Ð"ÑÐ Ñ œ _______________
d) 0Ð+Ñ œ _______________
e) 0Ð#BÑ œ _______________
f) 0ÐB Ñ& œ _______________
23. 0ÐBÑ œ #B# � B� $
a) 0ÐB � 2Ñ œ __________________________
b) 0ÐB � 2Ñ � 0ÐBÑ œ __________________________
c) 0ÐB�2Ñ�0ÐBÑ
2 œ __________________________
d) What does the answer to part c get closer and closer to as gets really teeny-tiny ?2
24. In the table below indicate with a 'yes' or 'no' whether the graph of the stated equation has the stated symmetry. The abbreviation 'wrt'
means 'with respect to' .
Equation of Curve Symmetric wrt axis Symmetric wrt axis Symmetric wrt Origin B C
B � (B � C
B
�
B
# #
# $
' #
œ
� "* � C œ
ÐB Ñ � $ � C œ
� " � C œ
0
0
1 0
0
#
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25. From the graph of the function, , shown below answer the following questions :0ÐBÑ
a) What symmetry if any does display?0ÐBÑ
b) For what intervals of the independent variable is increasing?B 0ÐBÑ
c) For what intervals of the independent variable is decreasing?B 0ÐBÑ
d) For what values of does have a relative or local minimum?B 0ÐBÑ
Sketch the following
e) f)
C œ C œ 0ÐB � #Ñ0ÐBÑ � "!
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In the next two problems indicate for the given function
i) the domain
ii) the range
then sketch the function .
26. 0ÐBÑ œ lB � #l � $
27. 0ÐBÑ œ $ � #ÐB � "Ñ#
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28. From the graph of the function, , shown below answer the following questions :0ÐBÑa) For what intervals of the independent variable is increasing?B 0ÐBÑ
b) For what intervals of the independent variable is decreasing?B 0ÐBÑ
c) Give an explicit (piecewise) definition of .0ÐBÑ
Sketch the following
d) e)
C œ � 0ÐBÑ C œ 0Ð � BÑ
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29. From the graphs of the functions, & , shown below answer the following questions:0ÐBÑ 1ÐBÑ
a) What symmetry if any does display?0ÐBÑ
b) For what intervals of the independent variable is increasing?B 0ÐBÑ
c) For what intervals of the independent variable is decreasing?B 0ÐBÑ
d) For what values of does have a relative or local minimum?B 0ÐBÑ
30. From the graphs above estimate the following:
a) 0Ð!Ñ † 1Ð!Ñ œ ________________________
b) Ð1 œ‰ 0ÑÐ"Þ&Ñ ________________________
c) Ð0 œ‰ 1ÑÐ!Ñ ________________________
d) Ð0 œ‰ 1ÑÐ"Ñ ________________________
Page 22 11 � �
31. A right circular cone, a right circular cylinder, and a hollow sphere are all filled with a liquid coming into the vessel at a fixed rate,
.i.e., the number of cubic centimeters per minute of liquid flowing into the vessel is a constant. Below are functions showing how the
height of liquid in the vessel depends on time. is time required to fill each vessel to the maximum height of liquid the vesseT l allows.
State which graph goes with which vessel and explain your choices.
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6 point Bonus Question: Absolute Madness
Consider the function where is a real number. The root or zeros of a real function, are0ÐBÑ œ , � lBl � ' � # , 0ÐBÑl ll l , ,
the real numbers, , for which < 0Ð<Ñ œ !.
a) For what values of does this function have no real roots?,
b) For , how many real roots does this function have?, & "!!!
c) Is it possible for this function to have exactly one real root?
d) For what values of does this function have exactly two real roots?,
e) Can this function ever have an odd total number of real roots? If so, for what value(s) of ?,
f) What is the maximum number of real roots of this function?
