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21.2E: Exercises


Practice Makes Perfect

Determine Whether an Ordered Pair is a Solution of a System of Equations

In the following exercises, determine if the following points are solutions to the given system of equations.

1. (left{ egin{array} {l} 2x−6y=0 3x−4y=5 end{array} ight.)

ⓐ ((3,1))
ⓑ ((−3,4))

Answer

ⓐ yes ⓑ no

2. (left{ egin{array} {l} −3x+y=8 −x+2y=−9 end{array} ight.)

ⓐ ((−5,−7))
ⓑ ((−5,7))

3. (left{ egin{array} {l} x+y=2 y=frac{3}{4}x end{array} ight.)

ⓐ ((87,67))
ⓑ ((1,34))

Answer

ⓐ yes ⓑ no

4. (left{ egin{array} {l} 2x+3y=6 y=frac{2}{3}x+2 end{array} ight.)

ⓐ ((−6,2))
ⓑ ((−3,4))

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

5. (left{ egin{array} {l} 3x+y=−3 2x+3y=5 end{array} ight.)

Answer

((−3,2))

6. (left{ egin{array} {l} −x+y=2 2x+y=−4 end{array} ight.)

7. (left{ egin{array} {l} y=x+2 y=−2x+2 end{array} ight.)

Answer

((0,2))

8. (left{ egin{array} {l} y=x−2 y=−3x+2 end{array} ight.)

9. (left{ egin{array} {l} y=frac{3}{2}x+1 y=−frac{1}{2}x+5 end{array} ight.)

Answer

((2,4))

10. (left{ egin{array} {l} y=frac{2}{3}x−2 y=−frac{1}{3}x−5 end{array} ight.)

11. (left{ egin{array} {l} x+y=−4 −x+2y=−2 end{array} ight.)

Answer

((−2,2))

12. (left{ egin{array} {l} −x+3y=3 x+3y=3 end{array} ight.)

13. (left{ egin{array} {l} −2x+3y=3 x+3y=12 end{array} ight.)

Answer

((3,3))

14. (left{ egin{array} {l} 2x−y=4 2x+3y=12 end{array} ight.)

15. (left{ egin{array} {l} x+3y=−6 y=−frac{4}{3}x+4 end{array} ight.)

Answer

((6,−4))

16. (left{ egin{array} {l} −x+2y=−6 y=−frac{1}{2}x−1 end{array} ight.)

17. (left{ egin{array} {l} −2x+4y=4 y=12x end{array} ight.)

Answer

no solution

18. (left{ egin{array} {l} 3x+5y=10 y=−frac{3}{5}x+1 end{array} ight.)

19. (left{ egin{array} {l} 4x−3y=8 8x−6y=14 end{array} ight.)

Answer

no solution

20. (left{ egin{array} {l} x+3y=4 −2x−6y=3 end{array} ight.)

21. (left{ egin{array} {l} x=−3y+4 2x+6y=8 end{array} ight.)

Answer

infinite solutions with solution set: (ig{ (x,y) | 2x+6y=8 ig})

22. (left{ egin{array} {l} 4x=3y+7 8x−6y=14 end{array} ight.)

23. (left{ egin{array} {l} 2x+y=6 −8x−4y=−24 end{array} ight.)

Answer

infinite solutions with solution set: (ig{ (x,y) | 2x+y=6 ig})

24. (left{ egin{array} {l} 5x+2y=7 −10x−4y=−14 end{array} ight.)

Without graphing, determine the number of solutions and then classify the system of equations.

25. (left{ egin{array} {l} y=frac{2}{3}x+1 −2x+3y=5 end{array} ight.)

Answer

1 point, consistent and independent

26. (left{ egin{array} {l} y=frac{3}{2}x+1 2x−3y=7 end{array} ight.)

27. (left{ egin{array} {l} 5x+3y=4 2x−3y=5 end{array} ight.)

Answer

1 point, consistent and independent

28. (left{ egin{array} {l} y=−12x+5 x+2y=10 end{array} ight.)

29. (left{ egin{array} {l} 5x−2y=10 y=52x−5 end{array} ight.)

Answer

infinite solutions, consistent, dependent

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

30. (left{ egin{array} {l} 2x+y=−4 3x−2y=−6end{array} ight.)

31. (left{ egin{array} {l} 2x+y=−2 3x−y=7 end{array} ight.)

Answer

((1,−4))

32. (left{ egin{array} {l} x−2y=−5 2x−3y=−4 end{array} ight.)

33. (left{ egin{array} {l} x−3y=−9 2x+5y=4 end{array} ight.)

Answer

((−3,2))

34. (left{ egin{array} {l} 5x−2y=−6 y=3x+3 end{array} ight.)

35. (left{ egin{array} {l} −2x+2y=6 y=−3x+1 end{array} ight.)

Answer

((−1/2,5/2))

36. (left{ egin{array} {l} 2x+5y=1 y=frac{1}{3}x−2 end{array} ight.)

37. (left{ egin{array} {l} 3x+4y=1 y=−frac{2}{5}x+2 end{array} ight.)

Answer

((−5,4))

38. (left{ egin{array} {l} 2x+y=5 x−2y=−15 end{array} ight.)

39. (left{ egin{array} {l} 4x+y=10 x−2y=−20 end{array} ight.)

Answer

((0,10))

40. (left{ egin{array} {l} y=−2x−1 y=−frac{1}{3}x+4 end{array} ight.)

41. (left{ egin{array} {l} y=x−6 y=−frac{3}{2}x+4 end{array} ight.)

Answer

((4,−2))

42. (left{ egin{array} {l} x=2y 4x−8y=0 end{array} ight.)

43. (left{ egin{array} {l} 2x−16y=8 −x−8y=−4 end{array} ight.)

Answer

((4,0))

44. (left{ egin{array} {l} y=frac{7}{8}x+4 −7x+8y=6 end{array} ight.)

