Midterm 2 Review Problems (the first 7 pages) Math Intermediate Algebra Online Spring 2013

Transcription

1 Midterm Review Problems (the first 7 pages) Math Intermediate Algebra Online Spring 01 Please note that these review problems are due on the day of the midterm, Friday, April 1, 01 at 6 p.m. in MER 1 at Hartnell College) Name: (Please Print) Problem 1: Find domain for this real function and label it on a number line or epress it in set 9 notation: f( ) 5 7 Problem : Find domain of the real function and label it on a number line or epress it in set notation: f ( ) 9 Problem : Find the domain of the real function and label it on a number line or epress it in set 4 notation: f( ) 6 4

2 Problem 4: Use vertical line test to determine whether the following form functions: Problem 5: Do the following divisions

3 Problem 6: Factor the following completely: a) 6 11y 10y b) 10 1 c) d) e) 6m 49n f) 16 1 p q g) 1 y 8 h) 15m 1 n 7

4 Problem 7: Solve the following equations by factoring: a) b) Problem 8: The area of rectangular garden is 00 square meters. The length of the garden is 1 meter more than times its width; find the length and the width of the garden.

5 Problem 9: a) Factor and simplify: b) Factor and simplify: 6 1 m m 7 5 m 7 m 1 4 Problem 10: Solve for k: k k 6 k k k 4k

6 Problem 11: a) Simplify: 50 b) Simplify: 5 15 a a 5a c) Simplify: 4 18 d) Rationalize the denominator: 5 5 Problem 1: (a) Simplify and epress in terms of positive eponents only: m p 4m p mp 4

7 (b) Simplify 1 y 9 y 6 Problem 1: a) Simplify and epress it in the form (a + ib): 6i 1 i (b) Simplify: i 011 END OF MIDTERM REVIEW

8 Midterm 1 Review Problems (the first 5 pages) Math Intermediate Algebra Online Spring 01 Please note that these review problems are due on the day of the midterm, Friday, March 1, 01 at 6 p.m. in MER 1 at Hartnell College) Name: (Please Print) Problem 1: Your cell/mobile phone plan charges you $75 per month which includes 500 minutes of talk time. Any additional minutes are charged at $0.40 per minute. Create a linear model for your phone bill as a function of the talk time where represents the number of minutes you talk over the regular 500 minutes included in the plan and y represents your monthly bill. Linear model: What is your bill for a month if in that month you talked for 700 minutes which included the regular 500 minutes of talk time that comes with the plan? Problem : Epress the equation y in the form y m b and find the slope and the coordinates of the y-intercept. Is the line going up or down as you go from left to right?

9 Problem : Find the equation of a line that passes through the point (-, ) and is parallel to the line y 9 Problem 4: Find the equation of a line that passes through the point (5, 8) and is perpendicular to the line 4y 1 Problem 5: A restaurant serves two types of buffet. Regular buffet cost $15 per person and kid s buffet cost $10 per kid. For a busy weekend the restaurant sold 750 meals which generated $9055 revenue for the weekend. How many kids visited the restaurant for the weekend?

10 Problem 6: Miture problem: At a party two types of mango Sherbets are served. One that contains 0% mango juice and the other contains 60% mango juice. However diabetes being the biggest killer on the planet some people have decided to cut down on their sugar consumption. So rather than having the 60% mango juice they wanted to make 40% mango juice for themselves from the given options. How much of each type should they mi in order to make 5 liters of 40% mango juice? Problem 7: A mother wants to invest $10,000 for her son s future education. She invests a portion of the money in the U.S. Government issued bond which earns 4% return per year and the remainder in a stable stock which on the average returns 8% per year. If the total return on investment (ROI) for the year is $60, how much money was invested in the stock?

11 Problem 8: You travel from San Francisco to Tokyo by air, a distance of about 8,400 kilometers. On your way to Tokyo it took 16 hours because of the headwind and on your way back it took 14 hours because of the tailwind. What are the speed of the wind and the speed of the plane? Problem 9: In a right triangle one of the acute angles is degrees more than times the other acute angle. Find the acute angles.

