Algebra Questions

lim f x x X X Left hand limit atx x lim f x lim f x h X lim f x x X X Continuity of Functions lim x h lim x h 2x h 20 x 0 0 RSITY OF DELE Since lim f x lim f x x Hence lim f x x Now the value of the function at x x is f x x from equation 1 and 2 we have lim f x f x x f x 0 Thus f x x is continuous at x x Since is an arbitrary element of R Therefore f x is continuous at each point of R Hence f x is continuous on R Example 4 Show that the function f x defined by Exfar if x 0 if x 0 is continuous on R Solution Given function is 1 xsin if x 0 X 0 if x 0 N

I Q 2 I Q 3 Theorem 1 Statement Only A function f X R is said to be continuous at a point EX if and only if for given any E neighborhood V f x of f x there exist a 5 neighborhood V x of x such that if x is any point of XV then f x belongs to V f x is f XV x V f x 5 Discontinuity of a Function at a Point Let X be a non empty subset of R let f X R and let EX Then the function f x is said to be discontinuous at if and only if it is not continuous at x 5 1 Types of Discontinuity Let f be a function defined as f X R and let x EX and let left hand limit right hand limit limit of the function and the value of the function at xare denoted by lim f x lim f x lim f x and f x X A respectively Then 9

Note The number of points of the complete bigraph Km n is m n and the number of lines is m n Example Problem Solution K4 2 3 Let V 1 2 3 n Let X i j i je V and are relatively prime The resulting graph V X is denoted by G Draw G and Gs K2 3 2 Graph of G Fig 1 11 1 2 DEGREES For G V 1 2 3 4 and X 1 2 1 3 1 4 2 3 3 4 For G V 1 2 3 4 5 and X 1 2 1 3 1 4 1 5 2 3 2 5 3 4 3 5 4 5 Fig 1 10 K3 3 2 Graph of Gs Fig 1 12 Definition The degree of a point v in graph G is the number of lines incident with v It is denoted by d v or deg v or d G v

9 Three questions are asked to some US households when the U S Bureau of Labor Statistics BLS conducts a nationwide random survey to determine who is employed and who is not employed Based on information below what is the unemployment rate What about the labor force participation rate Questions Yes 1 Are you under 16 or institutionalized 10 million 2 Are you currently working 40 million 3 Have you been actively seeking work in the past four 8 million weeks No 50 million 10 million 2 million

from Sines rule Sin A a T sin A Sin B Sinc Cos A 9448 In equation 1 6 9 2bc ak ab 79 ck b a b c 2x q b c 4x1 2 6 c 92 2x29bck Lbc Sinc c 9 b 2 ca bk ab sin c 9 b c 2 C 2 b 2 2 a c 9 b 9 46 c 2 abc k c 9 6 Cos B 209 9 8 c dab ck 55 2 1 ab sinc 4 9 b c 246

Prove the following using a proof by contrapositive For x yZ if xy2Z then x2Z or y2Z

a i Describe the set of points in the complex plane that satisfy the inequality 2 z 9 4i 3 ii Provide a hand sketch of this region iii Find the area of this region b Describe the set of points in the complex plane that satisfy z z 4

Explore 6 1 6 2 Create a hanger using all of these shapes 5 circles 4 triangles 3 squares 5X Determine the value of x that will balance it x 344 A4

16 Cem 15 Can x ya ndad r Cem Can n ya nda iken Can n do mas na 5 y l vard Buna g re Can n bug nk ya ka t r A 5 B 6 C 7 D 8 E 9

7 Find the domain of y 2 x 3 x 7

If tan 0 24 10 sin 0 equals cos 0 equals 8 qual 0 0 T 2 the

1 The complete graph of a function is shown below a What is the domain of the function b What is the range of the function 10 9 8 7 6 5 4 3 2 1 1 0 1 2 3 4 5 6 7 8 9 10