45. (left{ egin{array} {l} y=−frac{2}{3}x+5 2x+3y=11 end{array} ight.)

Answer

no solution

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination.

46. (left{ egin{array} {l} 5x+2y=2 −3x−y=0 end{array} ight.)

47. (left{ egin{array} {l} 6x−5y=−1 2x+y=13 end{array} ight.)

Answer

((4,5))

48. (left{ egin{array} {l} 2x−5y=7 3x−y=17 end{array} ight.)

49. (left{ egin{array} {l} 5x−3y=−1 2x−y=2 end{array} ight.)

Answer

((7,12))

50. (left{ egin{array} {l} 3x−5y=−9 5x+2y=16 end{array} ight.)

51. (left{ egin{array} {l} 4x−3y=3 2x+5y=−31 end{array} ight.)

Answer

((−3,−5))

52. (left{ egin{array} {l} 3x+8y=−3 2x+5y=−3 end{array} ight.)

53. (left{ egin{array} {l} 11x+9y=−5 7x+5y=−1 end{array} ight.)

Answer

((2,−3))

54. (left{ egin{array} {l} 3x+8y=67 5x+3y=60 end{array} ight.)

55. (left{ egin{array} {l} 2x+9y=−4 3x+13y=−7 end{array} ight.)

Answer

((−11,2))

56. (left{ egin{array} {l} frac{1}{3}x−y=−3 x+frac{5}{2}y=2 end{array} ight.)

57. (left{ egin{array} {l} x+frac{1}{2}y=frac{3}{2} frac{1}{5}x−frac{1}{5}y=3 end{array} ight.)

Answer

((6/−9,24/7))

58. (left{ egin{array} {l} x+frac{1}{3}y=−1 frac{1}{3}x+frac{1}{2}y=1 end{array} ight.)

59. (left{ egin{array} {l} frac{1}{3}x−y=−3 frac{2}{3}x+frac{5}{2}y=3 end{array} ight.)

Answer

((−3,2))

60. (left{ egin{array} {l} 2x+y=3 6x+3y=9 end{array} ight.)

61. (left{ egin{array} {l} x−4y=−1 −3x+12y=3 end{array} ight.)

Answer

infinitely many solutions with solution set: (ig{ (x,y) | x−4y=−1 ig})

62. (left{ egin{array} {l} −3x−y=8 6x+2y=−16 end{array} ight.)

63. (left{ egin{array} {l} 4x+3y=2 20x+15y=10 end{array} ight.)

Answer

infinitely many solutions with solution set: (ig{ (x,y) | 4x+3y=2 ig})

Choose the Most Convenient Method to Solve a System of Linear Equations

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

64.
ⓐ (left{ egin{array} {l} 8x−15y=−32 6x+3y=−5 end{array} ight.)

ⓑ (left{ egin{array} {l} x=4y−3 4x−2y=−6 end{array} ight.)

65.
ⓐ (left{ egin{array} {l} y=7x−5 3x−2y=16 end{array} ight.)

ⓑ (left{ egin{array} {l} 12x−5y=−42 3x+7y=−15 end{array} ight.)

Answer

ⓐ substitution ⓑ elimination

66.
ⓐ (left{ egin{array} {l} y=4x+95 x−2y=−21 end{array} ight.)

ⓑ (left{ egin{array} {l} 9x−4y=24 3x+5y=−14 end{array} ight.)

67.
ⓐ (left{ egin{array} {l} 14x−15y=−30 7x+2y=10 end{array} ight.)

ⓑ (left{ egin{array} {l} x=9y−11 2x−7y=−27 end{array} ight.)

Answer

ⓐ elimination ⓑ substitution

Writing Exercises

68. In a system of linear equations, the two equations have the same intercepts. Describe the possible solutions to the system.

69. Solve the system of equations by substitution and explain all your steps in words: (left{ egin{array} {l} 3x+y=1 2x=y−8 end{array} ight. )

Answer

Answers will vary.

70. Solve the system of equations by elimination and explain all your steps in words: (left{ egin{array} {l} 5x+4y=10 2x=3y+27 end{array} ight. )

71. Solve the system of equations (left{ egin{array} {l} x+y=10 x−y=6 end{array} ight.)

ⓐ by graphing ⓑ by substitution
ⓒ Which method do you prefer? Why?

Answer

Answers will vary.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


  • Weber imagined that an increasing rationalization of society would lead to man being trapped in a iron cage of rationality and bureaucracy.
  • Marx believed that capitalism resulted in the alienation of workers from their own labor and from one another, preventing them from achieving self-realization ( species being ).
  • Finally, Durkheim believed that industrialization would lead to decreasing social solidarity.
  • Bureaucracy is a type of organizational or institutional management that is based upon legal-rational authority. Weber believed that industrialization was leading to a growing influence of rational ideas and thought in culture, which, in turn, led to the bureaucratization of society.
  • Karl Marx understood species being to be the original or intrinsic essence of the species. A simplified understanding of species being is that it is a form of self-realization or self-actualization resulting from fulfilling or meaningful work.
  • Durkheim imagined that industrialization would lead to a decrease in social solidarity, which can be defined as a sense of community. He referred to this decrease in social solidarity as anomie, a French word for chaos.
  • Durkheim imagined that industrialization would lead to a decrease in social solidarity, which can be defined as a sense of community.
  • Durkheim referred to the decrease in social solidarity resulting from industrialization as anomie, a French word for chaos.
  • Industrializing societies would be characterized by specialization in that individuals would occupy different roles and occupations in a given society. According to Durkheim, specialization would lead to interdependence between the various components of society. He referred to this interdependence as organic solidarity.
  • Societies exhibit mechanical solidarity when the source of its cohesion is the homogeneity of its individuals in terms of their work, educational and religious training and lifestyles.
  • species being: Karl Marx understood species being to be the original or intrinsic essence of the species, which is characterized by pluralism and dynamism: all beings possess the tendency and desire to engage in multiple activities to promote their mutual survival, comfort and sense of inter-action. A simplified understanding of species being is that it is a form of self-realization or self-actualization resulting from fulfilling or meaningful work.
  • anomie: Alienation or social instability caused by erosion of standards and values.
  • alienation: Emotional isolation or dissociation.