12 Problem 10: Your uncle Juan has a piggy bank which has 110 coins in quarters and dimes only. He wants to give you all the money if you can tell him the eact number of quarters and the eact number of dimes given that there is $1.65 in the piggy bank. Problem 11: Do the following multiplications: (a) (b) 5 4 (c) 5 y z yz

13 (d) 5 5 END OF MIDTERM 1 REVIEW

14 Problem 4: Find domain for this real function and label it on a number line or epress it in set 9 notation: f( ) 5 7 Problem 5: Find domain of the real function and label it on a number line or epress it in set notation: f ( ) 9 Problem 6: Find the domain of the real function and label it on a number line or epress it in set 4 notation: f( ) 6 4 Problem 7: Use vertical line test to determine whether the following form functions:

15 Problem 17: Do the following addition/subtractions: a) 5 7 5

16 b) c) Problem 18: Do the following multiplications: a) 7 5 b) c) 5 5 d) Problem 19: Do the following divisions: a)

17 b) Problem 0: Factor the following completely: a) 6 11y 10y b) 10 1

18 c) d) e) 6m 49n f) 16 1 p q

19 g) 1 y 8 h) 15m 1 n 7 i) 64m 1 n j)

20 k) Problem 1: Solve the following equations by factoring: a) b) Problem : The area of rectangular garden is 00 square meters. The length of the garden is 1 meter more than times its width; find the length and the width of the garden.

21 Problem : a) Factor and simplify: b) Factor and simplify: 6 1 m m 7 5 m 7 m 1 4 Problem 4: Solve for k: k k 6 k k k 4k

22 Problem 5: a) Simplify: 50 b) Simplify: 5 15 a a 5a c) Simplify: 4 18 d) Rationalize the denominator: 5 5 Problem 6: (a) Simplify and epress in terms of positive eponents only: m p 4m p mp 4

23 (b) Simplify 1 y 9 y 6 Problem 7: a) Simplify and epress it in the form (a + ib): 6i 1 i (b) Simplify: i 011 Problem 8: Find the overlapping region for these inequalities by graphing the inequalities in the rectangular coordinate system given below: y y

24 L Overlapping region -6 L The first step is to graph the straight lines: y y To graph a straight line we need two different points on the line. In y we solve for corresponding y values: The line is labeled L1 y and pick any two convenient points for and find the y = ( )/ (, y) 0 1 (0, 1) 1 (, 1) To find the region of interest for L1 just pick a point that is not on the line. Since the line does not go through the origin (0, 0) I pick this as my reference point and check it on the inequality y which is a false statement. Hence for this line the region of interest is the region below the line L1. Also we make the line solid because it includes the boundary. Similarly in y we solve for the corresponding y values: y and pick any two convenient points for and find y = ( + 1)/ (, y) 0 (, 0) 4 1 (4, 1)

25 The line is labeled L. Once again we may use (0, 0) as our reference point since the point is not on the line. We have y which is a true statement; the region below L is the region of interest. The line is made dotted indicating that the boundary points are not included. Now if we compare the two regions of interest for y and y as outlined by L1 and L, we obviously notice that the overlapping region is as labeled in the figure above. Problem 9: Find the following for the parabola y (a) Does the parabola open up or down? Why? The parabola opens up because the coefficient of the square term 4 is 4 which is greater than zero. If the coefficient of the square term is negative then the parabola opens down. b (b) Find the coordinates of the verte: -coordinate of the verte is given by a b 16 so the we have a 4 To find the y-coordinate of the verte just plug in = in the equation of the parabola y and then solve for y. y y y y Hence the coordinates of the verte are: ( y, ) (, 1) b b 4ac (c) Find the -intercepts if they eist: (Use a To find the -intercept set the y = 0 and then solve for in Thus we have Using the quadratic equation solution formula we have if needed) y b b 4ac a and a = 4, b = 16, and c = 15 from

26 ( 16) ( 16) or So the -intercepts are at 5,0 4 and,0 Also note that if the solutions to the quadratic equation were to be comple conjugates that would mean that the parabola did not have -intercepts. (d) Find the y-intercept: To find the y-intercept just set = 0 in the equation y y y 15 So the y-intercept of the parabola is: (0, 15) y and solve for y. (e) Sketch the parabola: The above is a sketch (approimation) of the parabola y Problem 0: Complete the following to make it a perfect square:

