Page 1 College Algebra : Computing Lab 1 /5 Due 1/18/18

Transcription

1 Page 1 College Algebra : Computing Lab 1 /5 Name Due 1/18/18 Using your calculator evaluate the following for B œ % and C œ. 1. ( œ 2. Ð (Ñ œ 3. $ÐBCÑC &B$C B œ 4. C C B Š B " " œ Winplot is a free general-purpose plotting utility which can be downloaded from There is an online tutorial at 5. In Winplot use the command sequence View/view/set corners to make a 2-dim Window with & Ÿ B Ÿ & and! Ÿ C Ÿ!. From the View/Grid menu use the check radio button to indicate that you want a rectangular grid with scale values shown on the axes. Use a horizontal scale interval of 0.5 with a freq of 2. This labels every other horizontal "tic mark" on the B axis. Use a vertical interval of 2 with a freq of 1. In the Equa/1. Explicit menu enter the following two expressionsþ Only type what is displayed in the "box". C œ 0ÐBÑ œ "B "$B"% C œ 0ÐBÑ œ Ð$BÑÐ%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 File/Format with the width set to 16 cm, then use File/Print. How do the graphs of the two expressions compare? Explain why this is so.

2 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 N + Q Á g. 4. There are some terminating or repeating decimals which are also irrational numbers. 5. È*' is an irrational number. 6. For any real number B, ± B ± must always equal B. 7. For any real number B, ± B ± must always equal B. 8. For any real number B, ± B ± must always equal ± B ±. Simplify and/or evaluate the following : 9. ± * "" ± ± $ " ± œ 10. For B!, ± B ± B œ " & $ 11. Š ) ƒš œ 12. " & $ ) ƒš œ 13. Š$ & œ & 14. Š %* $ œ 15. Š ( & $ œ Simplify the following using only positive exponents. 16. ( ' D C D( C' œ

3 Page 3 2 $ $ $ $CB B 17. B& $C% œ " " 18. Š +,, + œ Simplify the following expressing all answers in simplest radical form with rationalized denominators. 19. É % &B $ œ B) 20. Ê & )C% œ È È & $ œ Express the following complex fraction as simple fraction with a rationalized numerator. 22. " " È B2 ÈB 2 œ Simplify the following giving the answer which is correct for any real value of x. 23. È $ B$ œ 24. È B œ Simplify the following using only positive rational exponents. 25. È % + %,% œ 26. È ' B* œ

4 Page 4 3 Factor the following polynomials as completely as possible over the integers : $ 27. "'B %!!B œ 28. B BC "&B$!C œ & 29. "&B B œ 30. "&C $C ) œ ( ' & 31. "B %)B %)B œ % $ 32. %B C 'BÐBCÑ %&B C œ Simplify the following rational expressions : 33. B B" ÐB "Ñ œ 34. $ B *B( B$ ( œ

5 Page 5 4 Perform the following operations and simplify the result: 35. B (B"& 'B (B"& $B (& %B )B$ œ 36. : :;; :; ; : ƒ : :;; œ 37. B B% B &B$ 'B& $B (B! $B& œ 38. Express the following complex fraction as a simple rational expression in reduced form. B" B" B" B" " " B B$ B œ 39. Find the quotient and remainder of the following division. % $ ( $B %!B %)B &B') ƒ ( B "%B') œ

6 Page 6 5 Solve the following equations for all real values which make them true. 40. ""B* œ % 41. $C $ÐC &Ñ œ $ÐC $Ñ Ð$C "Ñ! 42. 'Ð= Ñ %Ð= %Ñ œ $Ð%= $Ñ Ð= "!Ñ "! 43. )Ð$B(Ñ $Ð)B")Ñ œ "% $Ð% BÑ $B 44. ± B" ± œ ( 45. ± B' ± œ ± B ± Rearrange the following formulas for the specified variable. 46. C œ $B) C œ 47 Þ 'BC &B œ $ÐC BÑ C œ 48 Þ 'BC &B œ $ÐC BÑ B œ

7 Page 7 6 Solve and graph the solutions of the following inequalitiesþ 49. & B & "" 50. ÐB$Ñ ÐB"Ñ ' ÐBÑ 51. ± $B' ± * 52. ± %B' ± Ÿ "! 53. "! C Ÿ $C & & C

8 Page 8 7 Express the following complex numbers in standard rectangular form, +, È È & "' œ È "(& %* È & È% È* œ 56. Each year John contributes $ % 00 less than twiceas much money as Jim to charity. If together they contribute $ *&!, how much does each contribute? A square is deformed into a rectangle by increasing one of its sides by $Þ! m. The resulting rectangle has %Þ! m more area 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?