Page 24 College Algebra : Computing Lab 5 /5
Name Due 2/26/18________________________
1.Sketch the function . Using your calculator or Winplot estimate to the nearest 0ÐBÑ œ B$ #� B � &B � #hundredth the three roots of and the coordinates of the two turning points of Use the convention0ÐBÑ 0ÐBÑ .
that root #1 root #2 root #3 and turning pt. #1 turning pt. #2 .� � �
root #1 turning pt. #1œ œ Ð _____________ _______,_______)
root #2 turning pt. #2œ œ Ð _____________ _______,_______)
root #3 œ _____________
2. Use the Rational Root Theorem and synthetic division to determine the exact values of the roots of .0ÐBÑ
How do these numbers compare to your estimates in problem 1 ?
Page 25 College Algebra : Computing Lab 6 /5
Name Due 3/05/18________________________
Consider the two rational functions given below :
0ÐBÑ œ 1ÐBÑ œB B# #
# #B �" B �"
a) Sketch the graph of both functions.
0ÐBÑ 1ÐBÑ
b) In what ways are the graphs of the two functions different? What feature of the functions causes this difference?
c) In what ways are the graphs of the two functions the same? What feature of the functions causes this similarity?
Page 26 College Algebra : Project 3 /40
Name Due 3/07/18________________________
Problems 3, 10, 11 and 12 are each worth 3 points. All other problems are worth 2 points each.
For each of the following parabolas : i) Calculate the coordinates of the vertex.
ii) Find and display the coordinates of and intercepts.B C
iii) Find and display the equation of the axis of symmetry.
iv) Sketch the curve.
1. "! � #B � "#B � %C œ !#
2. %C � #B � )C � % œ !#
Page 27 2 � �
3. Write an equation which generates each of the following parabolas.
a) A parabola pointing to the left with vertex at which passes through the point Ð& � $Ñ Ð � #( "Ñ , , Þ
b) A parabola pointing down with vertex at and a intercept of .Ð � # "Ñ � "", C
c) A parabola pointing to the right with the line as the axis of symmetry and intercepts at , C œ � # Ð � $ß !Ñ Ð! � # � Ñ , ,È###
and .Ð! � # � Ñ , È###
4. Perform synthetic division to find the quotient and remainder of the following divisions.
a) $B �(B�*
B�#
#
œ ______________________
b) $B �$B �B �#B �#B�%& % $ #
B�" œ ______________________
c) %B �#B �$B �&B �'B�#
B�"
( & $ #
œ ______________________
5. Given the polynomial, , perform the requested evaluations using synthetic0ÐBÑ œ $B% $ #� %B � 'B � &B � $ division.
a) 0Ð"Ñ œ ______________________
b) 0Ð#Ñ œ ______________________
c) 0Ð!Þ&Ñ œ ______________________
Page 28 3 � �
6. Given and with , , , and all real, prove by explicit calculation thatD œ + � ,3 A œ - � .3 + , - . DA D † Aqq q q . œHere means the complex conjugate of u .
_ ?
7. Given that and that is a polynomial of degree 4 with real coefficients and has0Ð � #Ñ œ 0Ð$Ñ œ 0Ð$ � #3Ñ œ ! 0ÐBÑ
0Ð"Ñ œ � *' , determine a formula for 0ÐBÑ .
8. Assuming that all turning points are shown in the graph of the polynomial function below, answer the following:
a) Is the polynomial of odd or even degree?
b) What is the sign of the coefficient of the highest degree term in the polynomial?
c) What is the minimum degree of the polynomial?
d) Give the location of all real roots and indicate which roots must be multiple roots.
Page 29 4 � �
9. Assuming that all turning points are shown in the graph of the polynomial function below, answer the following:
a) Is the polynomial of odd or even degree?
b) What is the sign of the coefficient of the highest degree term in the polynomial?
c) What is the minimum degree of the polynomial?
d) Give the location of all real roots and indicate which roots must be multiple roots.
