course Mth 163 I will be turning in Assignment 1 tommorrow, January 22.
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10:47:00 `q001. Explain the difference between x - 2 / x + 4 and (x - 2) / (x + 4).
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RESPONSE --> The difference between x-2/x+4 and (x-2)/(x+4) is that the parentheses require that you perform those operations first whereas you may divide first in the problem without parentheses.
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10:49:45 The order of operations dictates that grouped expressions must be evaluated first, that exponentiation must be done before multiplication or division, which must be done before addition or subtraction. It makes a big difference whether you subtract the 2 from the 2 or divide the -2 by 4 first. If there are no parentheses you have to divide before you subtract: 2 - 2 / 2 + 4 = 2 - 1 + 4 (do multiplications and divisions before additions and subtractions) = 5 (add and subtract in indicated order) If there are parentheses you evaluate the grouped expressions first: (x - 2) / (x - 4) = (2 - 2) / ( 4 - 2) = 0 / 2 = 0.
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RESPONSE --> I did not remember that you had to divide before you subtract or add in the problem without parentheses. I did however remember that the order of operations requires that you perform operations in parentheses first.
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10:51:42 `q002. Explain the difference between 2 ^ x + 4 and 2 ^ (x + 4). Then evaluate each expression for x = 2. Note that a ^ b means to raise a to the b power. This process is called exponentiation, and the ^ symbol is used on most calculators, and in most computer algebra systems, to represent exponentiation.
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RESPONSE --> The difference in this problem is the first problem, 2^x + 4, has the 2 taken to a power of x. The second problem, 2^(x+4) takes to 2 to a power of x+4. 2^2 + 4 = 4 + 4 = 8 2^(2+4) = 2^6 = 64
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10:52:29 2 ^ x + 4 indicates that you are to raise 2 to the x power before adding the 4. 2 ^ (x + 4) indicates that you are to first evaluate x + 4, then raise 2 to this power. If x = 2, then 2 ^ x + 4 = 2 ^ 2 + 4 = 2 * 2 + 4 = 4 + 4 = 8. and 2 ^ (x + 4) = 2 ^ (2 + 4) = 2 ^ 6 = 2*2*2*2*2*2 = 64.
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RESPONSE --> I understand this concept. I did not say in my answer that you are required to add the x+4 first in the second problem.
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10:58:41 `q003. What is the numerator of the fraction in the expression x - 3 / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x? What is the denominator? What do you get when you evaluate the expression for x = 2?
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RESPONSE --> The numerator in this problem is 3. The denominator is [(2x-5)^2 * 3x +1]. When you evaluate the equation for x = 2: 2 - 3/[(2*2 - 5)^2 * 3(2) + 1] - 2 + 7(2) 2 - 3/[(4 - 5)^2 * 6 + 1] - 2 + 14 2 - 3/[(-1)^2 * 6 + 1] - 2 + 14 2 - 3/[1 * 6 + 1] - 2 + 14 2 - 3/7 - 2 + 14 14/7 - 3/7 - 14/7 + 98/7 95/7 is the answer.
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10:59:46 The numerator is 3. x isn't part of the fraction. / indicates division, which must always precede subtraction. Only the 3 is divided by [ (2x-5)^2 * 3x + 1 ] and only [ (2x-5)^2 * 3x + 1 ] divides 3. If we mean (x - 3) / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x we have to write it that way. The preceding comments show that the denominator is [ (2x-5)^2 * 3x + 1 ] Evaluating the expression for x = 2: - 3 / [ (2 * 2 - 5)^2 * 3(2) + 1 ] - 2 + 7*2 = 2 - 3 / [ (4 - 5)^2 * 6 + 1 ] - 2 + 14 = evaluate in parenthese; do multiplications outside parentheses 2 - 3 / [ (-1)^2 * 6 + 1 ] -2 + 14 = add inside parentheses 2 - 3 / [ 1 * 6 + 1 ] - 2 + 14 = exponentiate in bracketed term; 2 - 3 / 7 - 2 + 14 = evaluate in brackets 13 4/7 or 95/7 or about 13.57 add and subtract in order. The details of the calculation 2 - 3 / 7 - 2 + 14: Since multiplication precedes addition or subtraction the 3/7 must be done first, making 3/7 a fraction. Changing the order of the terms we have 2 - 2 + 14 - 3 / 7 = 14 - 3/7 = 98/7 - 3/7 = 95/7.
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RESPONSE --> I understand this concept. The order of operations really comes into play in this problem. You have to be very careful and make sure you are doing the problem in the correct order.
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11:02:23 `q004. Explain, step by step, how you evaluate the expression (x - 5) ^ 2x-1 + 3 / x-2 for x = 4.
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RESPONSE --> (x-5)^2x-1 + 3/x-2 evaluate for x if x=4 (4-5)^2(4)-1 + 3/4-2 (-1)^8-1 + 3/4-2 (-1)^7 + 3/2 (-1) + 1.5 so the answer would be 0.5 or 1/2
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11:04:18 We get (4-5)^2 * 4 - 1 + 3 / 1 - 4 = (-1)^2 * 4 - 1 + 3 / 4 - 2 evaluating the term in parentheses = 1 * 4 - 1 + 3 / 4 - 2 exponentiating (2 is the exponent, which is applied to -1 rather than multiplying the 2 by 4 = 4 - 1 + 3/4 - 2 noting that 3/4 is a fraction and adding and subtracting in order we get = 1 3/4 = 7 /4 (Note that we could group the expression as 4 - 1 - 2 + 3/4 = 1 + 3/4 = 1 3/4 = 7/4). COMMON ERROR: (4 - 5) ^ 2*4 - 1 + 3 / 4 - 2 = -1 ^ 2*4 - 1 + 3 / 4-2 = -1 ^ 8 -1 + 3 / 4 - 2. INSTRUCTOR COMMENTS: There are two errors here. In the second step you can't multiply 2 * 4 because you have (-1)^2, which must be done first. Exponentiation precedes multiplication. Also it isn't quite correct to write -1^2*4 at the beginning of the second step. If you were supposed to multiply 2 * 4 the expression would be (-1)^(2 * 4). Note also that the -1 needs to be grouped because the entire expression (-1) is taken to the power. -1^8 would be -1 because you would raise 1 to the power 8 before applying the - sign, which is effectively a multiplication by -1.
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RESPONSE --> I must have made one of those common errors. I included too much in the exponent, instead of just raising the (4-5) to the 2nd power, I raised it to the 2*4-1 or 7th power.