9 Page 9 College Algebra : Computing Lab 2 /5 Name Due 2/01/18 1. Sketch the function 0ÐBÑ œ B B( Þ Using your calculator or Winplot estimate to the nearest hundredth the two roots of 0ÐBÑ. Use the convention that root 1 root 2. root 1 œ root 2 œ Give the 'exact ' answers for the two roots of 0ÐBÑ. root 1 œ root 2 œ Below approximate the exact results as decimal numbers to at least six places. root 1 œ root 2 œ 2. From your graph and your knowledge of the roots of 0ÐBÑ, what is the solution set of 0ÐBÑ!? 3. What is the solution set of B ( B? 4. Solve and graph the solution set of the inequality B $B $B%.

10 Page 10 College Algebra : Computing Lab 3 /5 Name Due 2/06/18 1. Using a graphing calculator or Winplot enter the following functions : B C œ " ; C œ B C œ B" ; C œ $B Set Xmin œ Ymin œ &, Xmax œ Ymax œ &, then ZOOM to a square grid and graph the four functions. 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 C œ 7B, have an effect on whether this line is to any other line? 2. On your graphing calculator enter the following functions : B C œ " ; C œ B C œ B" ; C œ B Set Xmin œ Ymin œ &, Xmax œ Ymax œ &, then ZOOM to a square grid and graph the four functions. a) Which of the lines are ¼ (perpendicular)? 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 C œ 7B, have an effect on whether this line is ¼ to any other line? 3. Let _ be the line that passes through Ð $ß &Ñ and is parallel to the line $BC œ "). Let ` be the line that passes through Ð $ß &Ñ and is perpendicular to the line $BC œ "). Fill in the following : Line Slope C intercept B intercept _ ` Equation of _ in slope-intercept form: Equation of ` in slope-intercept form: Sketch the graph of the lines _ and `

11 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 œ "! ; Xmax œ "! and use a square zoom. In Winplot use View/view to set corners with left œ down œ "!, right œ up œ "!, then from View/Zoom select Square. From the Equa menu select User functions, then in the dialogue box enter f for "name" and x^2 for " name(x)= ", followed by pressing the "enter" button. Repeat this procedure to define 1ÐBÑ as 0ÐB%Ñ $ and then close the dialogue box. Enter the six curves one at a time by using the Explicit dialogue box from the Equa menu. Standard Notation TI-84 Curve 1. C œ 0ÐBÑ œ B Y 1 œ 2 X Curve 2. C œ 1ÐBÑ œ 0ÐB%Ñ $ Y 2 œ Y1ÐX 4Ñ 3 Curve 3. C œ 1Ð BÑ Y 3 œ Y2Ð X Ñ Curve 4. C œ 1ÐBÑ Y 4 œ Y2ÐX Ñ Curve 5. C œ 0ÐBÑ Y 5 œ Y1Ð2X Ñ Curve 6. C œ 0ÐBÑ Y 6 œ 2Y1ÐX Ñ 2. Repeat problem 1 for 0ÐBÑ œ lbl Graph of Problem 1 Graph of Problem 2

12 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. $ B "&B œ "&B)( $ 2. "B %B œ B 3. ± B$ ± œ B B" 4. Solve and graph the solutions of the inequality B %B Ÿ % B. 5. A rectangle is ( m longer than it is wide and has an area of "! m. What are the rectangle's dimensions? 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?

13 Page 13 2 Solve the following equations. 7. & 10 B " B œ B2 B 8. % B œ B' 9. B œ È&B% 10. ÈB ÈB& œ ",-B 11. Solve the formula, + œ,b., for B. 12. Solve and graph the solution of the inequality $ B B %.