Page 30 5 � �
10. A open-topped box is formed from a rectangular cm cm piece of sheet metal by cutting out identical squares (each of side"! ‚ 5length, ) from each of the four corners and then folding up the remaining four rectangular side pieces. The result is a box of B depth .Ba) Determine a formula for the volume of the box as a function of .B
b) What value(s) of give the most volume for the box ?B
c) What is the greatest volume in cm that the box can have ?3
11 Given Þ T ÐBÑ œ #B% $ #� $B � 'B � "#B � )
a) What is the maximum number of positive roots of ?T ÐBÑ
b) What is the maximum number of negative roots of ?T ÐBÑ
c) What is the set of possible rational roots of ?T ÐBÑ
d) Find and indicate all roots (both real and complex) of T ÐBÑ .
e) Sketch below.T ÐBÑ
Page 31 6 � �
12 Given Þ T ÐBÑ œ B& % $� &B � ""B � ""B � ()B# a) What is the maximum number of positive roots of ?T ÐBÑ
b) What is the maximum number of negative roots of ?T ÐBÑ
c) What is the set of possible rational roots of ?T ÐBÑ
d) Find and indicate all roots (both real and complex) of T ÐBÑ .
e) Sketch below.T ÐBÑ
Page 32 7 � �
In the following five problems indicate for the given function
i) the domain
ii) the range
iii) all real roots
iv) any asymptotes
v) the coordinates of any turning points to the nearest tenth, then sketch the function .
13. 0ÐBÑ œB �"$
#B �"
14. 0ÐBÑ œ #B#�$B�&B�#
Page 33 8 � �
15. 0ÐBÑ œBB
#
#��*
1
16. 0ÐBÑ œ B�"B �%#
Page 34 9 � �
17. 0ÐBÑ œ'B#
#�"$B�'
$B �B�#
18. For fixed electrical charge, the capacitance of a pair of charged parallel circular plates varies directly as the square of the
plate radius and inversely as the plate separation. If the capacitance is (micro farads) when the plate radius is $%Þ( "Þ!.F cm
and the plate separation is (microns, ), what is the plate separation, if for the same amount of%Þ! " œ !Þ!!!". .m m cm
charge and a plate radius of the capacitance is ?#Þ! $%(Þ!cm F.
21. (5 points)Bonus Problem
A standard 12 oz soda can has a volume of 355 ml. It can be modeled by a right circular cylinder with radius and height < 2 Þa) Determine a formula for the total (side plus top and bottom) surface area of the can as a function only of . The variable < 2 should
not appear in your answer.
b) Determine the value of that gives the least amount of surface area and therefore requires the least amount of metal.<
c) Do real soda cans have these dimensions? If not, give a reasonable explanation.
Page 35 College Algebra : Computing Lab 7 /5
Name Due 3/19/18________________________
1. Using WinPlot, investigate the computer plots of the following five curves. Use the window : left down ;œ œ � !Þ&right up 4 and then switch to a from the menu The function stands for the square root function.œ œ Zoom/Square View sqrt
Check " " to fix the domains as stated below. The designation x=f(t) is "2. Parametric" mode, while y=f(x) is "1. Exlock interval plicit"
mode as found under the menu.Equa
format Domain formulas to enter Equa
Curve 1. y=f(x) low x high x y x^2œ ! œ % œCurve 2. y=f(x) low x high x y xœ ! œ % œ
Curve 3. x=f(t) low t high t x t^2 ; y tœ ! œ % œ œCurve 4. y=f(x) low x high x y sqr(x)œ ! œ % œ
Curve 5. x=f(t) low t high t x sqr(t) ; y tœ ! œ % œ œ
a) What is true about the line (Curve 2) , with respect to Curve 1 and Curve 3?C œ B
b) How do the Curve 1 and Curve 5 compare? Explain why this is not surprising.
c) How do the Curve 3 and Curve 4 compare? Explain why this is not surprising.
For the following functions generate the graph and determine if the given function is one-to one. If it is, find the inverse function.
2. 3. 0ÐBÑ œ l#B � $l 0ÐBÑ œ "�"B$
Page 36 College Algebra : Computing Lab 8 /5
Name Due 3/22/18________________________
1. Use Winplot and attach the computer plot or sketch the results below. Use the window : left down ; right up .œ œ � & œ œ &To generate the "correct" shape of the curves select from the menu.Zoom/Square View
format Domain formulas to enter Equa
Curve 1. y=f(x) low x high x y e^xœ � & œ & œCurve 2. y=f(x) low x high x y xœ � & œ & œCurve 3. x=f(t) low t high t x e^t ; y tœ � & œ & œ œCurve 4. y=f(x) low x high x y ln(x)œ � & œ & œ
Curve 5. x=f(t) low t high t x ln(t) ; y tœ � & œ & œ œ
a) What is true about the line (Curve 2) with respect to the Curve 1 and Curve 3?C œ B ,
b) How do the Curve 1 and Curve 5 compare? Explain why this is not surprising.
c) How do the Curve 3 and Curve 4 compare? Explain why this is not surprising.