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y|Ӛ썡߰ Student Name: assignment #001 001. typewriter notation |ڤwb Student Name: assignment #002 002. Describing Graphs ɚʧܕҾMթ Student Name: assignment #003 003. PC1 questions ʯܻwǞtMǞ褹s| Student Name: assignment #001 003. PC1 questions
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11:10:01 `q001. It will be very important in this course for your instructor to see and understand the process of visualization and reasoning you use when you solve problems. This exercise is designed to give you a first experience with these ideas, and your instructor a first look at your work. Answer the following questions and explain in commonsense terms why your answer makes sense.
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RESPONSE --> Ok I will answer the questions and explain in common sense terms why the answer makes sense to me.
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11:13:56 For each question draw a picture to make sense out of the situation, and include a description of the picture. Samples Sample question and response Question: If a bundle of shingles covers 30 square feet, how many bundles are required to cover a 600 square foot roof? Response: We might draw a picture of a rectangle representing the area, dividing the rectangle into a number of smaller rectangles each representing the area covered by a single bundle. This makes it clear that we are dividing the roof area into 1-bundle areas, and makes it clear why we are going to have to divide. Reasoning this problem out in words, we can say that a single bundle would cover 30 square feet. Two bundles would cover 60 square feet. Three bundles would cover 90 square feet. We could continue in this manner until we reach 600 square feet. However, this would be cumbersome. It is more efficient to use the ideas of multiplication and division. We imagine grouping the 600 square feet into 30 square foot patches. There will be 600 / 30 patches and each will require exactly one bundle. We therefore require 600 / 30 bundles = 20 bundles. {}Your responses might not be as clear as the above, though they might be even more clear. I won't be looking for perfection, though I wouldn't object to it, but for a first effort at visualizing a situation and communicating a reasoning process. This is not something you are used to doing and it might take a few attempts before you can achieve good results, but you will get better every time you try. {}You might be unsure of what to do on a specific question. In such a case specific questions and expressions of confusion are also acceptable responses. Such a response must include your attempts to come up with a picture and reason out an explanation. For example your response might be Sample expression of confusion: I've drawn a picture of a pile of bundles and a roof but I'm not sure how to connect the two. I tried multiplying the number of bundles by the square feet of the roof but I got 18,000, and I know it won't take 18,000 bundles to cover the roof. How do you put the area covered by a bundle together with the roof area to get the number of bundles required? A poor response would be something like 'I don't know how to do #17'. This response reveals nothing of your attempt to understand the question and the situation. Nor does it ask a specific question. Incidentally, you might be tempted to quote rules or formulas about rates and velocities in answering these questions. Don't. This exercise isn't about being able to memorize rules and quote them. It is about expanding your ability to visualize, reason and communicate.
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RESPONSE --> I will make sure to describe in detail my reasoning process as to how I reached my answer. If I am confused by a problem, I will include a question as to what I do not understadn but will also include an attempt and my thoughts on what I have done to try and solve the problem.
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11:18:34 In your own words briefly summarize the instructions and the intent of this exercise.
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RESPONSE --> The instructions for this exercise are to get practice as to what the instructor is looking for when giving a response to questions. You should practice giving a detailed picture in words so that the instructor can visualize what you are doing to solve the problem. The intent of this exercise is to help you to get more detailed in your descriptions through practice.
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11:21:36 `q001. If you earn 50 dollars in 5 hours, at what average rate are you earning money, in dollars per hour?
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RESPONSE --> You are working 5 hours. You leave earning $50.00. In order to find the rate at which you are earning money you will need to divide the money you earn by the hours you have worked. 50 / 5 = 10. So you are earning money at a rate of $10.00 per hour.
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11:23:07 If you travel 300 miles in 6 hours, at what average rate are you traveling, in miles per hour?
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RESPONSE --> If you travel 300 miles, and you travel this in 6 hours, you will need to divide the miles travelled by the hours it took for you to go that far. So we see 300/6 = 50 So you travelled at an average speed of 50 mph.
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11:25:26 `q002. If a ball rolling down a grooved track travels 40 centimeters in 5 seconds, at what average rate is the ball moving, in centimeters per second?
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RESPONSE --> The ball travels 40 centimeters. It takes the ball 5 seconds to travel this length. So in order to find the average rate at which the ball is moving, we will divide the length travelled (40 cm) by the time it took the ball to get there (5 seconds). 40 / 5 = 8 So the ball is travelling at 8 cm per second.
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11:33:01 The preceding three questions illustrate the concept of a rate. In each case, to find the rate we divided the change in some quantity (the number of dollars or the distance, in these examples) by the time required for the change (the number of hours or seconds, in these examples). Explain in your own words what is meant by the idea of a rate.
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RESPONSE --> Rate is an average of something, whether it be money, speed, or some other measure we are looking to find the average of. We find the average by deciphering what it is we are looking to find and dividing to get the rate for that problem.
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11:35:37 `q003. If you are earning money at the average rate of 15 dollars per hour, how much do you earn in 6 hours?
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RESPONSE --> If you earn money at an average rate of $15/hour, and you work for 6 hours, you will mulitply the amount per hour (15) by the number of hours worked (6). 15 * 6 = 90 so you would earn an average of $90.00 for working 6 hours at an average rate of $15/hour.
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11:37:50 If you are traveling at an average rate of 60 miles per hour, how far do you travel in 9 hours?
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RESPONSE --> If we travel at an average rate of 60 miles per hour and travel for 9 hours......you would find the distance you would travel by multiplying the average rate of speed (60) by the time you travelled (9) 60 * 9 = 540 miles on average
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11:39:30 `q004. If a ball travels at and average rate of 13 centimeters per second, how far does it travel in 3 seconds?
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RESPONSE --> If the ball is travelling at 13 cm/sec and it rolls for 3 seconds, you would find the diustance travelled by mulitplying the average rate of speed (13) by the length of time it travels (3) 13 * 3 = 39 cm in 3 seconds
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11:40:43 In the preceding three exercises you turned the concept of a rate around. You were given the rate and the change in the clock time, and you calculated the change in the quantity. Explain in your own words how this increases your understanding of the concept of a rate.
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RESPONSE --> Rate can also be found by being given the original numbers, instead of being told how far we travelled, we figured out how far we would travel by multiplication instead of division.
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11:42:03 `q005. How long does it take to earn 100 dollars at an average rate of 4 dollars per hour?
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RESPONSE --> If you are making $4/hr and you need $100, you will divide the amount of money you need (100) by the rate at which you earn the money (4) 100/4 = 25, so it will take you 25 hours to earn $100.00.
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11:43:57 How long does it take to travel 500 miles at an average rate of 25 miles per hour?