14 Page 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, $Ñ. 15. Sketch the circle: B C 'B)C œ! 16. Find the slope, distance and midpoint between the points Ð ", &Ñ and Ð$, ( Ñ. slope œ distance œ mid point œ

15 Page Graph the line $B&C œ "! and fill in the following : slope œ B intercept œ C intercept œ 18. Graph the line *C œ ( and fill in the following: slope œ B intercept œ C intercept œ

16 Page Graph the line $B œ " and fill in the following : slope œ B intercept œ C intercept œ 20. Indicate which of the following BC relations define C as a function of B or B as a function of C. a) C œ B ( b) B œ C ( c) B C œ "' d) B C œ "' 21. The function 0ÐBÑ is defined as follows : The dependent variable (output) is three times the cube of the independent variable (argument or input).use order of operations! a) 0ÐBÑ œ b) 0ÐBÑ œ c) 0ÐB"Ñ œ d) 0ÐCÑ œ e) 0ÐB Ñ œ

17 Page 17 6 Evaluate the following : 22. 0ÐBÑ œ $B &B a) 0Ð!Ñ œ b) 0Ð"Ñ œ c) 0Ð0Ð"ÑÑ œ d) 0Ð+Ñ œ e) 0ÐBÑ œ & f) 0ÐB Ñ œ 23. 0ÐBÑ œ B B$ a) 0ÐB2Ñ œ b) 0ÐB2Ñ 0ÐBÑ œ c) 0ÐB2Ñ0ÐBÑ 2 œ d) What does the answer to part c get closer and closer to as 2 gets really teeny-tiny? 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 B axis Symmetric wrt C axis Symmetric wrt Origin B (BC œ 0 $ B "* C œ 0 ÐB 1Ñ $ C œ 0 ' B " C œ 0

18 Page From the graph of the function, 0ÐBÑ, shown below answer the following questions : a) What symmetry if any does 0ÐBÑ display? b) For what intervals of the independent variable B is 0ÐBÑ increasing? c) For what intervals of the independent variable B is 0ÐBÑ decreasing? d) For what values of B does 0ÐBÑ have a relative or local minimum? Sketch the following e) f) C œ 0ÐBÑ "! C œ 0ÐBÑ

19 Page 19 8 In the next two problems indicate for the given function i) the domain ii) the range then sketch the function ÐBÑ œ lbl $ 27. 0ÐBÑ œ $ ÐB"Ñ

20 Page From the graph of the function, 0ÐBÑ, shown below answer the following questions : a) For what intervals of the independent variable B is 0ÐBÑ increasing? b) For what intervals of the independent variable B is 0ÐBÑ decreasing? c) Give an explicit (piecewise) definition of 0ÐBÑ. Sketch the following d) e) C œ 0ÐBÑ C œ 0Ð BÑ

21 Page From the graphs of the functions, 0ÐBÑ & 1ÐBÑ, shown below answer the following questions: a) What symmetry if any does 0ÐBÑ display? b) For what intervals of the independent variable B is 0ÐBÑ increasing? c) For what intervals of the independent variable B is 0ÐBÑ decreasing? d) For what values of B does 0ÐBÑ have a relative or local minimum? 30. From the graphs above estimate the following: a) 0Ð!Ñ 1Ð!Ñ œ b) Ð1 0ÑÐ"Þ&Ñ œ c) Ð0 1ÑÐ!Ñ œ d) Ð0 1ÑÐ"Ñ œ

22 Page 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. T is time required to fill each vessel to the maximum height of liquid the vessel allows. State which graph goes with which vessel and explain your choices.

23 Page point Bonus Question: Absolute Madness l l Consider the function 0ÐBÑ œ, lbl ' where, is a real number. The root or zeros of a real function, 0ÐBÑ are the real numbers, <, for which 0Ð<Ñ œ!. a) For what values of, does this function have no real roots? l l,, 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?

24 Page 24 College Algebra : Computing Lab 5 /5 Name Due 2/26/18 1.Sketch the function 0ÐBÑ œ B $ B &B. Using your calculator or Winplot estimate to the nearest hundredth the three roots of 0ÐBÑ and the coordinates of the two turning points of 0ÐBÑ. Use the convention that root 1 root 2 root 3 and turning pt. 1 turning pt. 2. root 1 œ turning pt. 1 œ Ð, ) root 2 œ turning pt. 2 œ Ð, ) root 3 œ 2. Use the Rational Root Theorem and synthetic division to determine the exact values of the roots of 0ÐBÑ. root 1 œ root 2 œ root 3 œ Now give the decimal approximations of these exact values accurate to at least the sixth decimal place. root 1 œ root 2 œ root 3 œ How do these numbers compare to your estimates in problem 1?

25 Page 25 College Algebra : Computing Lab 6 /5 Name Due 3/05/18 Consider the two rational functions given below : 0ÐBÑ œ B B B " 1ÐBÑ œ 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?