27 (a) 0 ( ) Use the formula Compare y y ( y ) 0 ( ) with y y 10 (10 ) y y Now you can complete the square using the formula So 10 (10 ) ( 10) y y ( y ) To make it a complete square you need to put 100 in the blank. (b) ( ) 81 Use the formula Compare y y ( y ) ( ) 81with y y 9 9 y y Now you can complete the square using the formula y y ( y ) So 10 (10 ) ( 10) To make it a complete square you need to put 0 in the blank. (c) ( ) 0 5 Use the formula y y ( y ) Compare ( ) 0 5 with y y 5 5 y y Now you can complete the square using the formula So 5 (5 ) ( 5) To make it a complete square you need to put y y ( y ) in the blank. Problem 1: Solve for using the quadratic formula: b b 4ac a (a) In this problem a = 1, b = 5, and c = 1. By putting these values in the formula b b 4ac a we get:

28 Hence the solutions are 5 1 and 5 1 ; note that the solutions are conjugates. (b) 1 0 Here a = 1, b = 1, and c = 1. By putting these values in the formula b b 4ac a we get: i since i 1 1 i Hence the solutions are and conjugates. 1 i ; note that the solutions are comple (c) Here a = 1, b = 4, and c = 5. By putting these values in the formula b b 4ac a we get: ( 4) ( 4) i since i 1 ( i) i Hence the solutions are i and i; note that the solutions are comple conjugates. Problem : Solve for : (a) 5

29 Note that for eample 1 means that the distance of the point from the origin in the number line is less than 1. This says that 1 1; also note that since absolute value of a number is its distance from the origin it is always positive or zero, since a negative distance does not make sense. Graphically I mean the following: 1 1 ( 1,1) Using this concept I can simplify 5 8 i.e. 8 8 Add to all sides then you get 8 8 So the solution set for is: ( 6,10) In interval notation ( 6,10) (b) Note that for eample 1 means that the distance of the point from the origin in the number line is greater than 1. This says that either 1 or 1. Graphically I mean the following: 1 or 1 (, 1) (1, ) 1 Now simplify 5to 8by adding to both sides. Then 8 0 means that either 8 or 8which leads to 10 or 6; now label these inequalities on the number line. 6 or 10 (, 6) (10, )

30 In interval notation (, 6) (10, ) Problem : Properties of logarithm: y log a( y) loga loga y loga loga loga y loga yloga y (a) Simplify using the properties of logarithm: Using the property log log log y a a a y ln z y we have z ln ln( z ) ln y y Now using the property log a( y) log a log a y we can further simplify: 1 1 n n ln( z ) ln y ln ln z ln y ln ln z ln y (since a a ) Now use this property log 1 1 ln ln z ln y ln ln z ln y a y y log and simplify further: a 5 (b) Simplify using the properties of logarithm: log r Using the property loga loga loga y we have y 5 y log 5 log ( y ) log r r y Now using the property log a( y) log a log a y we can further simplify: log ( y) log r log log y log r n n y r a a log log log (since ) Now use this property log a y y log and simplify further: log log y log r log log y log r 5 a (c) Solve for t using the properties of logarithm: log 10( t 15) log10 t

31 We use the property log a log a y log a( y) to simplify log 10( t 15) log10 t log t( t 15) log ( t 15 t ) y Now use the definition of logarithm log t y a t to simplify a log 10( t 15 t ) t t t t t t t t t t ( 0) 5( 0) 0 ( t 0)( t 5) 0 t 0 0 or t 5 0 t 0 or t 5 But negative value of t is not acceptable since log is not defined for negative values, i.e. the argument of logarithm may not be negative. Hence t = 0. (d) Solve for : log ( 4) log ( ) 4 Using the property log log y log log ( 4) log ( ) 4 a a a y we have 4 log 4 y Now use the definition of logarithm log t y a t to simplify 4 a log log ( ) 16 ( ) (e) Solve for t: log t 1 log 5 log 11