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RESPONSE --> If you travel at 25 mph and want to go 500 miles, you will find out how long it will take to get there by dividing the distance you wish to travel (500) by the rate at which you will be travelling 500/25 = 20 so it will take 20 hours to go 500 miles at 25 mph.
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11:46:12 `q006. How long does it take a rolling ball to travel 80 centimeters at an average rate of 16 centimeters per second?
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RESPONSE --> If the ball is rolling at an average rate of 16 cm/sec and you want to know how long it will take to go 80 cm, you will divide the length you want the ball to go (80) by the average speed it travels (16) 80/16 = 5 so it would take 5 seconds to roll 80 cm at a rate of 16 cm/sec
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11:48:58 In the preceding three exercises you again expanded your concept of the idea of a rate. Explain how these problems illustrate the concept of a rate.
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RESPONSE --> Rate can be found by both division and multiplication. It applies to many different aspects of life, from how much you earn in the workplace to how long it will take to go and visit grandma.
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瘒Ծ[߈z縖 Student Name: assignment #001 001. Rates
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12:06:09 `q001. You should copy and paste these instructions to a word processor for reference. However you can always view them, as well as everything else that has appeared in this box, by clicking the 'Display Everything' button. 1. For the next question or answer, you click on 'Next Question / Answer' button above the box at top left until a question has been posed. Once a question has been posed you are to answer before you click again on this button. 2. Before clicking for an answer, type your best answer to the current question into the box to the right, then clip on the 'Enter Answer' button. 3. After entering your answer you will click on 'Next Question / Answer' to view the answer to the question. Do not tamper with the information displayed in the left-hand box. 4. If your answer was incorrect, incomplete or would otherwise require revision, you will enter a self-critique. If you learned something from the answer, you need to restate it in your own words in order to reinforce your learning. If there is something you feel you should note for future reference, you should make a note in your own words. Go to the response box (the right-hand box) and type in a self-critique and/or notes, as appropriate. Do not copy and paste anything from the left-hand box, since that information will be saved in any case. 5. If you wish to save your response to your Notes file you may choose to click on the 'Save As Notes' button rather than the 'Enter Answer' button. Doing so will save your work for your future reference. Your work will be saved in a Notes file in the c:\vhmthphy folder. The title of the Notes file will also include the name you gave when you started the program. 6. After clicking either the 'Enter Response' or the 'Save as Notes' button, click on 'Next Question / Answer' and proceed in a similar manner. In the right-hand box briefly describe your understanding of these instructions, then click 'Enter Answer'.
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RESPONSE --> I have copied the instructions into a word document for viewing at later times if needed. As you progress through the problems you only need to click on Next Question/Answer and to enter a response simply click on Enter Response. Your answer should represent your best effort to answer a question. If you fail to mention a point, you can have a second chance in the following box as you critique your answer. If the answer is incorrect altogether, you will need to find where you went wrong and explain so the instructor will know that you now understand the concept. You can save your responses to the Notes section for viewing at a later time.
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12:06:19 Your answer has been noted. Enter 'ok' in the Response Box and click on Enter Response, then click on Next Question/Answer for the first real question.
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RESPONSE --> Ok
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12:09:01 `q002. Note that there are 10 questions in this assignment. The questions are of increasing difficulty--the first questions are fairly easy but later questions are very tricky. The main purposes of these exercises are to refine your thinking about rates, and to see how you process challenging information. Continue as far as you can until you are completely lost. Students who are prepared for the highest-level math courses might not ever get lost. If you make $50 in 5 hr, then at what rate are you earning money?
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RESPONSE --> If you work for 5 hours and you earn $50, in order to find the rate at which you are earning money you will divide the amount you made (50) by the length of time it took for you to make it (5) 50 / 5 = 10 so you are earning money at a rate of $10 per hour.
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12:10:33 The rate at which you are earning money is the number of dollars per hour you are earning. You are earning money at the rate of 50 dollars / (5 hours) = 10 dollars / hour. It is very likely that you immediately came up with the $10 / hour because almosteveryone is familiar with the concept of the pay rate, the number of dollars per hour. Note carefully that the pay rate is found by dividing the quantity earned by the time required to earn it. Time rates in general are found by dividing an accumulated quantity by the time required to accumulate it. You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.
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RESPONSE --> I understand the concept of average rate of pay and how the rate is found given the amount earned and time it took to earn the amount.
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12:13:55 `q003.If you make $60,000 per year then how much do you make per month?
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RESPONSE --> If you earn an annual salary of $60,000 per year and you wish to know how much you earn per month, you simply need to divide the amount you earn per year (60,000) by the units in a year (12) 60,000/12 = 5,000 So earning $60,000 a year, you will earn $5,000 per month.
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12:14:59 Most people will very quickly see that we need to divide $60,000 by 12 months, giving us 60,000 dollars / (12 months) = 5000 dollars / month. Note that again we have found a time rate, dividing the accumulated quantity by the time required to accumulate it. You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.
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RESPONSE --> I understand that to find the amount earned per month given the amount earned in a year, you simply need to divide by the units in a year, or 12 months.
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12:18:28 `q004. Suppose that the $60,000 is made in a year by a small business. Would be more appropriate to say that the business makes $5000 per month, or that the business makes an average of $5000 per month?
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RESPONSE --> It would be more appropriate to say that a small business earns an average of $5,000 per month. Due to the nature of business, amounts earned vary month to month. Whereas an individual earns the same each month as salary, though if hourly the amount may vary as well though not as dramatically as in business.
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12:18:29 `q004. Suppose that the $60,000 is made in a year by a small business. Would be more appropriate to say that the business makes $5000 per month, or that the business makes an average of $5000 per month?
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RESPONSE -->
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12:19:26 Small businesses do not usually make the same amount of money every month. The amount made depends on the demand for the services or commodities provided by the business, and there are often seasonal fluctuations in addition to other market fluctuations. It is almost certain that a small business making $60,000 per year will make more than $5000 in some months and less than $5000 in others. Therefore it is much more appropriate to say that the business makes and average of $5000 per month. You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.
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RESPONSE --> I understand that the very nature of business itself will cause income to vary from month to month.
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12:21:23 `q005. If you travel 300 miles in 6 hours, at what average rate are you covering distance, and why do we say average rate instead of just plain rate?
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RESPONSE --> If you travel 300 miles in 6 hours and wish to find the average rate at which you were travelling, you will divide the distance driven (300) by the time it took to drive the distance (6). 300/6 = 50 So you drove at an average speed of 50 mph. We say an average rate due to the fact that there were times where you probably went slower due to a stop or traffic as well as times where you went somewhat faster.