26 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 B and C intercepts. 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!

27 Page Write an equation which generates each of the following parabolas. a) A parabola pointing to the left with vertex at Ð&, $Ñ which passes through the point Ð (, "Ñ Þ b) A parabola pointing down with vertex at Ð, "Ñ and a C intercept of "". È c) A parabola pointing to the right with the line C œ as the axis of symmetry and intercepts at Ð $ß!Ñ, Ð!, Ñ, È and Ð!, Ñ. 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, 0ÐBÑ œ $B % %B $ 'B &B$, perform the requested evaluations using synthetic division. a) 0Ð"Ñ œ b) 0ÐÑ œ c) 0Ð!Þ&Ñ œ

28 Page Given D œ +,3 and A œ -.3 with +,,, -, and. all real, prove by explicit calculation that qq DA œ qd q A. _ Here u means the complex conjugate of?. 7. Given that 0Ð Ñ œ 0Ð$Ñ œ 0Ð$ 3Ñ œ! and that 0ÐBÑ is a polynomial of degree 4 with real coefficients and has 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.

29 Page 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.

30 Page A open-topped box is formed from a rectangular "! cm 5 cm piece of sheet metal by cutting out identical squares (each of side length, B ) from each of the four corners and then folding up the remaining four rectangular sidepieces. The result is a box of depth B. a) Determine a formula for the volume of the box as a function of B. b) What value(s) of B give the most volume for the box? 3 c) What is the greatest volume in cm that the box can have? 11Þ Given TÐBÑ œ B % $B $ 'B "B) a) What is the maximum number of positive roots of TÐBÑ? b) What is the maximum number of negative roots of TÐBÑ? c) What is the set of possible rational roots of TÐBÑ? d) Find and indicate all roots (both real and complex) of TÐBÑ. e) Sketch TÐBÑ below.

31 Page Þ Given TÐBÑ œ B & &B % ""B $ ""B ()B a) What is the maximum number of positive roots of TÐBÑ? b) What is the maximum number of negative roots of TÐBÑ? c) What is the set of possible rational roots of TÐBÑ? d) Find and indicate all roots (both real and complex) of TÐBÑ. e) Sketch TÐBÑ below.

32 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 ÐBÑ œ B $ " B " 14. 0ÐBÑ œ B $B& B

33 Page ÐBÑ œ B 1 B * 16. 0ÐBÑ œ B" B %

34 Page ÐBÑ 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 $%Þ(.F (micro farads) when the plate radius is "Þ! cm and the plate separation is %Þ!. m (microns, ". m œ!þ!!!" cm), what is the plate separation, if for the same amount of charge and a plate radius of Þ! cm the capacitance is $%(Þ!. F? 21. Bonus Problem (5 points) 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 2Þ a) 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.

35 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 œ down œ!þ& ; right œ up œ 4 and then switch to a Zoom/Square from the View menu The function sqrt stands for the square root function. Check " lock interval" to fix the domains as stated below. The designation x=f(t) is "2. Parametric" mode, while y=f(x) is "1. Explicit" mode as found under the Equa menu. Equa format Domain formulas to enter Curve 1. y=f(x) low x œ! high x œ % y œ x^2 Curve 2. y=f(x) low x œ! high x œ % y œ x Curve 3. x=f(t) low t œ! high t œ % x œ t^2 ; y œ t Curve 4. y=f(x) low x œ! high x œ % y œ sqr(x) Curve 5. x=f(t) low t œ! high t œ % x œ sqr(t) ; y œ t a) What is true about the line C œ B (Curve 2), with respect to Curve 1 and Curve 3? 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. 0ÐBÑ œ lb$l 3. 0ÐBÑ œ " B $ "

36 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 œ down œ & ; right œ up œ &. To generate the "correct" shape of the curves select Zoom/Square from the View menu. Equa format Domain formulas to enter Curve 1. y=f(x) low x œ & high x œ & y œ e^x Curve 2. y=f(x) low x œ & high x œ & y œ x Curve 3. x=f(t) low t œ & high t œ & x œ e^t ; y œ t Curve 4. y=f(x) low x œ & high x œ & y œ ln(x) Curve 5. x=f(t) low t œ & high t œ & x œ ln(t) ; y œ t a) What is true about the line C œ B (Curve 2), with respect to the Curve 1 and Curve 3? 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 C œ 0ÐBÑ œ B, i) Give the domain, ii) Give the range, iii) Sketch the curve