2. For the following function i) Give the domain, ii) Give the range, iii) Sketch the curve C œ 0ÐBÑ œ #�B,
Page 37 College Algebra : Project 4 /40
Name Due 3/28/18________________________
Problems 1, 2, 3, 4, 5, 6, 7, 9, 25, 26, 27, 28 are each worth 2 points. All other problems are worth 1 point each.
1. Given and 0ÐBÑ œ #B � & 1ÐBÑ œ"
B�"a) What is the domain of ?0
b What is the range of ?Ñ 0
c) What is the domain of ?1
d) What is the range of ?1
2. Determine the following:
e) 0ÐBÑ † 1ÐBÑ œ ________________________
f) Ð1 ‰ 0ÑÐBÑ œ ________________________
g) Ð0 ‰ 1ÑÐBÑ œ ________________________
3. Using the functions and defined in problem 1, find the inverse functions 0ÐBÑ 1ÐBÑ
a) 0 œ�"ÐBÑ ________________________
b) 1 œ�"ÐBÑ ________________________
c) 0 œ�"Ð Ñ0ÐBÑ ________________________
d) 1 œÐ Ñ1 ÐBÑ�" ________________________
4. For the functions and defined in problem 1 and the inverse functions of problem 3 .0ÐBÑ 1ÐBÑ
a) What is the domain of ?0�"
b) What is the range of ?0�"
c) What is the domain of ?1�"
d) What is the range of ?1�"
e) Ð ÑÐBÑ0 ‰ 0 œ�" ________________________
f) Ð ÑÐBÑ1 ‰ 1 œ�" ________________________
Page 38 2 � �
For the following functions
i) Give the domain
ii) Give the range
iii) Sketch the curve
5. C œ 0ÐBÑ œ $B
6. C œ 0ÐBÑ œ � lnÐB � $Ñ
7. Change the following from exponential to logarithmic form:
a) "! œ "!!ß !!!ß !!!)