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12:26:43 The average rate is 50 miles per hour, or 50 miles / hour. This is obtained by dividing the accumulated quantity, the 300 miles, by the time required to accumulate it, obtaining ave rate = 300 miles / ( 6 hours) = 50 miles / hour. Note that the rate at which distance is covered is called speed. The car has an average speed of 50 miles/hour. We say 'average rate' in this case because it is almost certain that slight changes in pressure on the accelerator, traffic conditions and other factors ensure that the speed will sometimes be greater than 50 miles/hour and sometimes less than 50 miles/hour; the 50 miles/hour we obtain from the given information is clearly and overall average of the velocities. You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.
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RESPONSE --> I understand the concept of rate as related to speed in this problem as well as why it would be called an average rate as opposed to a rate.
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12:32:41 `q006. If you use 60 gallons of gasoline on a 1200 mile trip, then at what average rate are you using gasoline, with respect to miles traveled?
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RESPONSE --> Using 60 gallons of gas in 1200 miles, in order to find how much you are using per mile, you will need to visualize 60 gallons of gas and 1200 miles of road. We will need to divide the gallons of gas (60) by the number of miles travelled (1200) 60/1200 = 0.05 gallons per mile To check we can multiply 1200 by 0.05 and get an answer of 60.
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12:38:46 The rate of change of one quantity with respect to another is the change in the first quantity, divided by the change in the second. As in previous examples, we found the rate at which money was made with respect to time by dividing the amount of money made by the time required to make it. By analogy, the rate at which we use fuel with respect to miles traveled is the change in the amount of fuel divided by the number of miles traveled. In this case we use 60 gallons of fuel in 1200 miles, so the average rate it 60 gal / (1200 miles) = .05 gallons / mile. Note that this question didn't ask for miles per gallon. Miles per gallon is an appropriate and common calculation, but it measures the rate at which miles are covered with respect to the amount of fuel used. Be sure you see the difference. Note that in this problem we again have here an example of a rate, but unlike previous instances this rate is not calculated with respect to time. This rate is calculated with respect to the amount of fuel used. We divide the accumulated quantity, in this case miles, by the amount of fuel required to cover those miles. Note that again we call the result of this problem an average rate because there are always at least subtle differences in driving conditions that require the use of more fuel on some miles than on others. It's very important to understand the phrase 'with respect to'. Whether the calculation makes sense or not, it is defined by the order of the terms. In this case gallons / mile tells you how many gallons you are burning, on the average, per mile. This concept is not as familiar as miles / gallon, but except for familiarity it's technically no more difficult. You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.
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RESPONSE --> I understand the concept of finding the information with respect to a particular component such as we have in this problem.
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12:40:34 `q007. The word 'average' generally connotes something like adding two quantities and dividing by 2, or adding several quantities and dividing by the number of quantities we added. Why is it that we are calculating average rates but we aren't adding anything?
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RESPONSE --> We are given a total of miles travelled or money earned which we then divide into in order to find the average. If we know the total of money earned or total miles travelled there is no reason to add, only divide.
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12:43:10 The word 'average' in the context of the dollars / month, miles / gallon types of questions we have been answering was used because we expect that in different months different amounts were earned, or that over different parts of the trip the gas mileage might have varied, but that if we knew all the individual quantities (e.g., the dollars earned each month, the number of gallons used with each mile) and averaged them in the usual manner, we would get the .05 gallons / mile, or the $5000 / month. In a sense we have already added up all the dollars earned in each month, or the miles traveled on each gallon, and we have obtained the total $60,000 or 1200 miles. Thus when we divide by the number of months or the number of gallons, we are in fact calculating an average rate. You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.
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RESPONSE --> I have made note of a better way to explain this point, but I do understand that we are given the total amount so therefore we do not have to add any amounts together.
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12:54:37 `q008. In a study of how lifting strength is influenced by various ways of training, a study group was divided into 2 subgroups of equally matched individuals. The first group did 10 pushups per day for a year and the second group did 50 pushups per day for year. At the end of the year to lifting strength of the first group averaged 147 pounds, while that of the second group averaged 162 pounds. At what average rate did lifting strength increase per daily pushup?
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RESPONSE --> There are 2 groups 1st group - 10 push ups per day 2nd group - 50 push ups per day 1st group - new average - 147 lbs 2nd group - new average - 162 lbs 1st group - 10 push ups per day for 365 days with an average lifting strength of 147 lbs. we first must know how many push ups were done in a year......10(365) = 3650 push ups a year then we divide the average strength (147) by the number of push ups (3650) 147/3650 = 0.0402739726 or 0.04 per daily push up 2nd group - 50 push ups per day for 365 days with an average lifting strength of 162 lbs. we first must know how many push ups were done in a year....50(365) = 18,250 push ups per year then we divide the average strength (162) by the number of push ups (18,250) 162/18,250 = 0.008 per daily push up
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13:00:31 The second group had 15 pounds more lifting strength as a result of doing 40 more daily pushups than the first. The desired rate is therefore 15 pounds / 40 pushups = .375 pounds / pushup. You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.
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RESPONSE --> I did not follow up on the difference between the two groups. With the second group having 15 lbs more strength than the first group since they performed 40 more push ups. So we should divide the difference of 15 lbs by the number of additional push ups, 40. 15/40 = 0.375 So the difference was 0.375 lbs/push up
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13:05:10 `q009. In another part of the study, participants all did 30 pushups per day, but one group did pushups with a 10-pound weight on their shoulders while the other used a 30-pound weight. At the end of the study, the first group had an average lifting strength of 171 pounds, while the second had an average lifting strength of 188 pounds. At what average rate did lifting strength increase with respect to the added shoulder weight?
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RESPONSE --> The first group lifted 10 lbs additional during the 30 push ups The second group lifted an additional 30 pounds during the 30 push ups There is a difference of 20 lbs. The difference in the lifting is found by 188 - 171 = 17 so 17/20 = 0.85
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13:07:34 The difference in lifting strength was 17 pounds, as a result of a 20 pound difference in added weight. The average rate at which strength increases with respect added weight would therefore be 17 lifting pounds / (20 added pounds) = .85 lifting pounds / added pound. The strength advantage was .85 lifting pounds per pound of added weight, on the average. You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.
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RESPONSE --> I understood the question this time around and got the answer correct. This was another question that inlcuded ""with respect to"" which allows us to know what we need to solve for and therefore which number to divide into.
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13:10:15 `q010. During a race, a runner passes the 100-meter mark 12 seconds after the start and the 200-meter mark 22 seconds after the start. At what average rate was the runner covering distance between those two positions?