37 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 0ÐBÑ œ B& and 1ÐBÑ œ " B" a) What is the domain of 0? bñ What is the range of 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 0ÐBÑ and 1ÐBÑ defined in problem 1, find the inverse functions a) 0 " ÐBÑ œ b) 1 " ÐBÑ œ " c) 0 Ð0ÐBÑ Ñ œ Ð Ñ " d) 1 1 ÐBÑ œ 4. For the functions 0ÐBÑ and 1ÐBÑ defined in problem 1 and the inverse functions of problem 3. 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) Ð0 0 ÑÐBÑ œ " f) Ð1 1ÑÐBÑ œ

38 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) B C œ 2

39 Page Change the following from logarithmic to exponential form : a) log % Ð'%Ñ œ $ b) log & Ð!Þ!%Ñ œ c) log B ÐVÑ 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 B CÐC Ñ œ e) / ln( $U ) œ 10. Express the following as a single logarithm with a coefficient of one. a) lnð&ñ ln Ð%Ñ œ ln Ð Ñ Ð Ñ Ð Ñ œ * "! "! "! b) log B $ log B log Ð Ñ c) " " $ lnð)ñ ' lnð'%ñ œ ln Ð Ñ 11. Compute the following logarithms to 4 places. a) log Ð"!!Ñ b) ln Ð"!!Ñ c) log Ð"!!Ñ œ œ œ

40 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, C œ & lnð$ñ, or you may give the numerical answer C œ %Þ&&""*'"$$ÞÞÞ ) œ &' + œ 13. log CÐ&'Ñ œ ) C œ 14. "! B œ "!ß!!! B œ 15. "! D œ (!ß!!! D œ 16. log 3ÐCÑ œ $ C œ 17. ln ÐBÑ œ B œ 18. & B œ %Þ* B œ 19. & B œ %Þ* B œ 20. && œ '!Ð" / B Ñ B œ $ 21. ln Š (B œ $ B œ 22. C œ log %Ð"!!Ñ C œ 23. < œ log 1Ð(*Ñ log Ð$Ñ 1 < œ

41 Page logð'%b Ñ œ log Ð"!!Ñ B œ 25. Solve and graph the solution set of the inequality log ÐB Ñ Ÿ ". Hint: Consider where C œ log ÐB Ñ crosses C œ ". 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 E œ 0Ð>Ñ œ E ÐÑ >! X, where A is the amount of material left after a time > has elapsed, A! is the starting amount of material, and X is the half-life of the material. If in$&!! years time, '!% of 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!!!?

42 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 isresponsible for data observation and or recording, who will do the algebra, who will check the work, who will write up thewhat parts of the report. It is possible that in one group a single individual writes the entire report, while in another group everyonewrites 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 one report for a given lab which should include the following : 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.

43 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 P of the string. Holding the string at this distance, let the lead weight swing freely from an initial position that makes a! angle Ð) œ! Ñ with the vertical. Measure the time for 10 full swings of the weight. Divide this time by 10 to obtain the period, X, the time for one full swing. Repeat this procedure for each of the specified values of P. Then repeat the experiment for ) 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 Time for 10 Swings Period X Time for 10 Swings Period X "!Þ! cm "&Þ! cm!þ! cm &Þ! cm $!Þ! cm $&Þ! cm %!Þ! cm %&Þ! cm &!Þ! cm &&Þ! cm '!Þ! cm '&Þ! cm (!Þ! cm (&Þ! cm )!Þ! cm

44 Page 44 2 Data Table for Second Weight ( If Balance available mass of weight = ) ) œ! ) œ & P Time for 10 Swings Period X Time for 10 Swings Period X "!Þ! cm "&Þ! cm!þ! cm &Þ! cm $!Þ! cm $&Þ! cm %!Þ! cm %&Þ! cm &!Þ! cm &&Þ! cm '!Þ! cm '&Þ! cm (!Þ! cm (&Þ! cm )!Þ! cm Give a brief but accurate description of the procedure you followed in obtaining your data. Use diagrams where necessary and identify all pertinent variables. II. Data Analysis (15 points) ( Feel free to use a spreadsheet 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 X versus PÞLabel all axes and label each curve as to the weight and angle Ð) Ñ used. You may if you wish put 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?