32 Problem 4: Sales of a new product is given by S( t) 100 0log (t 1) where t is in years. (a) What were the sales after 1 year? Substitute t = 1 in S( t) 100 0log (t 1) S( t) 100 0log (t 1) S(1) log ( 1 1) S(1) log S(1) (since log 1) S(1) 10 (b) What were the sales after 1 years? Substitute t = 1 in S( t) 100 0log (t 1) S( t) 100 0log (t 1) S(1) log ( 1 1) S(1) log 7 S (1) 100 0log ( ) S(1) log (since log 1 and log y log y) S(1) S (1) 190 Problem 5: Find the amount of money in an account after 1 years if $5000 is deposited at 7% annual interest compounded as follows. (a) Quarterly (b) Daily (c) Continuously A( t) P 1 A() t Pe rt r n nt a a

33 r (a) Quarterly: You would need to find all the values in the formula A( t) P 1 n As given in the problem t = 1 years, r = 0.07, P = $5,000 and n is the number of periods in a year which in this case is 4, since there are 4 quarters in a year, so n = 4. Note that n = 4 means that interest is collected 4 times a year. Put the values in the formula: A( t) P 1 r n nt A(1) $5, A(1) $5, A(1) $11, nt r (b) Daily: You would need to find all the values in the formula A( t) P 1 n As given in the problem t = 1 years, r = 0.07, P = $5,000 and n is the number of periods in a year which in this case is 65, since there are 65 days in a year, so n = 65. Note that n = 65 means that interest is collected 65 times a year. Put the values in the formula: A( t) P 1 r n nt A(1) $5, A(1) $5, A(1) $11, (c) Continuously: If interest is collected continuously, i.e. at every instant then you would use rt this formula A() t Pe As given in the problem t = 1 years, r = 0.07 and P = $5,000. Put the values in the formula: rt A() t Pe A(1) $5, 000 e A(1) $11,581.8 Note the difference in the amount of interest paid for different compounding periods as indicated by the total amount received after 1 years: nt

34 Principal Amount Compounding Period Interest Charged Total after 1 years $5,000 Quarterly $6,498 $11,498 $5,000 Daily $6,581 $11,581 $5,000 Continuously $6,58 $11,58 Problem 6: A sample of 400g of lead 10 decays to polonium 10 according to the function 0.0t defined by A( t) 400e where t is in years. (a) How much lead will be left in the sample after 5 years? Substitute t = 5 years and simplify A( t) 400e A( t) 400e A( t) 179.7g (b) How long will it take the initial sample to decay to half of its original amount? This amount of time needed for the initial amount of 400g to reduce to 00g is called half-life, a very important concept in radioactive decay. 0.0t Use the formula A( t) 400e and set A(t) = 00g e 0.0t t 400e t e Now to solve for t use the definition of ln (natural log) or equivalently use the inverse function of e which is ln t e ln 1 0.0t ln e (Apply ln to both sides) 1 ln(1) ln 0.0 t (Since ln ln(1) ln ) t t ln 0.0t ln years

35 Problem 7: Carbon-14 has a half-life of about 5700 years. An ancient artifact found has about 0.87g of Carbon-14 left whereas it is estimated that at the time the artifact was built had 5 grams of Carbon-14 in it. How old is the artifact? This problem is a very important one. For solving a problem similar to this a Nobel Prize in chemistry was awarded and this discovery revolutionized archeology and our understanding of the history of the planet. The key to solving this problem is to find the decay rate. This could be done by using the fact that half-life of Carbon-14 is 5700 years. The general formula for eponential growth or decay kt is A() t A0e where A 0 is the initial amount and k is the eponential growth or decay. Positive value of k means eponential growth and negative value of k is eponential decay. kt In this problem A 0 = 5 grams plug this value in A() t A0e and then solve for k using the halflife concept. kt A() t A e 0 5 k e We use the concept that the initial amount of 5 grams will become 5/ grams in 5700 years and kt plugged these values in the general formula A() t A0e Hence we have the following: 5 k e 1 k 5700 e (by dividing both sides by 5) 1 k 5700 ln ln e (Applied natural log (ln) to both sides) 1 ln(1) ln k 5700 (Since ln ln(1) ln ) ln k 5700 So the formula for eponential decay becomes: Now the amount left is 0.87g hence we have A( t) 5e ln t 5700