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RESPONSE --> The first position time was 100/12 = 8.33 meters per second The second position was reached at a rate of 200/22 = 9.09 meters per second. 8.33 + 9.09 = 17.12 17.12/2 = 8.56 meters per second average speed between the last two points.
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13:12:54 The runner traveled 100 meters between the two positions, and required 10 seconds to do so. The average rate at which the runner was covering distance was therefore 100 meters / (10 seconds) = 10 meters / second. Again this is an average rate; at different positions in his stride the runner would clearly be traveling at slightly different speeds. You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.
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RESPONSE --> I made the mistake of using the average of the times it took to run the first position and then the second position and divided by 2. I should have taken the difference between the two points - 100 meters and the diffrence in the time - 10 seconds - and divided the distance 100m divided by the time 10s. 100 / 10 = 10 meters per second average speed
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13:19:38 `q011. During a race, a runner passes the 100-meter mark moving at 10 meters / second, and the 200-meter mark moving at 9 meters / second. What is your best estimate of how long it takes the runner to cover the 100 meter distance?
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RESPONSE --> passes 100 m at 10m/sec passes 200 m at 9 m/sec 100 m difference in distance 1 second difference in speed This is where I get confused. There is a difference of 100 m in the distance travelled and 1 second difference in the speed. So I would divide 1/100 to get the amount to multiply by in order to get the answer.
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13:22:59 At 10 meters/sec, the runner would require 10 seconds to travel 100 meters. However the runner seems to be slowing, and will therefore require more than 10 seconds to travel the 100 meters. We don't know what the runner's average speed is, we only know that it goes from 10 m/s to 9 m/s. The simplest estimate we could make would be that the average speed is the average of 10 m/s and 9 m/s, or (10 m/s + 9 m/s ) / 2 = 9.5 m/s. Taking this approximation as the average rate, the time required to travel 100 meters will be (100 meters) / (9.5 m/s) = 10.5 sec, approx.. Note that simply averaging the 10 m/s and the 9 m/s might not be the best way to approximate the average rate--for example we if we knew enough about the situation we might expect that this runner would maintain the 10 m/s for most of the remaining 100 meters, and simply tire during the last few seconds. However we were not given this information, and we don't add extraneous assumptions without good cause. So the approximation we used here is pretty close to the best we can do with the given information. You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.
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RESPONSE --> The speed slows from 10 m/sec to 9 m/sec. Take the average of the two times 10 + 9/ 2 = 19 / 2 = 9.5 meters per second. Then we would divide 100 by 9.5 100 / 9.5 - 10.5 seconds per meter as an approximate rate of speed.
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13:24:13 `q012. We just averaged two quantities, adding them in dividing by 2, to find an average rate. We didn't do that before. Why we do it now?
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RESPONSE --> As we were not given the total of the quantities this time and the speed changed, we were required to add the two speeds and then divide by 2 in order to get a good average rate of speed.
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13:25:11 In previous examples the quantities weren't rates. We were given the amount of change of some accumulating quantity, and the change in time or in some other quantity on which the first was dependent (e.g., dollars and months, miles and gallons). Here we are given 2 rates, 10 m/s and 9 m/s, in a situation where we need an average rate in order to answer a question. Within this context, averaging the 2 rates was an appropriate tactic. You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.
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RESPONSE --> We had 2 rates here that we needed to get an average for so we had to average the 2 speeds and divide by 2.
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zS|~nyI姸 Student Name: assignment #002 002. Describing Graphs
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13:47:54 `q001. You will frequently need to describe the graphs you have constructed in this course. This exercise is designed to get you used to some of the terminology we use to describe graphs. Please complete this exercise and email your work to the instructor. Problem 1. We make a table for y = 2x + 7 as follows: We construct two columns, and label the first column 'x' and the second 'y'. Put the numbers -3, -2, -1, -, 1, 2, 3 in the 'x' column. We substitute -3 into the expression and get y = 2(-3) + 7 = 1. We substitute -2 and get y = 2(-2) + 7 = 3. Substituting the remaining numbers we get y values 5, 7, 9, 11 and 13. These numbers go into the second column, each next to the x value from which it was obtained. We then graph these points on a set of x-y coordinate axes. Noting that these points lie on a straight line, we then construct the line through the points. Now make a table for and graph the function y = 3x - 4. Identify the intercepts of the graph, i.e., the points where the graph goes through the x and the y axes.
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RESPONSE --> y = 3x - 4 x y -3 -13 -2 -10 -1 -7 0 -4 1 -1 2 2 3 5 The graph is a straight line that has intercepts of (1.5, 0) and (0, -4).
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13:49:09 The graph goes through the x axis when y = 0 and through the y axis when x = 0. The x-intercept is therefore when 0 = 3x - 4, so 4 = 3x and x = 4/3. The y-intercept is when y = 3 * 0 - 4 = -4. Thus the x intercept is at (4/3, 0) and the y intercept is at (0, -4). Your graph should confirm this.
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RESPONSE --> I understand the concept addressed in this problem. I had the same intercepts as shown here of (1.5, 0) and (0, -4). The graph I made for this equation did confirm the correct points.
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13:51:17 `q002. Does the steepness of the graph in the preceding exercise (of the function y = 3x - 4) change? If so describe how it changes.
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RESPONSE --> The graph does become steeper as it goes toward the positive side of the x axis. There is a smaller difference in the numbers as we approach the x values of 0 - 3.
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13:52:17 The graph forms a straight line with no change in steepness.
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RESPONSE --> I must have had a bad graph as it seemed that the numbers became smaller as I approached the positive side of the x axis.
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13:54:00 `q003. What is the slope of the graph of the preceding two exercises (the function ia y = 3x - 4;slope is rise / run between two points of the graph)?
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RESPONSE --> The slope could be found by using the two intercepts of (0, -4) and (1.5, 0) (0-4) / (1.5-0) = -4 / 1.5 = -2.67 as the slope
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13:57:04 Between any two points of the graph rise / run = 3. For example, when x = 2 we have y = 3 * 2 - 4 = 2 and when x = 8 we have y = 3 * 8 - 4 = 20. Between these points the rise is 20 - 2 = 18 and the run is 8 - 2 = 6 so the slope is rise / run = 18 / 6 = 3. Note that 3 is the coefficient of x in y = 3x - 4. Note the following for reference in subsequent problems: The graph of this function is a straight line. The graph increases as we move from left to right. We therefore say that the graph is increasing, and that it is increasing at constant rate because the steepness of a straight line doesn't change.