45 Page 45 3 Graphs of X versus P Construct a graph of lnðxñ versus lnðpñþlabel all axes and label each curve as to the weight and angle Ð) Ñ used. You may if you 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. LnÐPÑ and LnÐX Ñ Data LnÐPÑand LnÐXÑData for First Weight LnÐXÑData for Second Weight Ln ÐXÑ LnÐXÑ P lnðpñ ) œ! ) œ & ) œ! ) œ & "!Þ! cm "&Þ! cm!þ! cm &Þ! cm $!Þ! cm $&Þ! cm %!Þ! cm %&Þ! cm &!Þ! cm &&Þ! cm '!Þ! cm '&Þ! cm (!Þ! cm (&Þ! cm )!Þ! cm

46 Page 46 4 Graphs of ln ÐX Ñversus lnðpñ III. Interpretation (15 points) Is the relationship between X and P linear? Explain your answer. Is the relationship between lnðx Ñ and lnðpñ linear? Explain your answer. From elementary physics the period of a simple pendulum for small initial angles satisfies a 'power law' relationship to the length. : That is X œ E P Þ Where E is a constant independent of P. From your data obtain an estimate of :Þ Explain how you obtained this estimate. Hint: The"best" or regression fit of a straight line through a set of 8 data points ÐB3ßC3 Ñ is given by the linear model C œ 7B,, where 7 is the slope and, is the C intercept. The slope of the regression line is given by 8 B 3C3 B3 C 3 B 3 C3 B3 B3C3 7 œ 8B2, œ 3 Ð B3Ñ2 while the C intercept can be calculated as 8B2 3 Ð B3Ñ2. Here we are using summation notation where B 3C3 œ B " C" B C B $ C$ ÞÞÞ B 8C8 ß B B B B B C C C C œ ÞÞÞ ß œ ÞÞÞ C,and B œ B B B ÞÞÞ B Þ 3 " $ 8 3 " $ 8 3 " $ 8 Estimated value of : œ For the same weight you used estimate the period if actually measuring the period. P œ "!! cm. If you have the equipment you may wish to check this estimate by Estimated period for a length of "!! cm œ

47 Page 47 5 IV. Application (5 points) Below is a table of data on the electrical resistance, V, for a 1 meter length of different gauge copper wire. The wire's diameter is.. Copper Wire Resistance Data Gauge.ÐcmÑ VÐmHÑ 30!Þ!$"(& $Þ( 25!Þ!&&&' ($Þ! 20!Þ!*&& %Þ* 15!Þ"()' (Þ" 12!Þ(() Þ* 10!Þ$&( "Þ) 9!Þ$*'* "Þ% 8!Þ%$'' "Þ 7!Þ%('$ "Þ! 6!Þ&"&*!Þ) 5!Þ&&&'!Þ( 4!Þ&*&$!Þ' According to theory the resistance is related to diameter by a 'power law'. That is V œ E. where E is a constant independent of.þ From the table obtain an estimate of :Þ Explain how you obtained this estimate. :, 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

48 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' lnðbñ œ B$. One method of solving it consists of finding the B coordinates of the points of intersection of the two curves: C œ 1ÐBÑ œ lnðbñ C œ 0ÐBÑ œ B$ Using your calculator, graph both curves. Sketch the result. How many real solutions does lnðbñ œ B$ have? Then using either a calculator or Winplot determine approximations to these solutions accurate to the nearest hundredth.

49 Page 49 College Algebra : Computing Lab 10 /5 Name Due 4/16/18 1. In Winplot open the 3-dim window and solve each of the following three equations for D. Enter each formula for D as an Explicit function in Equa/Explicit menu. From View pick Axes to get a better 3D perspective of the surfaces. Be sure to use a different color for each surface. This is chosen under color of the Equa/Explicit menu. Solve this system of equations and enter the coordinates of the solution using the Equ/Point/Cartesianmenu. Be sure the color of this point is distinct from the color of the three surfaces. 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 D. Enter each formula for D as an Explicit function in Equa/Explicit menu. From View pick Axes to get a better 3D perspective of the surfaces. Be sure to use a different color 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 D œ > where > is any real number. Solve for B and C in terms of >. In Winplot use the Equa/Curve menu to enter the form of the infinetly many solutions. Set t lo = 5 and t hi = 5. Set the pen width to 2 and choose a dominant 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?