Match each of the following with the correct statement A The series is absolutely convergent C The series converges but is not absolutely convergent D The series diverges NOTE Make sure that you can provide the proper arguments to support your answers 1 2 3 DO 5 iMiMiMiMiM 1 3 1 3 1 n 2 10 n 4 1 2 8n n 1 522 1 4n h

not monotone Can you find a monotone one to one function that is not onto Or a monotone onto function that is not one to one Notes and Remarks

A term of the binomial expansion mx y where m is a positive integer is 640x The value of mis 640x y C mx 6 2 y 3 4 Jm3

Problem 1 Enter the following 3 matrices into Matlab Problem 1 a Problem 1 b 27 28 48 38 47 18 44 7 A 14 35 30 14 32 13 44 33 17 23 17 2 B ATCB 26 1 18 25 41 46 30 C 2 1 6 14 2 6 44 5 44 38 To check that you have correctly entered these matrices into Matlab verify that 62675 3337 44933 109842 42703 24803 68129 49438 24542 11410 34418 31620 33815 2227 24953 58782 and that tr ACB 13508 a Let D CBAT Find d42 i e find the entry in row 4 column 2 of the matrix D b Let E ABTCT Find tr E c Let F be the reduced row echelon form of AFCB Find f 4 i e find the entry in row 1 column 4 of the reduced row echelon form of ATCB

e Page 67 4 2 Determine whether something is living or nonliving based on the charac Which of the following are living scientists such as Daniel Gibson a rocky outcropping a petri dish bacteria like Mycoplasma mycoides nonliving living tics

5 points Let n and i be nonnegative integers Prove the following equality by nduction on n n 1 2 i 1