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RESPONSE --> Ok I am apprently rusty in the idea of slope. I should have substituted a number into the equation such as the 2 as done on the side. where x = 2 y = 3*2 - 4 = 2 y = 6 - 4 = 2 where x = 8 y = 3*8 - 4 y = 24 - 4 = 20 The rise is found 20 - 2 = 18 The run is found by 8 - 2 = 6 The slope is found by 18 / 6 = 3
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14:08:19 `q004. Make a table of y vs. x for y = x^2. Graph y = x^2 between x = 0 and x = 3. Would you say that the graph is increasing or decreasing? Does the steepness of the graph change and if so, how? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
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RESPONSE --> y = x^2 x y 0 0 1 1 2 4 3 9 After making the graph I would say that it is increasing. The graph is becoming steeper as the x values grow larger. I woule say that the graph is increasing at an increasing rate.
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14:09:06 Graph points include (0,0), (1,1), (2,4) and (3,9). The y values are 0, 1, 4 and 9, which increase as we move from left to right. The increases between these points are 1, 3 and 5, so the graph not only increases, it increases at an increasing rate.
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RESPONSE --> I understand this concept, just needed to take more time on the graph so it would reflect correctly the actual steepness of the line.
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14:14:14 `q005. Make a table of y vs. x for y = x^2. Graph y = x^2 between x = -3 and x = 0. Would you say that the graph is increasing or decreasing? Does the steepness of the graph change and if so, how? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
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RESPONSE --> y = x^2 x y -3 9 -2 4 -1 1 0 0 I would say that the graph is increasing. The steepness of the graph increases as the y values become larger. I would say the graph is increasing at an increasing rate.
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14:16:31 From left to right the graph is decreasing (points (-3,9), (-2,4), (-1,1), (0,0) show y values 9, 4, 1, 0 as we move from left to right ). The magnitudes of the changes in x from 9 to 4 to 1 to 0 decrease, so the steepness is decreasing. Thus the graph is decreasing, but more and more slowly. We therefore say that the graph is decreasing at a decreasing rate.
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RESPONSE --> Ok I misunderstood. I thought since we went from -3 to 0 and the graph became smaller, however I was looking at the graph going from right to left instead of left to right. So since it was larger at the value of -3 and smaller at the 0 the graph would be decreasing. Also because it is not as steep from the -2 value to the 0 it is decreasing at a decreasing rate.
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14:20:31 `q006. Make a table of y vs. x for y = `sqrt(x). [note: `sqrt(x) means 'the square root of x']. Graph y = `sqrt(x) between x = 0 and x = 3. Would you say that the graph is increasing or decreasing? Does the steepness of the graph change and if so, how? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
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RESPONSE --> y = 'sqrt(x) x y 0 0 1 1 2 1.41 3 1.73 I would say the graph is increasing. The graph is steep in the beginning and then seems to become more rounded as it goes up. I would say that the graph is increasing at a decreasing rate.
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14:21:41 If you use x values 0, 1, 2, 3, 4 you will obtain graph points (0,0), (1,1), (2,1.414), (3. 1.732), (4,2). The y value changes by less and less for every succeeding x value. Thus the steepness of the graph is decreasing. The graph would be increasing at a decreasing rate.{}{} If the graph respresents the profile of a hill, the hill starts out very steep but gets easier and easier to climb. You are still climbing but you go up by less with each step, so the rate of increase is decreasing. {}{}If your graph doesn't look like this then you probably are not using a consistent scale for at least one of the axes. If your graph isn't as desribed take another look at your plot and make a note in your response indicating any difficulties.
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RESPONSE --> I understand this concept. I was able to determine that the graph was increasing though becoming less steep as it rises, it is increasing at a decreasing rate.
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14:27:31 `q007. Make a table of y vs. x for y = 5 * 2^(-x). Graph y = 5 * 2^(-x) between x = 0 and x = 3. Would you say that the graph is increasing or decreasing? Does the steepness of the graph change and if so, how? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
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RESPONSE --> y = 5 * 2^(-x) x y 0 5 1 2.5 2 1.25 3 0.625 I would say the graph is decreasing. The graph is steep with the initial point of (0, 5) and then seems to slope more gently. I would say the graph is decreasing at a decreasing rate.
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14:28:22 ** From basic algebra recall that a^(-b) = 1 / (a^b). So, for example: 2^-2 = 1 / (2^2) = 1/4, so 5 * 2^-2 = 5 * 1/4 = 5/4. 5* 2^-3 = 5 * (1 / 2^3) = 5 * 1/8 = 5/8. Etc. The decimal equivalents of the values for x = 0 to x = 3 will be 5, 2.5, 1.25, .625. These values decrease, but by less and less each time. The graph is therefore decreasing at a decreasing rate. **
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RESPONSE --> I understand this concept of the graph steepness and the idea of increasing and dereasing graphs. As well as the idea of decreasing or increasing at a decreasing or increasing rate.
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14:30:56 `q008. Suppose you stand still in front of a driveway. A car starts out next to you and moves away from you, traveling faster and faster. If y represents the distance from you to the car and t represents the time in seconds since the car started out, would a graph of y vs. t be increasing or decreasing? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?
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RESPONSE --> y = distance from you to car t = time in seconds since the car started out A graph of y vs. t in this instance would be increasing as the speed of the car increases and the distance from you to the car also increases. I would say the graph would increase at an increasing rate.
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14:31:45 ** The speed of the car increases so it goes further each second. On a graph of distance vs. clock time there would be a greater change in distance with each second, which would cause a greater slope with each subsequent second. The graph would therefore be increasing at an increasing rate. **
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RESPONSE --> I understand this concept of the car getting further away with each second as well as the speed of the car growing with each passing second.
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ԓRȍvۢxdӻ Student Name: assignment #003 003. PC1 questions
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14:35:38 `q001 A straight line connects the points (3, 5) and (7, 17), while another straight line continues on from (7, 17) to the point (10, 29). Which line is steeper and on what basis to you claim your result?
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RESPONSE --> (3, 5) and (7, 17) 17 - 5 / 7 - 5 = 12/2 = 6 (7, 17) and (10, 29) 29 - 17/10 - 7 = 12 / 3 = 4 The line connecting (3, 5) and (7, 17) is steeper as it has a larger slope.
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14:37:30 The point (3,5) has x coordinate 3 and y coordinate 5. The point (7, 17) has x coordinate 7 and y coordinate 17. To move from (3,5) to (7, 17) we must therefore move 4 units in the x direction and 12 units in the y direction. Thus between (3,5) and (7,17) the rise is 12 and the run is 4, so the rise/run ratio is 12/4 = 3. Between (7,10) and (10,29) the rise is also 12 but the run is only 3--same rise for less run, therefore more slope. The rise/run ratio here is 12/3 = 4.