As Western societies transitioned from pre-industrial economies based primarily on agriculture to industrialized societies in the 19 th century, some people worried about the impacts such changes would have on society and individuals. Three early sociologists, Max Weber, Karl Marx, and Emile Durkheim, envisioned different outcomes of the Industrial Revolution on both the individual and society and described these effects in their work.


2 Solutions to Exercises

If you have not already attempted the Exercises, you are encouraged to do so before reviewing the answers below.

There is often more than one approach to the exercises. Do not be concerned if your approach is different than the solution provided.

The following libraries were loaded to run the example solutions.

  1. Use the following code to load the warpbreaks data set and examine the variables in the data set.

  1. Use the Poisson family and fit breaks with wool, tension, and their interaction.
  1. Check to see if this is an appropriate model. If not, choose a more appropriate model form.

  1. Use the backward selection method to reduce your model, if possible. Use your model from the prior problem as the starting model.

No terms can be dropped from the model.

The negative binomial model is closer to means of the loess line than to the quasi-Poisson model. The range of values predicted by the model for the predicted values below 20 appears to be overstated by this model, though not as much as by the quasi-Possion model.


9.2E: Null and Alternative Hypotheses (Exercises)

  • Contributed by Barbara Illowsky & Susan Dean
  • Statistics at De Anza College
  • Sourced from OpenStax

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. What is the random variable? Describe in words.

The random variable is the mean Internet speed in Megabits per second.

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. State the null and alternative hypotheses.

The American family has an average of two children. What is the random variable? Describe in words.

The random variable is the mean number of children an American family has.

The mean entry level salary of an employee at a company is $58,000. You believe it is higher for IT professionals in the company. State the null and alternative hypotheses.

A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the proportion is actually less. What is the random variable? Describe in words.

The random variable is the proportion of people picked at random in Times Square visiting the city.

A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the claim is correct. State the null and alternative hypotheses.

In a population of fish, approximately 42% are female. A test is conducted to see if, in fact, the proportion is less. State the null and alternative hypotheses.

Suppose that a recent article stated that the mean time spent in jail by a first&ndashtime convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was 3 years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. If you were conducting a hypothesis test to determine if the mean length of jail time has increased, what would the null and alternative hypotheses be? The distribution of the population is normal.

A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. If you were conducting a hypothesis test to determine if the population mean time on death row could likely be 15 years, what would the null and alternative hypotheses be?

The National Institute 9.2.14 of Mental Health published an article stating that in any one-year period, approximately 9.5 percent of American adults suffer from depression or a depressive illness. Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. If you were conducting a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population, what would the null and alternative hypotheses be?


Think Java 2e

This is the home page for the second edition of Think Java, by Allen Downey and Chris Mayfield.

The code examples in this book are available for download from this repository on GitHub. Instructions for working with the code are in the preface of the book.

Description

Think Java is a hands-on introduction to computer science and programming used by many universities and high schools around the world. Its conciseness, emphasis on vocabulary, and informal tone make it particularly appealing for readers with little or no experience. The book starts with the most basic programming concepts and gradually works its way to advanced object-oriented techniques.

In this fully updated and expanded edition, authors Allen Downey and Chris Mayfield introduce programming as a means for solving interesting problems. Each chapter presents material for one week of a college course and includes exercises to help you practice what you’ve learned. Along the way, you’ll see nearly every topic required for the AP Computer Science A exam and Java SE Programmer I certification.

Discover one concept at a time: tackle complex topics in a series of small steps with multiple examples

Understand how to formulate problems, think creatively about solutions, and develop, test, and debug programs

Learn about input and output, decisions and loops, classes and methods, strings and arrays, recursion and polymorphism

Determine which program development methods work best for you, and practice the important skill of debugging

Think Java is a free textbook available under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Readers are free to copy and distribute the text they are also free to modify it, which allows them to adapt the book to different needs, and to help develop new material. The LaTeX source for this book is in the ThinkJava2 repository on GitHub.


Exercise of Options

Wheelsure announces that it has today received an exercise notice from Gerhard Dodl, Chief Executive of the Group, for the exercise of options over 60,000 new ordinary shares in the Group, at an exercise price of 1.625p per share (the "New Shares").

Application has been made for the admission of the New Shares to trading on the AQSE Growth Market ("Admission"). It is ‎expected that Admission will take place at 8.00 a.m. (London time) on or around 18 June 2021. The New Shares will, when issued, be credited as fully paid and will rank pari passu in all respects ‎with the Company’s existing issued ordinary shares.

Following Admission of the New Shares, the Company's issued share capital will consist of 4,013,428 ordinary shares. This number may be used by shareholders in the Company as the denominator for the calculations by which they will determine if they are required to notify their interest in, or a change in their interest in, the share capital of the Company under the FCA's Disclosure Guidance and Transparency Rules. The Company does not hold any shares in treasury.

Following the above exercise and Admission, Gerhard Dodl is interested in 102,150 ordinary shares, representing 2.55 per cent. of the Group’s issued share capital.

This announcement contains inside information for the purposes of the UK Market Abuse Regulation and the Directors of the Company are responsible for the release of this announcement.

For further information, please contact:

Wheelsure Holdings plc

Cairn Financial Advisers LLP

Jo Turner / Liam Murray / Mark Rogers

Notification of a Transaction pursuant to Article 19(1) of Regulation (EU) No. 596/2014

Details of the person discharging managerial responsibilities/person closely associated


The Adrenaline GTS 21 is the ever popular anti-pronation road running shoe from Brooks, designed for your everyday training. This excellent trainer can be used for any distance from 2 miles to marathon or even ultra distance.