b) % œ "!#%&
c) Z œ /#>
d) "!!! œ !Þ"!� "$
e) BC œ 2
Page 39 3 � �
8. Change the following from logarithmic to exponential form :
a) log%Ð'%Ñ œ $
b) log&Ð!Þ!%Ñ œ � #
c) logBÐVÑ œ >
9. Fill in the following :
a) log #Ð!Þ"#&Ñ œ _____________
b) log "'Ð'%Ñ œ _____________
c) log %Ð!Þ&Ñ œ _____________
d) log CÐC ÑB œ _____________
e) / œln( )$U _____________
10. Express the following as a single logarithm with a coefficient of one.
a) ln ln ln _______Ð#&Ñ � Ð%Ñ œ Ð Ñ
b) log log log _______"! "! "!Ð Ñ Ð Ñ œB � $ B Ð Ñ* #
c) " "$ 'ln ln ln _______Ð)Ñ � Ð'%Ñ œ Ð Ñ
11. Compute the following logarithms to 4 places.
a) log Ð"!!Ñ œ ___________
b) ln Ð"!!Ñ œ ___________
c) log #Ð"!!Ñ œ ___________
Page 40 4 � �
Solve the following equations for all real roots. If no real solution exists, write ' No Solution' . You may leave answers in a
form which involves an irrational base- or natural logarithm. For example, , or you may give the numerical"! C œ&Ð$Ñln
answer C œ %Þ&&""*'"$$ÞÞÞ
12. +) œ #&' + œ ___________
13. log CÐ#&'Ñ œ ) C œ ___________
14. "! B œB œ "!ß !!! ___________
15. "! D œD œ (!ß !!! ___________
16. log 3ÐCÑ œ $ C œ ___________
17. ln ÐBÑ œ � # B œ ___________
18. & B œB œ %Þ* ___________
19. & B œB œ � %Þ* ___________
20. && œ '!Ð" � / Ñ œ�B B ___________
21. ln Š ‹#(B œ � $ B œ$ ___________
22. C œ log %Ð"!!Ñ C œ ___________
23. < œ log
log1
1
Ð(#*ÑÐ$Ñ < œ ___________
Page 41 5 � �
24. log log #Ð'%B Ñ œ Ð"!!Ñ B œ# ___________
25. Solve and graph the solution set of the inequality Consider where crosses .log . log # #Ð Ñ Ÿ � " Ð ÑB � # C œ B � # C œ � "Hint:
26. Solve the following equation for > ÐE & Z & ! , Ñ Z œ EÐ" � / Ñ �+>
27. The decay of a radioactive material is expressed by the function where is the amount of material left aE œ 0Ð>Ñ œ E Ð#Ñ!� >
X , A fter
a time has elapsed, is the starting amount of material, and is the half-life of the material. If in years time, o> X A! $&!! '! % f the material
decays (leaving of the material still present), what is the half-life?%!%
28. Six thousand dollars is deposited in a savings account with a fixed yearly interest rate of .$Þ(&%a) After five years how much money is in the account if the it is compounded quarterly?
b) After five years how much money is in the account if the it is compounded continuously?
c) In this situation how much money does continuous compounding gain versus quarterly compounding?
d) If the account is compounded quarterly, how many years would it take for the amount to reach ?$"!ß !!!
Page 42 Instructions for the Group Labs
The following guidelines should be adhered to in forming your lab group, performing the experiments, and writing up the labs.
Group Requirements:
Each group must consist of at least two individuals but no more than four individuals. You are free to form your own groups, but if you
can't find a partner see me and I'll assign you to a group. Two class periods will be devoted to doing each lab, but probably some of the
report writing will have to be done outside of class. It is up to the group to decide any internal division of labor, eg., who is responsible
for data observation and or recording, who will do the algebra, who will check the work, who will write up the what parts of the report.
It is possible that in one group a single individual writes the entire report, while in another group everyone writes up a different part.
It is in your own best interest to insist that you understand the entire lab report. You are free to use any written resources or computing
technology in doing your analysis.
Report Requirements:
Each group must hand in report for a given lab which should include the following :one
1. The names of all group participants. If the report writers feel an individual did not perform his/her assigned task, you are free to delete
that person's name from the report. I will arbitrate all appeals on such disagreements and reserve the right to give either a written or oral
exam to decide the issue.
2. The conclusions stated neatly in sentences which are both concise and complete.
3. The work attached in a way which is both neat and clear. Answers should be presented in the same order as the associated questions.
Grading:
1. Each person in the group will receive the same point total out of 50 that the lab report receives. Appeals on this are permitted, but I
reserve the right to then administer either an oral or written exam to such an individual to replace the group score. Thus, it is the
responsibility of everyone in the group to review the analysis, conclusions and answers to all of the questions.
2. Grades will be based on both the quality of the data taken and the correctness of the methods used to analyze the data. Thus, a correct
conclusion arrived at by accident using faulty mathematics will not count for much. Points will be deducted for incomplete, illegible,
sloppy or incomprehensible answers.
Page 43 College Algebra : Lab 1 : The Simple Pendulum
Lab Scheduled 4/02/18 & 4/03/18 Lab Report : Due 4/12/18
/50
Name _______________________ Name _______________________
Name _______________________ Name _______________________
Purpose : To investigate the relationship between the length of a simple pendulum and the time it takes to complete a full swing.
Equipment : String, stop watch, weights, meter stick, protractor, ( a balance if available).
General Procedure : Tie one of the weights to the end of the string. From the center of the weight measure off the specified
length of the string. Holding the string at this distance, let the lead weight swing freely from an initial position that makeP s a #!°
angle with the vertical . Measure the time for full swings of the weight. Divide this time by to obtain the periodÐ) œ #! Ñ° 10 10 , ,Xthe time for one full swing. Repeat this procedure for each of the specified values of . Then repeat the experiment for °.P œ #&)Finally, pick a second, different mass weight and repeat the entire set of measurements.