36 A( t) 5e e ln t 5700 ln t e ln ln e ln t 5700 ln t 5700 ln ln t ln t ln 5 t ln 5700 t 14,00 years e Hence the artifact is about 14,00 years. Problem 8: Sketch the following eponential function and label the y-intercept. 1 f( ) 4 Simplify the given function: f ( ) f ( ) 4 f ( ) 4 f ( ) f( ) 4 1 f( ) 4 So we can graph the simplified function: f( ) 4 as is done below:

37 Problem 9: Consider the following functions: (a) Find f g( ) f ( ) 5 g( ) 5 By definition

38 f g( ) f ( g( )) f Hence f g( ) f ( g( )) (b) Find g f ( ) By definition: g f ( ) g( f ( )) g 5 ( 5 ) Hence g f ( ) g( f ( )) (c) Compare the results you found in parts (a) and (b). Can you draw a conclusion about the functions? Parts (a) and (b) tell me that f g( ) f ( g( )) and g f ( ) g( f ( )) i.e. f g( ) g f ( ) So the functions are inverses of each other. Also note that if two functions are inverses of each other, then their graphs are reflections of each other about the line y =. Problem 40: (a) Determine whether the points are the vertices of a right triangle:,5,,1,,7 1, 4,,8, 1,1

39 (b) Determine whether the points are the vertices of a right triangle: 1, 4, 1, 4, 1, 9 (c) Find the centers and the radii of the circles: 4 y 1y 10 0 Problem 41: (a) Epress the following in the standard form for ellipse and sketch: 5 4y 100 0

40 (b) Sketch the following hyperbola: y Problem 4: Two ships leave port at the same time, one heading due south and the other heading due east. Several hours later, they are 170 miles apart. If the ship traveling south traveled 70 miles farther than the other ship, how many miles did they each travel? Quadratic formula:

41 b b 4ac a Problem 4: Solve for y: y 1y 6 0 Problem 44: (a) On a bright sunny day you (6 ft) tall produce a shadow that is 8 ft long. A tree in front of you produces a shadow that is 150 ft long. What is the height of the tree?

42 (b) The intensity of light varies inversely as the square of the distance. If the intensity is 100 lumens at distance of 7 ft. then find the intensity are distance of 10 ft. Problem 45: Two cars are 00 miles apart and are traveling toward each other on the same road. They meet each other after driving hours. One car is traveling 0 mph slower than the other. Find the speeds of the cars.

43 Problem 46: Maya can do a job in 7 hours and her sister Roana can do the same job in 8 hours. Maya joins her sister Roana after Roana has worked on the job for hours and they work together to finish the rest of the job. How long did Maya and Roana work together? END OF FINAL EXAM REVIEW Midterm 1 Review Problems Math 1 Intermediate Algebra Fall 01 Name: (Please Print) Problem 1: Find the -intercept, y-intercept, and graph the straight line 4y 1

44 Problem : A cell phone company charges $40 per month and gives 400 minutes of free talk time anywhere in the U.S. Any additional talk time is charged at $0.5 per minute. (a) Create a linear model, i.e. a linear equation for this scenario. (b) Graph it.

45 (c) What will be the monthly payment if a person subscribing to this plan talks for 500 minutes in a month? Problem : Determine if the lines through the following pairs of points are parallel, perpendicular, or neither. Show your work to justify your answer. (a) 1 1,,, 1 6,,, 5 5

46 (b) ,,.75, ,1.5,, (c) 1, and,4 5,6 and 0,0 Problem 4: (a) Find the equation of a line that passes through the point (1, ) and is parallel to the line 5 4y 1 (b) Find the equation of a line that passes through the point (1, ) and is perpendicular to the line 5 4y 1

47 Problem 5: Vertical line test Problem 6: (a) Find domain for this function: f( ) 1 (b) Find domain: f ( ) 5

48 Problem 7: For a show at the opening night, 406 tickets were sold. Students paid $1.50 each, while non-students paid $.50 each. If a total of $ was collected, how many students and how many non-students attended? Problem 8: Miture problem: A solution has 60% acid in it, and another solution has 90% acid in it. You take 0 liters of 60% solution, how many liters of 90% solution do you need in order make a solution that is an 80% solution?