Brooks use DNA Loft cushioning in their premium running shoes. Now it has been implemented to the full length of the Adrenaline's midsole to promote softer comfort and improve the absorption of impact from running on hard roads.

The renovated Engineered Air Mesh upper with improved moisture-management ventilates your feet to keep them cool. The 3D Fit Print details give this shoe a faster look and also offer you additional structure.

The revolutionary GuideRail support technology has been well received and appreciated by many runners. It diminishes excessive rolling down of your arch and keeps your feet and knees in a natural strait alignment. The flexibility aspect of the midsole creates a smooth heel-to-toe transition to promote a smooth ride.


Cambridge Extension 2 solutions available (1 Viewer)

As of last Friday Cambridge University Press have been releasing solutions to their new Extension 2 book.

But they are coming out in bits and pieces.

So far they have the enrichment questions for chapters 1-3.

They also say on their website “Solutions for selected foundation and development questions are in preparation, as are solutions for all enrichment questions in chapters 4-6.”

More specifically, Cambridge Extension 2 solutions currently available:

1A Q17, 18, 19, 20
1B Q17, 18, 19
1C Q15, 16, 17
1D Q23, 24, 25
1E Q22, 23, 24
1F Q19, 20, 21
1G Q19, 20, 21, 22, 23
2A Q16, 17 (incorrectly numbered 15, 16)
2B Q20, 21, 22, 23, 24
2C Q16, 17, 18, 19
2D Q19, 20, 21
2E Q22, 23, 24, 25

(2F Q17, 18, 19, 20, 21 missing)

3A Q18
3B Q15
3C Q12, 13, 14, 15
3D Q16, 17, 18, 19
3E Q16, 17

It also says in the book "No-one should try to do all the questions".

So which ones to do? I would suggest seeing which ones they publish - and do the rest (at least . + enough more to cover the syllabus if necessary).

I think their intention is to publish all the enrichment ones, but maybe not all the others.

EDIT 1: Solutions for enrichment questions for chapters 4-6 have since been released. Scroll down to subsequent posts to see which numbers are available.

EDIT 2: Development and Foundation questions are starting to be released too now.


MATH246 Exercise Environment (beta)

(mathcal = smathcal - y(0)) for (s>0) whenever (y(t)) is bounded.

Exercise 2

Exercise 3

Exercise 4

Exercise 5

For problems # 6 – 11 find the function which has the given Laplace Transform

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10

Exercise 11

Solve the given initial value problems using the Laplace Transform

Exercise 12

Exercise 13

Exercise 14

Use the Laplace transform to solve (y''-y = f(t)) where (f(t) = 1) for (t<1) and 0 everywhere else and (y(0)=2) and (y'(0) = 3) .

Exercise 15

[ ext < Let >f(v)= left<egin sin(v) & ext < if >0 leq v < 2pi v - 2pi & ext < if >v geq 2pi end ight.] and consider the initial-value problem [y'' + y' - 6y = f(v) , y(0) = 1, , y'(0) = 2.]

Use the Laplace Transform to find the Green function for the given differential operator

Exercise 16

Exercise 17

Exercise 18

Exercise 19

Exercise 20

Find a solution to the following initial value problem:

Exercise 21

>) Here we compare and contrast the methods of Laplace transforms with Green functions for obtaining the general solution to a second-order constant coefficient non-homogeneous differential equation, with prescribed initial conditions.

a) Using Green’s functions, show that the solution to the initial value problem [w'' + 2w' + 2w = f(v), w(0) = 0, w'(0) = 0,] is the following: [w(v) = int_<0>^ e^<-(v - s)>f(s) sin(v - s) ds.]

b) Show that if (f(v) = delta(v - pi)) , then the solution in part a) becomes (w(v) = u_pi (v) e^<-(v - pi)>sin(v - pi). )

c) Now use the method of Laplace transforms to solve the initial value problem prescribed in a) with (f(v) = delta (v - pi)) , and show that the solution obtained in this case agrees with the solution derived in b).

Exercise 22

Suppose that (g(u) = int_<0>^ f(s) ds) . If (G(s)) and (F(s)) are the Laplace transforms of (g(u)) and (f(u)) respectively, show that (G(s) = frac.)

Exercise 23

(f) Here we explore the fact that Laplace transform might not be useful in solving homogeneous equations with non-constant coefficients, especially when the coefficients at play are not linear functions of the independent variable. We explore this observation in the following two examples below.

By taking the Laplace transform of differential equations with prescribed initial conditions below, show that the differential equation for (Y(s) = LL(y(v))) is of first order in part a), and of second order in part b).

b) ((1 - v^2) y'' - 2vy' + alpha(alpha + 1)y = 0, y(0) = 0, y'(0) = 1. )

Exercise 24

Consider the following equation: [w(t) + int_<0>^h(t - s)w(s) ds = g(t),] where (h(t)) and (g(t)) are functions known a priori, and (w(t)) is our function to be determined. This type of equation belongs to the class of integral equations, because the unknown function (w(t)) also appears in integral form. Take the Laplace transform of the equation above, and obtain a closed form solution for (LL(w(t))) in terms of the Laplace transforms for (LL(g(t))) and (LL(h(t))) . The inverse transform of (LL(w(t))) would then yield the true solution, (w(t)) of the original integral equation.

Exercise 25

Consider the following integral equation: [v(t) + int_<0>^ (t - s)v(s) ds = sin(2s).]

a) Solve the integral equation using Laplace transform, using the method outlined in the previous problem.

b) Differentiate the differential equation (v(t) + int_<0>^ (t - s)v(s) ds = sin(2s)) twice and write down the differential equation that you obtain.

Also show that the initial conditions satisfy the following: [v(0) = 0, v'(0) = 2.]

c) Solve the initial value problem in b). How does it compare to the solution you derived in part a)?