I. Data Collection (15 points)
Data Table for First Weight ( If Balance available mass of weight = __________ ) ) )œ #! œ #&° °
P X XTime for 10 Swings Period Time for 10 Swings Period
"!Þ!
"&Þ!
#!Þ!
#&Þ!
$!Þ!
$&Þ!
%!Þ!
%&Þ!
&!Þ!
&&Þ!
'!Þ!
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
'&Þ!
(!Þ!
(&Þ!
)!Þ!
Page 44 2 � �
Data Table for Second Weight ( If Balance available mass of weight = __________ )
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
) )œ #! œ #&° °
P X XTime for 10 Swings Period Time for 10 Swings Period
"!Þ!
"&Þ!
#!Þ!
#&Þ!
$!Þ!
$&Þ!
%!Þ!
%&Þ!
&!Þ!
&&Þ!
'!Þ! cm
cm
cm
cm
cm
'&Þ!
(!Þ!
(&Þ!
)!Þ!
Give a brief but accurate description of the procedure followed in obtaining your data. Use diagrams where necessary andyou
identify all pertinent variables.
II. Data Analysis spreadsheet (15 points) ( Feel free to use a to perform the required calculations and plot the graphs. )
What were the relevant variables in this experiment?
Which variables were independent and which were dependent?
Construct a graph of versus all axes and label each curve as to the weight and angle used. You may if you wish put X P Þ Ð ÑLabel ) all
four curves on the same graph. You may use either the grid provided or your own graph paper.
In general, how did the period depend on the initial angle ?)
In general, how did the period depend on the weight used ?
Page 45 3 � �
Graphs of versus X P
Construct a graph of versus all axes and label each curve as to the weight and angle used. You may if youln lnÐX Ñ ÐPÑ Þ Ð ÑLabel )wish put all six curves on the same graph. You may use either the grid provided or your own graph paper. Record the data for these
graphs in the table provided.
Ln and Ln DataÐPÑ ÐXÑ
Ln and Ln Data for First Weight Ln Data for Second Weight
Ln Ln
ln
ÐPÑ ÐX Ñ ÐX Ñ
ÐX Ñ ÐX Ñ
P ÐPÑ
° ° ° °
cm
) ) ) )œ #! œ #& œ #! œ #&
"!Þ!
"&Þ!
#!Þ!
#&Þ!
$!Þ!
$&Þ!
%!Þ!
%&Þ!
&!Þ!
&&Þ!
'!Þ!
'&Þ!
(!Þ!
(&Þ!
)!Þ!
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
cm
Page 46 4 � �
Graphs of ln versus lnÐX Ñ ÐPÑ
III. Interpretation (15 points)
Is the relationship between and linear? Explain your answer.X P
Is the relationship between and linear? Explain your answer.ln lnÐX Ñ ÐPÑ
From elementary physics the period of a simple pendulum for small initial angles satisfies a 'power law' relationship to the length.
That is Where is a constant independent of From your data obtain an estimate of Explain how you obtainedX œ E † P Þ E P :Þ:
. this estimate.
Hint: The"best" or regression fit of a straight line through a set of data points is given by he linear model 8 ÐB ß C Ñ3 3 t C œ 7B� , ,
where is the slope and is the intercept. The slope of the regression line is given by7 , C
7 œ , œ8 B B� �
3 3 3 3
3 33 3
#3 3 3 3 3C � C
8 B 8 BB C � C
�� ��B� Ð B Ñ � Ð B Ñ
B† †�� �� �2 22 2 while the intercept can be calculated as C .
Here we are using where summation notation �B B B B B3 3 " " # # $ $ 8 8C œ C � C � C � ÞÞÞ � C ß � � �B B B B B C C C C B B B B B3 " # $ 8 3 " # $ 8 3 " # $# #
For the same weight you used estimate the period if If you have the equipment you may wish to check this estimate byP œ "!! cm . actually measuring the period.
Estimated period for a length of "!! cm œ _____________
Page 47 5 � �
IV. Application (5 points)
Below is a table of data on the electrical resistance, for a 1 meter length of different gauge copper wire. The wire's diameV , ter is ..