49 Problem 9: A mother wants to invest $5000 for her son s future education. She invests a portion of the money in a bank certificate of deposit (CD account) which earns 4% and the remainder in a savings bond that earns 7%. If the total interest earned after one year is $00, how much money was invested in the CD account? Problem 10: It takes 5 hours for a boat to go a distance of 15 miles against the current and hours in the direction of the current. What is the speed of the boat in still water and what is the speed of the current? (Guessing will give no credit. You need to setup the system of linear equations using two variables and solve it.)

50 Problem 11: In a right triangle one of the acute angles is 10 degrees less than times the other acute angle. Find the acute angles. Problem 1: Assume that you have a piggy bank with 66 coins. Some are quarters, and the rest are half-dollars. If the total value of the coins is $0.50, how many of each denomination do you have?

51 Print Name: Midterm Review Problems Math 1 Fall 01 Problem 1: Two cars are 00 miles apart and are traveling toward each other on the same road. They meet each other after driving hours. One car is traveling 0 mph slower than the other. Find the speeds of the cars. Problem : Divide: 4 a) 4 5 b)

52 Problem : a) Factor completely: b) Factor completely: 8 7 c) d) Factor completely: 9 y 5 y 4 e) Factor completely: f) Solve by factoring:

53 Problem 4: Factor and simplify Problem 5: Simplify: p p p p Problem 6: Simplify: 6 1 m m 7 5 m 7 m

54 Problem 7: Simplify: z z Problem 8: Solve for k: k k 6 k k k 4k Problem 9: Simplify and epress in terms of positive eponents only: m p 4m p mp m p

55 Problem 10: Simplify 1 y 9 y 6 Problem 11: Simplify completely and epress only in terms of positive eponents. 4n p 8n p np 4n p

56 Problem 1: Simplify the epression below using the rules of eponents, so that you have positive eponents only. 5 1 y z y 11 y z Problem 1: Simplify: 4 81y y Problem 14: Simplify: Simplify 15: 5 5

57 Problem 16: Solve for : 4 4 Problem 17: Simplify and epress it in the form (a + ib): 6i 1 i Problem 18: Simplify: i 011 Problem 19: Is ( + i) a solution to the equation 4 1 0?

58 Ignore anything below this for this eam.

59 Print Name: Final Eam Review Problems Math 1 Fall 011 Problem 1: Find the center and the radius of the circle given by y 6y 15 0

60 Problem : Find the overlapping region for these inequalities by graphing the inequalities in the rectangular coordinate system given below: y y Problem : Find the following for the parabola y (a) Does the parabola open up or down? Why? (b) Find the coordinates of the verte: -coordinate of the verte is given by b a (c) Find the -intercepts if they eist: (Use b b 4ac a if needed) (d) Find the y-intercept: (e) Sketch the parabola:

61 Problem 4: Solve for and label the result on the number line or epress it in the interval notation: 5 Problem 5: Properties of logarithm: y log a( y) loga loga y loga loga loga y loga yloga y Simplify using the properties of logarithm: ln z y

62 Problem 6: Find the amount of money in an account after 1 years if $5000 is deposited at 7% annual interest compounded as follows. (a) Quarterly (b) Daily (c) Continuously A( t) P 1 A() t Pe rt r n nt Problem 7: A sample of 400g of lead 10 decays to polonium 10 according to the function 0.0t defined by A( t) 400e where t is in years. (a) How much lead will be left in the sample after 5 years? (b) How long will it take the initial sample to decay to half of its original amount? This amount of time needed for the initial amount of 400g to reduce to 00g is called half-life, a very important concept in radioactive decay. Problem 8: Carbon-14 has a half-life of about 5700 years. An ancient artifact found has about 0.87g of Carbon-14 left whereas it is estimated that at the time the artifact was built had 5 grams of Carbon-14 in it. How old is the artifact?