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RESPONSE --> I made a mistake. I subtracted the 5 in the (3, 5) instead of the 3 in the x value and therefore came up with a value of 6 (12/2) and therefore I thought it had a larger slope. I now see what I did in the preceeding problem.
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14:42:32 `q002. The expression (x-2) * (2x+5) is zero when x = 2 and when x = -2.5. Without using a calculator verify this, and explain why these two values of x, and only these two values of x, can make the expression zero.
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RESPONSE --> (x-2) * (2x + 5) when x = 2 (2-2) * (2*2 + 5) 0 * (4 + 5) 0 * 9 = 0 when x = -2.5 (-2.5 - 2) * (2 * -2.5 + 5) (-4.5) * (-5 + 5) (-4.5) (0) = 0 The reason that these two values cause the equation to be zero is that each one cancels out one side of the equation. The 2 will cancel out the (x-2) portion of the equation causing it to become 0 which will cause the equation to equal 0. The -2.5 will cancel out the (2x +5) making that portion of the equation to be 0, so the equation will equal 0. Any other number will leave a balance on either side therefore causing the equation not to equal zero.
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14:43:42 If x = 2 then x-2 = 2 - 2 = 0, which makes the product (x -2) * (2x + 5) zero. If x = -2.5 then 2x + 5 = 2 (-2.5) + 5 = -5 + 5 = 0.which makes the product (x -2) * (2x + 5) zero. The only way to product (x-2)(2x+5) can be zero is if either (x -2) or (2x + 5) is zero. Note that (x-2)(2x+5) can be expanded using the Distributive Law to get x(2x+5) - 2(2x+5). Then again using the distributive law we get 2x^2 + 5x - 4x - 10 which simplifies to 2x^2 + x - 10. However this doesn't help us find the x values which make the expression zero. We are better off to look at the factored form.
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RESPONSE --> I understand that either side of the equation must equal zero before the equation can equal zero.
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14:45:43 `q003. For what x values will the expression (3x - 6) * (x + 4) * (x^2 - 4) be zero?
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RESPONSE --> For the equation (3x - 6) * (x + 4) * (x^2 - 4) the following values will cause it to equal zero (3x - 6).........2 (x + 4)..........-4 (x^2 - 4).........2 So the two numbers that will cause this expression to equal zero are 2 and -4.
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14:46:34 In order for the expression to be zero we must have 3x-6 = 0 or x+4=0 or x^2-4=0. 3x-6 = 0 is rearranged to 3x = 6 then to x = 6 / 3 = 2. So when x=2, 3x-6 = 0 and the entire product (3x - 6) * (x + 4) * (x^2 - 4) must be zero. x+4 = 0 gives us x = -4. So when x=-4, x+4 = 0 and the entire product (3x - 6) * (x + 4) * (x^2 - 4) must be zero. x^2-4 = 0 is rearranged to x^2 = 4 which has solutions x = + - `sqrt(4) or + - 2. So when x=2 or when x = -2, x^2 - 4 = 0 and the entire product (3x - 6) * (x + 4) * (x^2 - 4) must be zero. We therefore see that (3x - 6) * (x + 4) * (x^2 - 4) = 0 when x = 2, or -4, or -2. These are the only values of x which can yield zero.**
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RESPONSE --> I forgot the mention the -2. This can be used in the final group of number since a negative number squared will yield a positive number. Other than that I understand this concept.
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15:02:41 `q004. One straight line segment connects the points (3,5) and (7,9) while another connects the points (10,2) and (50,4). From each of the four points a line segment is drawn directly down to the x axis, forming two trapezoids. Which trapezoid has the greater area? Try to justify your answer with something more precise than, for example, 'from a sketch I can see that this one is much bigger so it must have the greater area'.
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RESPONSE --> The trapeziod that has the points (10, 2) and (50, 4) will have the greater area. It may not be as tall as the trapeziod containing point (3, 5) and (7, 9), but it is much wider and therefore has more units included in its area than the other.
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15:03:37 Your sketch should show that while the first trapezoid averages a little more than double the altitude of the second, the second is clearly much more than twice as wide and hence has the greater area. To justify this a little more precisely, the first trapezoid, which runs from x = 3 to x = 7, is 4 units wide while the second runs from x = 10 and to x = 50 and hence has a width of 40 units. The altitudes of the first trapezoid are 5 and 9,so the average altitude of the first is 7. The average altitude of the second is the average of the altitudes 2 and 4, or 3. So the first trapezoid is over twice as high, on the average, as the first. However the second is 10 times as wide, so the second trapezoid must have the greater area. This is all the reasoning we need to answer the question. We could of course multiply average altitude by width for each trapezoid, obtaining area 7 * 4 = 28 for the first and 3 * 40 = 120 for the second. However if all we need to know is which trapezoid has a greater area, we need not bother with this step.
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RESPONSE --> I understand this concept and have made further notes as to the math featured in this problem.
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15:08:58 * `q005. Sketch graphs of y = x^2, y = 1/x and y = `sqrt(x) [note: `sqrt(x) means 'the square root of x'] for x > 0. We say that a graph increases if it gets higher as we move toward the right, and if a graph is increasing it has a positive slope. Explain which of the following descriptions is correct for each graph: As we move from left to right the graph increases as its slope increases. As we move from left to right the graph decreases as its slope increases. As we move from left to right the graph increases as its slope decreases. As we move from left to right the graph decreases as its slope decreases.
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RESPONSE --> For graph of y = x^2 x y 0 0 1 1 2 4 3 9 For this graph, the description ""As we move from left to right the graph increases as its slope increases."" would be correct. For the graph of y = 'sqrt(x) x y 0 0 1 1 2 1.41 3 1.73 For this graph, the description ""As we move from left to right the graph increases as its slope decreases."" would be correct.
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15:10:16 For x = 1, 2, 3, 4: The function y = x^2 takes values 1, 4, 9 and 16, increasing more and more for each unit increase in x. This graph therefore increases, as you say, but at an increasing rate. The function y = 1/x takes values 1, 1/2, 1/3 and 1/4, with decimal equivalents 1, .5, .33..., and .25. These values are decreasing, but less and less each time. The decreasing values ensure that the slopes are negative. However, the more gradual the decrease the closer the slope is to zero. The slopes are therefore negative numbers which approach zero. Negative numbers which approach zero are increasing. So the slopes are increasing, and we say that the graph decreases as the slope increases. We could also say that the graph decreases but by less and less each time. So the graph is decreasing at a decreasing rate. For y = `sqrt(x) we get approximate values 1, 1.414, 1.732 and 2. This graph increases but at a decreasing rate.