sider the below matrix A which you can copy and paste directly into Matlab The matrix contains 3 columns which represent the final grades in a course with 3 sections of equal size The first column is C01 the second is CO2 and the third is C03 A 100 100 29 87 17 91 20 07 66 17 63 35 86 56 89 68 68 06 100 55 75 37 91 30 26 65 93 23 78 71 66 100 84 07 14 71 40 11 15 48 72 34 36 31 76 22 25 79 34 89 50 68 49 43 41 61 94 9 57 32 100 94 77 99 28 100 100 100 73 73 72 96 13 65 100 57 09 100 70 67 100 34 73 95 3 100 80 76 40 03 100 54 01 99 22 100 17 66 54 74 100 87 95 100 100 97 04 71 29 65 18 96 6 63 67 47 44 15 97 40 72 44 35 69 96 74 26 35 75 56 27 52 19 100 97 4 100 83 5 100 100 16 47 55 12 100 29 83 84 45 88 65 100 95 99 25 95 20 59 55 22 100 84 73 65 02 61 85 81 62 97 56 67 76 15 57 100 94 75 100 12 76 45 91 53 71 77 95 55 19 17 63 24 22 100 17 75 100 31 28 44 98 84 65 100 62 21 58 91 25 41 75 68 68 66 96 35 19 34 38 39 61 32 93 48 49 13 100 68 45 50 5 100 78 03 94 63 42 24 58 93 56 3 100 55 04 57 7 80 43 78 85 36 79 100 70 40 46 100 89 96 50 08 85 42 86 9 93 31 56 56 15 3 29 98 73 58 32 78 17 91 40 37 36 94 53 71 94 95 17 45 41 17 63 07 55 34 97 53 79 5 75 01 72 87 100 70 19 94 63 31 31 25 68 54 15 81 26 71 31 28 4 55 48 43 09 77 56 90 36 87 84 42 45 82 67 80 5 12 07 99 13 42 51 45 100 63 14 37 59 100 59 81 71 2 66 44 98 35 60 8 67 17 38 45 100 89 78 24 44 100 75 56 100 34 91 76 39 60 16 74 52 100 44 58 82 16 72 05 27 16 78 08 100 15 59 100 74 01 22 73 93 94 100 100 100 59 26 100 43 27 91 8 72 94 28 45 85 48 38 7 99 78 34 82 62 42 21 43 97 53 35 41 58 85 68 32 100 100 81 15 71 68 51 7 38 23 31 76 67 92 100 41 2 35 82 100 100 22 44 27 51 15 53 51 31 21 41 99 47 52 49 28 36 78 7 16 48 100 20 73 100 41 62 65 92 49 36 100 100 76 3 100 100 91 09 41 76 45 04 100 63 61 66 81 71 48 86 51 51 05 19 24 88 14 16 51 20 61 94 52 100 88 95 69 56 55 81 22 83 100 49 92 25 78 67 5 90 65 100 68 51 100 60 39 62 57 99 37 75 81 100 83 08 38 63 60 5 51 12 53 65 73 41 39 37 44 13 98 65 79 24 82 17 60 13 67 2 59 74 100 100 83 19 92 65 32 72 53 16 78 18 96 56 27 92 91 08 99 5 54 22 85 44 100 27 56 98 48 100 56 71 88 02 54 67 44 25 21 81 100 44 83 31 57 89 71 100 39 09 42 05 95 15 100 25 58 74 44 100 82 92 59 2 100 95 1 82 56 37 68 83 46 89 23 80 64 25 95 97 39 25 29 21 82 77 21 95 52 67 39 69 82 65 78 35 08 93 38 48 87 100 70 36 79 35 85 72 100 75 8 77 31 97 77 56 13 41 28 45 05 74 100 94 57 85 57 85 17 51 89 70 24 89 76 18 53 21 07 41 5 15 39 41 77 96 59 100 88 95 73 60 78 100 100 88 2 66 62 42 28 19 94 100 100 66 19 48 18 41 3 32 54 100 16 64 49 45 79 18 33 59 61 03 88 45 77 73 63 07 100 100 38 68 49 92 18 52 50 98 34 33 75 83 91 89 3 19 51 76 9 47 24 20 11 100 80 75 19 45 50 18 100 61 46 65 03 84 52 54 03 74 84 89 36 43 62 76 18 100 55 81 60 68 100 100 60 06 77 33 86 73 42 24 25 74 100 33 22 68 44 100 99 88 91 91 81 77 31 15 82 76 100 48 08 100 68 54 97 3 100 100 100 49 39 62 57 59 26 95 36 49 34 100 79 02 16 27 91 61 33 45 87 5 63 06 100 91 81 66 33 22 57 72 27 100 73 94 100 88 93 100 100 100 53 15 61 41 54 46 56 53 76 72 37 18 100 34 6 79 9 100 a Use the colon operator to extract the marks from C03 and then find the average of those marks using the sum and length commands b Use vertical concatenation to create a single column vector consisting of all of the marks from all 3 sections

Graph the following elements on a number line then determine the maximum minimum and mean of these data 2 7 6 Graph 2 10 8 3 BOOK 0

Problem 2 a Problem 2 b Hint Use the colon operator to produce the vector 2 5 8 113 and then find the product of the elements of that vector b Find the value of the following sum 8 13 18 428 Just Save Problem 2 Your Answer Submit Problem 2 for Grading Attempt 1 2 a 2143 Attempt 2 X 113 2 a 766 Attempt 3 2 a YEY

1 Find the solution set to each inequality Express the solution in set notation AND graphically on the number line a 2x 10 ZI X 5 Solution Set 4 3 2 c 6x 9 1 5x49 4x ax 4x 2 3xx9 3 M Im b Solution Set 3 3 t 2 1 30 b 15x 45 15 15 2 X 31 32 2 5 1 0 Solution Set d 6x 5 7x 4 7x F7X 18 5847 Solution Set T 10

Use logarithms to evaluate 895 73 486

Identify the vertex and leading coefficient Then write the expression as f x ax bx c f x x 4 2 5