Exercise 26

The gamma function (Gamma(p)) is defined by the following integral [Gamma(p + 1) = int_<0>^e^<-x>x^p dx.] Let us now study some of its properties.

a) Show that for (p > 0) , (Gamma(p + 1) = pGamma(p).) Subsequently, show that (Gamma(1) = 1) .

b) Show that (Gamma(n + 1) = n!) , where n is a positive integer.

c) Show that for (p > 0) , (p(p + 1)(p + 2) . (p + n - 1) = frac.) It is possible to show that (Gamma(frac<1><2>) = sqrt.) Knowing this fact, find (Gamma(frac<3><2>)) and (Gamma(frac<11><2>)) .

Exercise 27

Now that we’ve defined and explored some of the properties of the gamma function, let us consider the Laplace transform of (x^p) , for positive p.

b) Show that (LL(x^n) = dfrac<>>) , for (n) a positive integer and (s > 0) .


Petersgiles / 2Eto5E-conversion-summary.md

This is the third in my series of conversion documents for D&D 5th Edition. (You can find the one for Next here and the one for 3.5/3E/Pathfinder here.) I created this document by comparing the 5E versions of monsters from the Monster Manual with their 2E counterparts. I also referred to two sources for converting monsters from 2E to 3E: Wizards’ official Conversion Manual and the Dragon Magazine article “How to Create a Monster”.

I will once again include two warnings:

If you compare the 2E and 5E versions of monsters yourself, you will

notice this conversion does not produce perfectly identical results. I went for approximation, not precision.

I am fallible, so there may be mistakes. If you find any, let me

The end of this document also explains how to adjust these guidelines for converting monsters from 1E.

You will need access to at least the D&D Basic Rules and the Monster Manual to make full use of this. The Dungeon Master’s Guide, while not strictly required, is highly recommended for its monster creation guidelines (pages 273-283).

Also, thanks to Russ Morrissey for a simplification on dragon AC, and the people in this ENWorld thread for general help!

Challenge Rating

Before you get started, you should choose an initial Challenge Rating for the creature. This is an estimate of the monster’s CR, which you can use to guide the rest of your conversion. There are two ways to do this:

Use the CR of a similar 5E creature. If you use this approach, the

creature should also serve as your “reference monster” (see Ability Scores).

Use the CR Estimator in Appendix 1 to convert its AD&D 2E XP Value.

When you are finished converting your monster to 5E, you may wish to adjust this initial CR for a better fit. The best way to do this is to use the rules for determining CR in the DMG, but if you’re pressed for time, you can just compare it to similar 5E creatures once again.

5E appears to use 3.5’s size ranges (slightly trimmed). These do not perfectly match 2E’s size categories, so convert them as follows:

2E Size 5E Size

Tiny (2’ or less) Tiny (2 feet or less) Small (2’-4’) Small (2-4 feet) Medium (4’-7’) Medium (4-8 feet) Large (7’-8’) Medium (4-8 feet) Large (8’-12’) Large (8-16 feet) Huge (12’-16’) Large (8-16 feet) Huge (16’-25’) Huge (16-32 feet) Gargantuan (25’-32’) Huge (16-32 feet) Gargantuan (32’ or more) Gargantuan (32 feet or more)

You can also simply use the 2E size category as is, if you don’t mind being off by a few feet.

2E did not have creature types, so you will need to assign those yourself, based on the monster’s description. The types are all explained in the 5E Basic Rules.

Appended to the creature type in 5E, these can be determined from the monster’s description. Typical tags include titan (for creatures like the tarrasque), shapechanger, specific types of fiends (such as demon or yugoloth), and specific humanoid races (elf, thri-kreen, etc.) If your creature isn’t likely to have any of the tags above, skip this step.

Generally, keep this the same. However, creatures in 5E may also be unaligned, meaning they operate on instinct - so you may want to change neutral to unaligned for some creatures.

Armor Class

It’s easiest to recalculate AC from scratch, based on their Dexterity bonus and any armor worn. (Make sure to check the rules for determining AC in the Basic Rules PDF.)

If the creature did not wear armor, and had an AC of 6 or below, they should have natural armor. In that case, give them a +2 bonus to their new AC.

Dragons appear to have stronger natural armor in 5E. For now, I suggest estimating their 5E AC by subtracting it from 19.

Example: A dragon of some sort has AC -1. Subtract that from 19: 19-(-1) = 19+1 = 20.

In 5E, the type of hit die is determined by a creature’s size. Tiny creatures use d4 hit dice. Small creatures use d6 hit dice. Medium creatures use d8 hit dice. Large creatures use d10 hit dice. Huge creatures use d12 hit dice. Gargantuan creatures use d20 hit dice.

In 2E, Hit Dice may be displayed in formats like “1-1” or “5+3”. The first number is the number of Hit Dice ignore the later number.

Tiny creatures should keep the same number of hit dice as they had in 2E.

Small or Medium creatures should add one hit die. For example, 1d6 should become 2d6.

Large, Huge, and Gargantuan creatures should add two hit dice. For example, 3d10 should become 5d10.

Average hit points for creatures should be recalculated as follows:

Xd4 - Multiply X by 2.5 (round down), then add their Constitution bonus, times X.

Xd6 - Multiply X by 3.5 (round down), then add their Constitution bonus, times X.

Xd8 - Multiply X by 4.5 (round down), then add their Constitution bonus, times X.

Xd10 - Multiply X by 5.5 (round down), then add their Constitution bonus, times X.

Xd12 - Multiply X by 6.5 (round down), then add their Constitution bonus, times X.

Xd20 - Multiply X by 10.5 (round down), then add their Constitution bonus, times X.

Example: A monster has 3d8 HD and a Constitution of 14 (+2 bonus). So their average hit points are 4.5 times 3, rounded down: 13. Then you multiply their Con bonus by their HD, for a total of +6. 13+6 = 19.

Oozes should probably keep their current hit dice.