63 Problem 9: Sketch the following eponential function and label the y-intercept. 1 f( )

64 Problem 10: Consider the following functions: (a) Find f g( ) f ( ) 5 g( ) 5 (b) Find g f ( ) (c) Compare the results you found in parts (a) and (b). Can you draw a conclusion about the functions? Problem 11: For a show at the opening night, 406 tickets were sold. Students paid $1.50 each, while non-students paid $.50 each. If a total of $ was collected, how many students and how many non-students attended? Problem 1: Two cars are 00 miles apart and are traveling toward each other on the same road. They meet each other after driving hours. One car is traveling 0 mph slower than the other. Find the speeds of the cars.

65 Problem 1: Divide: Problem 14: a) Factor completely: b) Factor completely: 8 7 c) Factor completely: d) Factor completely: 9 y 5 y 4

66 4 e) Factor completely: f) Solve by factoring: Problem 15: Simplify: 6 1 m m 7 5 m 7 m Problem 16: (a) Simplify and epress in terms of positive eponents only: m p 4m p mp m p

67 (b) Simplify 1 y 9 y 6 Problem 17: a) Rationalize the denominator: 5 5 b) Simplify and epress it in the form (a + ib): 6i 1 i (c) Simplify: i 011 Problem 18: Find equation of a line that is perpendicular to the line 4 7y 9 0 and goes through the point (1,1).

68 Problem 19: (a) Determine the domain and range for this relation:,, 1,0,,5, 1,0 (b) Given that f 7 find f(-) (c) Use vertical line test to determine whether the following figure forms a function: (d) Do the following pairs form a function? Eplanation required. (-, 1), (1, 1), (, ), (5, 4), (-, -1) (e) Find the domain for the following function and label it on the number line. y ( 5)

69 Problem 0: Solve the following system of linear equations by using the Gaussian elimination method. Write the operations you perform in each step. 4y 5y 1 END OF FINAL EXAM REVIEW Midterm Review Problems Math 1 Fall 011 Print Name:

70

71 Print Name: Midterm 1 Review Problems Math 1 Fall 011 Problem 1: Find the slopes of the lines and determine whether the lines are parallel, L1 :, 4 and, 1 perpendicular, or neither: L : 1,6 and 1, 1 Problem : Find equation of a line that is parallel to the line 5y 0 and goes through the point (1,1). Problem : Find equation of a line that is perpendicular to the line 4 7y 9 0 and goes through the point (1,1).

72 Problem 4: Find the overlapping region for these inequalities by graphing the inequalities in the rectangular coordinate system given below: y y Problem 5: (a) Determine the domain and range for this relation:,, 1,0,,5, 1,0 (b) Given that f 7 find f(-) (c) Use vertical line test to determine whether the following figure forms a function:

73 (d) Do the following pairs form a function? Eplanation required. (-, 1), (1, 1), (, ), (5, 4), (-, -1) (e) Find the domain for the following function and label it on the number line. y ( 5) Problem 6: For a show at the opening night, 406 tickets were sold. Students paid $1.50 each, while non-students paid $.50 each. If a total of $ was collected, how many students and how many non-students attended?

74 Problem 7: Miture problem: A solution has 60% acid in it, and another solution has 90% acid in it. You take 0 liters of 60% solution, how many liters of 90% solution do you need in order make a solution that is an 80% solution? Problem 8: Two cars are 00 miles apart and are traveling toward each other on the same road. They meet each other after driving hours. One car is traveling 0 mph slower than the other. Find the speeds of the cars.

75 Problem 9: It takes 5 hours for a boat to go a distance of 15 miles against the current and hours in the direction of the current. What is the speed of the boat in still water and what is the speed of the current? (Guessing will give no credit. You need to setup the system of linear equations using two variables and solve it.) Problem 10: Divide: 4 a) 4 5 b)

76 Problem 11: a) Factor completely: b) Factor completely: 8 7 c) d) Factor completely: 9 y 5 y 4

77 4 e) Factor completely: f) Solve by factoring: Problem 1: Solve the following system of linear equations by using the Gaussian elimination method. Write the operations you perform in each step. 4y 5y 1

78 Problem 1: (15 points) Solve the system of linear equations using Cramer s rule. You need to find the solution for only; do not find the answers for y and z. y z y z y z 7