Instructions Find the missing probability P B P A B 9 20 P A and B 50 X 100

Instructions Find the missing probability P B P A and B 3 25 P A B

Make m the subject Nw tm I 1 r

3 Angles of Elevation Depression Be sure to show any and all work a 5 pts If we have a point 6 14 on a graph and a line that passes through it at an angle of elevation at 60 degrees what is the function created by that line b 5 pts We know that the distance between us and a pole is 50 yards and the height of the pole is 20 yards as well What is the angle of elevation between us and the pole c 5 pts How close are we to a wall if our eyes are 5 ft above the ground creating an angle of depression to the ground that is 59 Degrees d 5 pts What are the angles of both corners in a right triangle if both sides have equal length

exact values below If applicable click on Undef 225 22 8 X Une

Solve the equation Check your solution 6 1 2 12

Solve the equation Check your solution 2a 9 a 2 10 3

Solve the equation Check your solution 4x 9 4x 33

Solve the equation Check your solution 5x 6 1 5 18 Simanl 6 TYDO 6 teger or a frac

3 Raul has 56 bouncy balls He puts three times many balls into red gift bags as he puts into gr gift bags If he puts the same number of balls i each bag how many balls does he put into ea green bag

For the following right triangle find the side length x X 6 8 0

Contestar lo siguiente a Hallar un ngulo entre 0 y 360 que sea coterminal con 1170 b Hallar un ngulo entre 0 y 2 que sea coterminal con Dar valores exactos en sus respuestas a b radianes X JT 5 19T 10

Hallar los valores exactos siguientes Si procede hacer clic en No est definido sin 11x 3 0 sec 225 0 8 Indefinido X JU

Let 5 4 be a point on the terminal side of 0 Find the exact values of sin 0 sec 0 and tan 0 sin 0 seco 8 X 0

12 x L Arrastre para mostrar el ngulo X

Find the exact value of cos 315 cos 315 0 3 16

Write an equation of the circle with center 9 3 and radius 4 0 8 0 0

1 15 pts For the following questions draw a right triangle with the values in their corresponding positions Once you have the triangle drawn find the missing values be sure to show work try out different methods to solve for the missing values too a The angle next to a length of 5 inches is 60 degrees and the hypotenuse is 10 inches b The two sides of a right triangle are 6 meters and 8 meters puro of a right triangle is 15 created with angles 0 36 87 and

y e x 28 23 37 a Solve out for the values of y x 0 and a a b What would the angle created between the hypotenuse of 28 and line 23 be c What would the angle created between the hypotenuse of 37 and line y be

5 5 f x x 4x 1 Find 1 A 7 C 1 B 1 D 4 7 f x x 2x 1 Find 1 6 8 6 1x 3x 4 4 Find 0 A 1 0 B 8 D 3 8 tx x 4 1 Find 10

9 7 f x x 2x 1 Find 1 9 x x 4x 3 Find 2 8 10 8 f x x 4 1 Find 10 10 x 3 x 4 2 Find 6

Which system of inequalities is represented by the graph O y 3x 6 y x 6 y x y 3x 6 y x 6 y x y x 6 y 3x 6 y x y x 6 y 3x 6 y x 10 8 6 4 10 N 8 6 4 2 94 4 b 6 8 10 2 4 6 xX 8 10

1 3 4 5 ach radical and write in simplified radical form 6 863 9b 4 32n 80n 3m 3 4m4 21uv 8 42u v 6 4 2x y 64x y 2 3 60x 15x y

Which ordered pair is a solution to the system of inequalities Jy 2 x y 4 O 1 5 O 1 3 O 2 3 O 0 4

Which graph represents the solution set to the system of inequalities 2x y 2 4 y 2x y 3 x 10 8 6 12 8 6 4 2 2 4 y 10 8 6 6 X 8 10 ca 10 8 6 4 Gi y 6 4 2 4 10 8 2 4 6 X 8 10

Simplify 27x6y Assume all variables are nonnegative Enter your answer in the box