Dragons have been upgraded a lot since 2E, so any 2E dragon converted like other monsters is likely to be comparatively weak. But since we have only one example, I can only suggest you convert them like other creatures of their size.

Use the 2E Movement stat, then convert it as follows for each movement type (in feet).

Movement up to 12: Divide by 3, multiply by 10, round to nearest 10.

Movement above 12: Multiply by 2, round to nearest 10.

For reference, the 2E abbreviations for special movement are Fl (flying), Sw (swimming), Br (burrowing), Cl (climbing), and Wb (moving across webs).

Example: A creature has Movement 9, Fl 21. Its new ground movement is 9/3 = 3, 3x10 = 30 feet. Its new flight speed is 21x2 = 42, rounded to 40 feet.

A creature that can move across webs (Wb) should replace that speed with the trait Web Walker (see Giant Spider in the Monster Manual).

Ability Scores

The only ability score provided in most 2E stat blocks is Intelligence. You can use this statistic as given - if there’s a range, go for the average or highest score. If only the Intelligence “rating” is provided, use the below for reference:

Non- (0) [which must be increased to at least 1 for 5E]

If you are really lucky, the monster’s description may describe other ability scores. Use them as given if they are provided, with the exception of Strength, which should be converted as follows:

2E Strength 5E Strength

1-18 Same 18/01-18/99 18 18/00 (2E ogre) 19 19 (2E hill giant) 20-21 20 (2E stone giant) 22-23 21 (2E frost giant) 23-24 22 (2E fire giant) 25-26 23 (2E cloud giant) 27-28 24 (2E storm giant) 29 25 30

For any ability scores missing at this point, you will need to choose a “reference monster”. This is an existing monster similar to the one you’re converting. (For example, a skeletal undead could use the skeleton as a “reference”.) If you based the monster’s initial CR on an existing 5E monster, you already have your “reference monster”!

Use the ability scores of the “reference monster” to fill in any blanks, possibly tweaking them as needed to fit the monster’s concept or known ability scores.

When choosing a “reference monster”, use these sources, in this order of preference:

Monsters from D&D Next material.

D&D 3.5’s Monster Manual. You can use this site as a quick

The “How to Create a Monster” article (linked above).

Note that in 5E, creatures always have all six ability scores. If you use a 3.5 “reference monster” and it is missing any of these, you should look at the next closest Next or 5E monster, or use the guidelines for filling in blanks from my 3.5 conversion document.

Alternatively, the 2E sourcebook Dungeon Master Option: High-Level Campaigns provides a method for determining monster ability scores, if you want to fill in the blanks from the 2E side. They recommend rolling randomly and consulting a table, but I would use the average result on the table instead (results 9-12). Another 2E resource to consider is The Complete Book of Humanoids. Note that any resulting Strength scores would still need to be converted to 5E.

Saving Throws

Ignore the 2E rules for monster saving throws. In 5E, saves are associated with each of the six ability scores, so most creatures just use their ability bonus for saves.

However, a few 5E monsters do appear to apply their proficiency bonus to these saves. You may wish to do the same for your converted creature, if their 2E description suggests it should.

In most cases, though, you should probably pass on giving your creature proficiency with saves - it’s much easier without it.

The majority of monsters will have few or no skills, especially monsters driven by instinct. Consult the 2E monster’s description to see if it has any talents that might match a particular 5E skill. For reference, the 5E skills are:

Athletics (Strength)

Acrobatics, Sleight of Hand, and Stealth (Dexterity)

Arcana, History, Investigation, Nature, and Religion (Intelligence)

Animal Handling, Insight, Medicine, Perception, and Survival (Wisdom)

Deception, Intimidation, Performance, and Persuasion (Charisma)

Full descriptions of the skills can be found in the Basic Rules PDF.

If a creature has proficiency in a skill, their bonus with that skill is equal to the relevant ability bonus, plus their proficiency bonus.

Example: A monster has a Dexterity of 15 (+2 bonus) and proficiency in Stealth. This means that they should have a +4 bonus with Stealth (Dexterity) checks. If they are a more powerful creature, they may have a +5 bonus instead.

If a creature is hard to surprise, it should probably have proficiency in Perception (Wisdom). If the creature is good at surprising enemies, it should probably have proficiency in Stealth (Dexterity).

Some creatures seem to have a higher proficiency in one skill than the others, typically double their normal proficiency bonus (+4 or +6). For example, doppelgangers have a +4 on Deception (Charisma) checks. You may wish to do the same with a creature’s “signature” skill.

Vulnerabilities, Resistances, Immunities

Look at the 2E creature’s Special Defenses and description - this should give you an idea what the creature is vulnerable against (listed under Damage Vulnerabilities in 5E), what the creature is resistant against (listed under Damage Resistances), and what the creature is immune against (listed under Damage Immunities or Condition Immunities).

Some 2E creatures can only be harmed by magical weapons of a certain bonus (+1, +2, etc.). This should be listed under Damage Resistances as follows:

bludgeoning, piercing, and slashing from nonmagical weapons

If the creature can only be harmed by some other material, add the following:

. nonmagical weapons that aren’t (silvered or adamantine or [other substance])

Modify the above as needed for other resistances.

When you note Condition Immunities, remember that they may overlap with resistances or damage immunities. For example, a creature immune to poison damage should also note that they are immune to the poisoned condition.

The innate immunities of undead in 5E are:

Condition Immunities: poisoned

Only add other immunities to an undead creature if they are separate from its undead nature.

Incorporeal creatures should probably have the following resistances and immunities:

Damage Resistances: acid, cold, fire, lightning, thunder bludgeoning, piercing, and slashing from nonmagical weapons

Condition Immunities: grappled, paralyzed, petrified, poisoned, prone, restrained

Magic resistance has also been simplified, and is now listed as a trait - see the flameskull for an example.

There are only four senses established in 5E at this point: blindsight, darkvision, tremorsense, and truesight. 2E’s infravision, ultravision, and similar senses are equivalent to darkvision. Any non-visual replacement for sight should be blindsight or tremorsense. Any creature with innate “true seeing” should have truesight.