50 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!". BC $ C B œ "' œ 1

51 Page 51 2 In the following problems solve and classify (consistent and independent, inconsistent, or dependent) the following systems of equations Þ If 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 (

52 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 ""

53 Page 53 4 Perform the following matrix operations: 10. Ô % & " Ô %! " $ % $ $ & " Ö Ù " $ Ö % & $ " Ù Õ % $ & ( Ø Õ " $ Ø œ & % 11. $ % œ 12. Ô $ " $ Ô $ $ & " œ Õ " ØÕ " Ø % $ $ 13. $ $ % œ., ,- +.,- +, - + œ -. +.,- +.,- 15. Ô $ " Ô " " Ö $ & ÙÖ! $ Ù œ Õ! " ØÕ & " Ø

54 Page Ô " " Ô $ " Ö! $ ÙÖ $ & Ù œ Õ & " ØÕ! " Ø +B,C œ / +, B / 17. The system of equations can be written as the matrix multiplication -B.C œ 0 -. œ. C 0 +, Use the result of problem 14 to write the solution of this system. Hint: What is -. "? B C œ Evaluate the following determinants: ( & 18. º $ % º œ 19. % " " $ ' â$ " â œ

55 Page !!,, + â+ +, 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!

56 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 Each problem is worth 5 points. 1. Expand and evaluate the following sums. Name a) ' 3œ Š 3 $ œ b) " 5œ" k ln Š k1 œ %!! $** c) j $ 3 m œ 4œ" 7œ3 2. Write in summation notation using a single symbol a) œ b) ÞÞÞ œ 5 jœ0 c) Ð Ñ j! + 6! + 7! + 8! + 9! + 10! œ 8 d) Ð Ñ 4œ" " " " 4$ 8& 8( œ

57 Page For the following sequences, fill in the next (missing term) and either give a formula or explain how the 8'th term isevaluated. a) &ß (ß *ß ""ß ß ÞÞÞ + œ 8 b) $ * ( % ß "' ß '% ß ß ÞÞÞ + œ 8 c) ß 9 ß 27 ß ß ÞÞÞ + œ 8 d) " $ ( "& $" '$ ß % ß ) ß "' ß $ ß '% ß ß ÞÞÞ + œ 8 4. Given the following explicit generating functions for a sequence, compute the term requested. + 8 "!! a) œ Ð8 "Ñ $& Find the 100'th term a œ b) + œ Ð "Ñ 8 Ð8 &Ñ Find the 40'th term a œ 8 %! 5. Given the following recursion formula for a sequence, compute the term requested. a) + 8" œ + 8 $ à + " œ '!! Find the 1500'th term a "&!! œ $ b) + 8" œ + 8 à + " œ Find the 6'th term a ' œ + c) + 8" œ 8 à + " œ %!*' Find the 15'th term a "& œ 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.. $ * ( a) ", %, "', '%, ÞÞÞ b) + œ % $Ð8 "Ñà8 œ "ß ß $ß ÞÞÞ 8

58 Page 58 3 * c) + 8 œ *& Š "" à 8 8 œ "ß ß $ß ÞÞÞÞ d) "%ß *ß %ß "ß 'ß ÞÞÞ Compute the following sums. 7. a) & ( * "" "$ "& "( "* ÞÞÞ $*%& œ b) The sum of all the even integers between )$%& and "!!)& œ c) " % ) "' $ '% ") &' &" "!% ÞÞÞ '&&$' œ d)!" 5œ) Š $5 % œ " " " " " " e) "' ) % " % ) "' $ '% ÞÞÞ œ 8. R 4œ" 4 a) for B Á "à B œ R 4œ" 4 b) for B œ "à B œ 4œ" 4 c) for ± B ± "à B œ d) Š " " & 8 "! 8 œ 8œ! e) 6œ! / 16 œ qqq f) Express!Þ&$"&$"&$"&$"&$"ÞÞÞ œ!.&$" as a ratio of two integers œ

59 Page Use the binomial theorem to expand out the following: % a) Ð$BCÑ œ b) Ð+,Ñ ( œ c) % % ÐB2Ñ B 2 œ 10. Construct a proof of the following using mathematical induction : a) 8 4œ" 4 $ œ 8 Ð8" Ñ %

60 Page 60 5 b) 8 4œ! " 8" Ð4"ÑÐ4$Ñ œ 8$ c) The complex conjugate of a product of 8 terms equals the product of the 8 complex conjugates.