Copper Wire Resistance Data
Gauge cm m
30
25
20
15
12
10
9
8
7
6
. Ð Ñ V Ð ÑH
!Þ!$"(& ##$Þ(
!Þ!&&&' ($Þ!
!Þ!*&#& #%Þ*
!Þ"()' (Þ"
!Þ#(() #Þ*
!Þ$&(# "Þ)
!Þ$*'* "Þ%
!Þ%$'' "Þ#
!Þ%('$ "Þ!
!Þ&"&* !Þ)
5
4
!Þ&&&' !Þ(
!Þ&*&$ !Þ'
According to theory the resistance is related to diameter by a 'power law' . That is where is a constant independentV œ E † . E:,
of From the table obtain an estimate of Explain how you obtained this estimate..Þ :Þ
Estimated value of : œ _____________
Estimate the resistance for a 1 meter length of 35 gauge copper wire ( .. œ !Þ!"*)% cm)
Estimated resistance for a 1 m length of 35 gauge copper wire œ _____________
Page 48 College Algebra : Computing Lab 9 /5
Name Due 4/09/18________________________
1.Solve the system of equations C œ $ � #B C œ B# � #B � #
a) First sketch the two curves and indicate any points of intersection.
b) Solve the system algebraically by substitution
2. Consider the 'Transcendental Equation' . One method of solving it consists of finding the coordinates of the polnÐBÑ œ B � $ B ints
of intersection of the two curves: C œ 1ÐBÑ œ ÐBÑln
C œ 0ÐBÑ œ B � $Using your calculator, graph both curves. Sketch the result. How many real solutions does have?lnÐBÑ œ B � $
Then using either a calculator or Winplot determine approximations to these solutions accurate to the nearest hundredth.
Page 49 College Algebra : Computing Lab 10 /5
Name Due 4/16/18________________________
1. In Winplot open the window and solve each of the following three equations for Enter each formula for as an 3-dim ExpliciD D. t
function in menu. From pick to get a better 3D perspective of the surfaces. Be sure to use a different cEqua/Explicit View Axes olor
for each surface. This is chosen under color of the menu. Solve this system of equations and enter the coordinatesEqua/Explicit of
the solution using the menu. Be sure the color of this point is distinct from the color of the three surfacesEqu/Point/Cartesian .
Use PgUp (or the View Memu) to "zoom in" and the left and right arrow keys to rotate the graph until you get a "good view"
which clearly shows the three surfaces and the solution point. Print and attach your graph.
B � C � D œ "B � #C � D œ � "#B � C � #D œ )
What kind of geometric surfaces does each equation describe?
What is the significance of the solution point with respect to the three surfaces?
2. In Winplot open the 3-dim window and solve each of the following three equations for Enter each formula for as an D D. Explicit
function in menu. From pick to get a better 3D perspective of the surfaces. Be sure to use a different cEqua/Explicit View Axes olor
for each surface. Show that this system is consistent but dependent with infinitely many solutions along a line. Determine the form
of this solution. In this form let where is any real number. Solve for and in terms of . In Winplot use the D œ > > B C > Equa/Curve
menu to enter the form of the infinetly many solutions. Set = 5 and = 5. Set the to 2 and choose a dominantt lo t hi pen width�color to make this line more prominent. Use Use PgUp (or the View Menu) to "zoom in" and the left and right arrow keys rotate the
graph until you get a "good view" which clearly shows the three surfaces and the line of solutions. Print and attach your graph.
B � C � D œ "B � #C � D œ � "$B � $C � D œ � "
What kind of geometric surfaces does each equation describe?
What is the significance of the line of solutions with respect to the three surfaces?
Page 50 College Algebra : Project 5 /40
Name Due 4/18/18________________________
Problems 1, 2, 18 and 19 are each worth 1 point. All other problems are worth 2 points each.
In the following problems solve and classify (consistent and independent, inconsistent, or dependent) the following systems of equationsÞ
1.
$B � #C œ (%B � $C œ � #
2.
%B � $C œ � #� )B � 'C œ '
Solve the following systems of non-linear equations for all real solutions.
3.