Also, all creatures have a passive Perception score. This is equal to 10 plus their total bonus to Perception (Wisdom) checks.

Refer to the Basic Rules PDF for the known languages in 5E. If no language for your creature is listed there, give them the native language provided in their 2E description.

To determine a creature’s Traits, look at the 2E monster’s description. When possible, you should use equivalent traits from the Monster Manual if this is not possible, either base the converted trait on a similar 5E trait, or simply use the original text as is.

A list of the known Traits in 5E can be found here.

Spellcasting is a special case. In addition to being used for 2E monsters with levels in a spellcasting class, lists of spell-like powers also tend to be translated as Spellcasting. (The only exception should be specific spell-like powers which are really important to a creature’s concept, in which case they are either listed as Innate Spellcasting or broken out as individual traits, actions, or reactions.)

If the creature lists a caster level, use that. If no caster level is provided, use their 2E Hit Dice. Make sure to use the 5E rules for spellcasting, and replace spells with closest equivalents if necessary refer to the Basic Rules for more details. (For Innate Spellcasting, the spellcasting ability is typically Wisdom or Charisma.)

Actions with an Attack Roll

Actions should be taken from a 2E creature’s normal attacks, Special Attacks, or description. You need to convert three things to make these into 5E actions - the “to hit” bonus, the damage inflicted by elements of the attack, and any saving throws required.

A list of the known Actions in 5E can be found here.

Ignore the THAC0 and recalculate this from scratch. In 5E, the “to hit” for an action is based on the proficiency bonus. This is combined with the relevant ability bonus (usually Strength or Dexterity).

Note that most natural attacks (claw, bite, etc.) appear to be finesse attacks, meaning they can use either Strength or Dexterity with attack rolls. Unless you think your creature should be a weak melee combatant, you should choose the highest of either Strength or Dexterity for “to hit” with natural attacks.

Unusual attacks like the wraith’s Life Drain appear to either have a higher bonus, or are using something other than Strength or Dexterity. I don’t have any specific recommendations for this right now, but feel free to experiment with other ability scores (like Constitution or Charisma) if that seems appropriate. You’re probably safer avoiding that, however.

In 2E, damage is often expressed not in dice, but as a range, like 1-6, 3-12, or 2-5. In such cases, you will have to figure out how many dice that is before converting. (In the above cases, it’s 1d6, 3d4, and 1d4+1.)

For attacks that use a weapon, refer to the damage listed in the Basic Rules PDF or Starter Set. If the creature is large, increase the weapon’s damage die by one (i.e. 1d6 becomes 2d6). If the creature is huge, increase the weapon’s damage die by two (i.e. 1d6 becomes 3d6). Some especially huge creatures, and gargantuan creatures, may increase the damage die by three (i.e. 1d6 becomes 4d6), but you may be safer sticking with two.

For natural attacks by tiny or small creatures, you should probably leave the damage dice alone. However, the minimum damage die appears to be 1d4, so any weaker attacks should probably upgrade to that. (The same goes for damage dice below 1d4 for other creatures.)

For natural attacks by medium or larger creatures, use the following table:

2E Damage Dice 5E Damage Dice

1d4 1d6 (or 2d4) 1d6 1d8 (or 2d6) 1d8 1d10 (or 2d8) 1d10 1d12 (or 2d10) 1d12 2d8 (or 2d12)

If an attack inflicts more than one die of damage, increase the 5E damage die by that number. For example, 2d4 should become 2d6 (or 3d4).

Some creatures’ strongest natural attacks seem to add both one die of damage and increase the die type by one - for example, 1d6 becoming 2d8. This should be used carefully, however, and shouldn’t be used on more than one attack.

Don’t forget to add the monster’s Strength bonus to melee damage and Dexterity bonus to ranged damage. You can calculate average damage through the method above under Hit Dice.

Other types of damaging attacks (like the flameskull’s fire ray) seem to be treated much like the “strongest natural attacks” above, but there aren’t many examples yet. Still, that guideline may work for now. Dragon breath weapons are upgraded much more than that, but until we have more dragons, I’m reluctant to recommend any guidelines there.

Saving Throws for Actions

5E saving throws are very different from 2E. They should probably be converted as follows:

2E Saving Throw 5E Saving Throw

Paralyzation, Poison, or Death Magic Constitution Rod, Staff, or Wand Wisdom (or Dexterity) Petrification or Polymorph Constitution (or Wisdom) Breath Weapon Constitution (or Dexterity) Spell Wisdom

Feel free to substitute a different ability score for the save, if it seems more appropriate.

You will need to calculate a Difficulty Class (DC) for the save your creature inflicts. The usual way to calculate this is 8 + proficiency bonus + the ability bonus of the creature for the relevant save. For example, a CR 2 creature with Constitution 13 (+1 bonus) that inflicts a Constitution save effect should require a DC of 11 (8+2+1).

The wraith’s Life Drain should be used in place of 2E’s energy drain attack.

Some actions also have a recharge. Two examples are the giant spider’s Web and the young green dragon’s Poison Breath, which both have a recharge of 5-6.

Recharge seems to cover strong effects that, in 2E, could only be used a limited number of times per day, or required a number of rounds between uses.

My recommendation is to use “Recharge 5-6” for most limited-use actions, and keep “X per day” for especially strong attacks that should only be used once in a battle.

Actions without an Attack Roll

Many 5E creatures - such as the doppelganger, flameskull, grick, nothic, owlbear, and young green dragon - have Multiattack, allowing them to take more than one action per turn. If your 2E monster could attack more than once, you should probably give them Multiattack in 5E. Note that most creatures can only make two attacks - the only exception is the young green dragon, which can make three.

There are also a few creatures that can take actions not requiring an attack roll or saving throw. If your monster could do this in 2E, you can probably use them as is in 5E.