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RESPONSE --> I understand this concept. I was able to determine the graphs correct descriptions and the coordinates.
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15:16:01 `q006. If the population of the frogs in your frog pond increased by 10% each month, starting with an initial population of 20 frogs, then how many frogs would you have at the end of each of the first three months (you can count fractional frogs, even if it doesn't appear to you to make sense)? Can you think of a strategy that would allow you to calculate the number of frogs after 300 months (according to this model, which probably wouldn't be valid for that long) without having to do at least 300 calculations?
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RESPONSE --> If the population of frogs were to increase at a rate of 10% each month, after the first 3 months you would have approximately 27 frogs. 20 * 10% = 2 22 * 10% = 2.4 24.2 * 10% = 2.42 I am not sure as to how to find the amount of frogs after 300 months. I would use an approximate method and assume that every 3 months we would have another 7 frogs on average. However, this will not work since the pool of frogs grows ever larger causing the amount represented by 10% to be raised each time.
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15:19:03 At the end of the first month, the number of frogs in the pond would be (20 * .1) + 20 = 22 frogs. At the end of the second month there would be (22 * .1) + 22 = 24.2 frogs while at the end of the third month there would be (24.2 * .1) + 24.2 = 26.62 frogs. The key to extending the strategy is to notice that multiplying a number by .1 and adding it to the number is really the same as simply multiplying the number by 1.1. 10 * 1.1 = 22; 22 * 1.1 = 24.2; etc.. So after 300 months you will have multiplied by 1.1 a total of 300 times. This would give you 20 * 1.1^300, whatever that equals (a calculator will easily do the arithmetic). A common error is to say that 300 months at 10% per month gives 3,000 percent, so there would be 30 * 20 = 600 frogs after 30 months. That doesn't work because the 10% increase is applied to a greater number of frogs each time. 3000% would just be applied to the initial number, so it doesn't give a big enough answer.
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RESPONSE --> I have made notes as to how to find the amount after 300 months. You will have multiplied the 1.1 factor 300 times, This gives us 20(the original number of frogs) * 1.1^300
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15:26:35 `q007. Calculate 1/x for x = 1, .1, .01 and .001. Describe the pattern you obtain. Why do we say that the values of x are approaching zero? What numbers might we use for x to continue approaching zero? What happens to the values of 1/x as we continue to approach zero? What do you think the graph of y = 1/x vs. x looks for x values between 0 and 1?
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RESPONSE --> 1/x where x = 1, 0.1, 0.01 and 0.001 1/1 = 1 1/0.1 = 10 1/0.01 = 100 1/0.001 = 1000 As the value underneath the 1 decreases, the answer for the problem becomes larger. There is a zero added for every digit to the right of the decimal point. I do not know what it means that the values of x are appraoching zero. The values of 1/x will continue to increase as we continue to approach zero. The graph of y = 1/x for values between 0 and 1 would be a decreasing graph and be decreasing at a decreasing rate. A graph of x for the same numbers would be an increasing graph at a steady rate.
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15:28:54 If x = .1, for example, 1 / x = 1 / .1 = 10 (note that .1 goes into 1 ten times, since we can count to 1 by .1, getting.1, .2, .3, .4, ... .9, 10. This makes it clear that it takes ten .1's to make 1. So if x = .01, 1/x = 100 Ithink again of counting to 1, this time by .01). If x = .001 then 1/x = 1000, etc.. Note also that we cannot find a number which is equal to 1 / 0. Deceive why this is true, try counting to 1 by 0's. You can count as long as you want and you'll ever get anywhere. The values of 1/x don't just increase, they increase without bound. If we think of x approaching 0 through the values .1, .01, .001, .0001, ..., there is no limit to how big the reciprocals 10, 100, 1000, 10000 etc. can become. The graph becomes steeper and steeper as it approaches the y axis, continuing to do so without bound but never touching the y axis. This is what it means to say that the y axis is a vertical asymptote for the graph .
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RESPONSE --> I have made notes as to approaching zero. I now understand the basic concept. The 1, 0.1, 0.01, etc....grow smaller yet they are unlimited as they become smaller the answer they yield becomes larger and there is no bounds as to how small or large the numbers may become.
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15:31:12 * `q008. At clock time t the velocity of a certain automobile is v = 3 t + 9. At velocity v its energy of motion is E = 800 v^2. What is the energy of the automobile at clock time t = 5?
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RESPONSE --> v = 3t +9 The energy of the auto at clock time t = 5 would be found by: v = 3(5) + 9 v = 15 + 9 v = 24 So E = 800 v^2 E = 800 * 24^2 E = 800 * 576 E = 460,800
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15:31:28
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RESPONSE --> Enter, as appropriate, an answer to the question, a critique of your answer in response to a given answer, your insights regarding the situation at this point, notes to yourself, or just an OK. ok
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15:31:57 For t=5, v = 3 t + 9 = (3*5) + 9 = 24. Therefore E = 800 * 24^2 = 460800.
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RESPONSE --> I understand this concept and did arrive at the same answer as you have here.
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15:32:04
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RESPONSE --> Enter, as appropriate, an answer to the question, a critique of your answer in response to a given answer, your insights regarding the situation at this point, notes to yourself, or just an OK.ok
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15:32:59 * `q009. Continuing the preceding problem, can you give an expression for E in terms of t?
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RESPONSE --> t = E/v^2 This equation will give us the clock time t/
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15:35:34 Since v = 3 t + 9 the expression would be E = 800 v^2 = 800 ( 3t + 9) ^2. This is the only answer really required here. For further reference, though, note that this expression could also be expanded by applying the Distributive Law:. Since (3t + 9 ) ^ 2 = (3 t + 9 ) * ( 3 t + 9 ) = 3t ( 3t + 9 ) + 9 * (3 t + 9) = 9 t^2 + 27 t + 27 t + 81 = 9 t^2 + 54 t + 81, we get E = 800 ( 9 t^2 + 54 t + 81) = 7200 t^2 + 43320 t + 64800 (check my multiplication because I did that in my head, which isn't always reliable).
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RESPONSE --> I am confused as to what was asked, though I have taken down notes as to what was supposed to have been done. I thought we were trying to find the value of t from the E equation. So we would find t in relation to E by' v = 3t + 9 and E = 800v^2 so 800 (3t + 9)^2 would be the answer.
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