BC œ "'
C � B œ !$
4. Using either a computer program or a graphing calculator solve the following non-linear system for all real solutions.
Report answers to at least the nearest !Þ!" .
1
BC œ "'
C � B œ$
Page 51 2� �
In the following problems solve and classify (consistent and independent, inconsistent, or dependent) the following systems of equations ÞIf the solutions are dependent, give the linear "form" of the infinite number of solutions.
5.
B � C � &D œ � '$B � #C � D œ )%B � $C � #D œ !
6.
B � C � D œ $B � #C � $D œ &$B � &C � (D œ "!
7.
B � C � D � #A œ ##B � $C � #D � $A œ "!B � C � #D � 'A œ "B � 'D � "$A œ (
Page 52 3 � �
Graph the region in the plane which satisfies the given constraints:
8.
B � C � "
%B � C � B "#
9.
B � C Ÿ "C � B Ÿ &B � C & $B � C � ""
Page 53 4� �
Perform the following matrix operations:
10.
Ô × Ô ×Ö Ù Ö ÙÖ Ù Ö ÙÖ Ù Ö ÙÕ Ø Õ Ø
# � % & " % ! � " $% # $ � $ � & # � " #
� # # � " $ % & � $ "% � $ & � ( � # # " $
� # œ
11. ” •” •
& # %$ � % � #
œ
12. Ô ×Ô ×Õ ØÕ Ø
$ � " $ $$ # � & "
� # # � " � "œ
13. ” •” •� % � $ # $$ # � $ � %
œ
14. – —” •. �,+.�,- +.�,-
�- ++.�,- +.�,-
+ ,- .
œ
15.
Ô ×Ô ×Ö ÙÖ ÙÕ ØÕ Ø
# $ � " # � " "� $ # & ! # $! � " # & " � #
œ
Page 54 5� �
16.
Ô ×Ô ×Ö ÙÖ ÙÕ ØÕ Ø
# � " " # $ � "! # $ � $ # && " � # ! � " #
œ
17. The system of equations can be written as the matrix multiplication +B � ,C œ / + , B /-B � .C œ 0 - . C 0
œ .
” •” • ” •Use the result of problem 14 to write the solution of this system. What is Hint: ” •+ ,
- .
?
�"
” •BC
œ
Evaluate the following determinants:
18. º º( � &� $ %
œ
____________
19.
â ââ ââ ââ ââ ââ ââ â% " � "# $ '$ � # � "
œ ____________
Page 55 6� �
20.
â ââ ââ ââ ââ ââ â+, , � ++ + ,
! !
œ ____________
Solve the following systems of equations by Cramer's Rule:
21.
$B � #C œ %&B � C œ ""
22.
B � C � #D œ "#B � #C � D œ � (B � #D œ !
Page 56 College Algebra : Lab 2 : Sequences and Series
Lab Scheduled 5/01/18 Lab Report : Due Final Exam: 5/08/18
/50
Name _______________________ Name _______________________
Name _______________________ Name _______________________
Each problem is worth 5 points.
1. Expand and evaluate the following sums.
a) �'3œ#
Š ‹3 � $ œ#
_______________________________
b) �"#5œ"
ln Š ‹kk�1
œ _______________________________
c) � �%!! $**
4œ" 7œ
j m $ � œ3
3 _______________________________
2. Write in summation notation using a single symbol.�a) 1 � � � � � œ1 1 1 1 1 1
4. Given the following explicit generating functions for a sequence, compute the term requested.
a) Find the 100'th term +8 "!!œ � #Ð8 � "Ñ � $& œ a _________________
b) Find the 40'th term + œ Ð � Ð#8 � &Ñ œ8 %!"Ñ8 a _________________
5. Given the following recursion formula for a sequence, compute the term requested.
a) Find the 1500'th term + + œ8�" œ + � $ à œ � '!!8 " "&!! a _________________
b) Find the 6'th term + + + œ8�" '8œ à œ#" a
$# _________________
c) Find the 15'th term + + œ8�" "&œ � à œ %!*'+#8 a " _________________
6. The following sequences are either geometric or arithmetic. Identify which are which. For geometric sequences state the value of <for arithmetic sequences state the value of . .