Anything that could be converted into an Action, but requires some sort of trigger, should be categorized as a Reaction. Otherwise, they should be converted like other Actions.

A list of the known Reactions in 5E can be found here.

Final Notes

Everything not mentioned above, like Morale, should probably be dropped in 5E. Of course, if you feel it’s very important to the creature’s concept, feel free to port it over anyway.

Appendix 1: CR Estimator

The below conversion follows a very broad comparison of the 5E CR and 2E XP Value for the same creatures in both editions. It’s not perfect, but it works as a rough starting point.

5E CR 2E XP

0 7 ⅛ 15 to 35 ¼ 65 to 120 ½ 175 to 270 1 420 2 650 3 975 to 1400 4 2000 5 3000 to 4000 6 5000 to 6000 7 7000 8 8000 +1 +1000

Appendix 2: 1E

While there were several underlying changes between 1E and 2E, the monster statistics can be converted more or less the same as 2E creatures. There are a few differences, however, which are mostly cosmetic.

Movement rates are the same as 2E, but they are usually displayed in the following format:

1E only recognizes three size categories: S (small), M (human-sized), and L (large). “M” is defined as 5-7 feet “S” is anything smaller than 5 feet “L” is anything larger than 7 feet. If a specific size in feet is given, you may use that number with the 2E size conversion if not, you will have to choose an appropriate size yourself, based on the creature’s description.

XP Value is calculated differently from 2E. As such, if you want to use their XP Value to determine their 5E CR, you cannot use Appendix 1’s CR Estimator. Instead, use the following alternative table (ignoring the extra XP per monster hp):

5E CR 1E Basic XP Value

0 5 or less ⅛ 5 to 20 ¼ 20 to 50 ½ 50 to 100 1 100 to 200 2 200 to 350 3 350 to 650 4 650 to 900 5 900 to 1300 6 1300 to 1700 7 1700 to 2100 8 2100 to 2500 9 2500 to 3000 10 3000 to 3500 11 3500 to 4000 12 4000 to 4500 13 4500 to 5000 14 5000 to 5500 15 5500 to 6000 16 6000 to 6500 17 6500 to 7000 18 7000 to 7500 19 7500 to 8000 20 8000 to 9000 21 9000 to 10,000 22 10,000 to 15,000 23 15,000 to 20,000 +1 +5000

(Note that the original 1E Monster Manual does not provide XP Values at all you have to refer to the 1E Dungeon Masters Guide.)

Psionic Ability is listed for many 1E monsters (as well as a smaller number of 2E monsters). Psionics rules are beyond the scope of this conversion, but you may wish to import any described psionic powers as Innate Spellcasting (see the Mind Flayer for an example), or as other traits, actions, or reactions as appropriate.

FYI, the 1E Monster Manual II has a list converting the damage ranges (2-8, etc.) to dice rolls, which may come in handy for both 1E and 2E conversions.


12.2E: Linear Equations (Exercises)

Use the following information to answer the next three exercises. A vacation resort rents SCUBA equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.

What are the dependent and independent variables?

dependent variable: fee amount independent variable: time

Find the equation that expresses the total fee in terms of the number of hours the equipment is rented.

Graph the equation from Exercise.

Figure (PageIndex<4>).

Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late, and $5 a day each day the payment remains unpaid.

Find the equation that expresses the total fee in terms of the number of days the payment is late.

Graph the equation from Exercise.

Figure (PageIndex<5>).

Is the equation (y = 10 + 5x &ndash 3x^<2>) linear? Why or why not?

Which of the following equations are linear?

(y = 6x + 8), (4y = 8), and (y + 7 = 3x) are all linear equations.

Does the graph show a linear equation? Why or why not?

Figure (PageIndex<6>).

Table contains real data for the first two decades of AIDS reporting.

Adults and Adolescents only, United States
Year # AIDS cases diagnosed # AIDS deaths
Pre-1981 91 29
1981 319 121
1982 1,170 453
1983 3,076 1,482
1984 6,240 3,466
1985 11,776 6,878
1986 19,032 11,987
1987 28,564 16,162
1988 35,447 20,868
1989 42,674 27,591
1990 48,634 31,335
1991 59,660 36,560
1992 78,530 41,055
1993 78,834 44,730
1994 71,874 49,095
1995 68,505 49,456
1996 59,347 38,510
1997 47,149 20,736
1998 38,393 19,005
1999 25,174 18,454
2000 25,522 17,347
2001 25,643 17,402
2002 26,464 16,371
Total 802,118 489,093

Use the columns "year" and "# AIDS cases diagnosed. Why is &ldquoyear&rdquo the independent variable and &ldquo# AIDS cases diagnosed.&rdquo the dependent variable (instead of the reverse)?

The number of AIDS cases depends on the year. Therefore, year becomes the independent variable and the number of AIDS cases is the dependent variable.

Use the following information to answer the next two exercises. A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is (y = 50 + 100x).

What are the independent and dependent variables?

What is the y-intercept and what is the slope? Interpret them using complete sentences.

The (y)-intercept is 50 ((a = 50)). At the start of the cleaning, the company charges a one-time fee of $50 (this is when (x = 0)). The slope is 100 ((b = 100)). For each session, the company charges $100 for each hour they clean.

Use the following information to answer the next three questions. Due to erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is (y = 12,000x).

What are the independent and dependent variables?

How many pounds of soil does the shoreline lose in a year?

What is the (y)-intercept? Interpret its meaning.

Use the following information to answer the next two exercises. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is (y = 15 &ndash 1.5x) where (x) is the number of hours passed in an eight-hour day of trading.

What are the slope and y-intercept? Interpret their meaning.

The slope is -1.5 ((b = -1.5)). This means the stock is losing value at a rate of $1.50 per hour. The (y)-intercept is $15 ((a = 15)). This means the price of stock before the trading day was $15.

If you owned this stock, would you want a positive or negative slope? Why?


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