course Mth 164 Hello
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Given Solution: `aThe 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 x or divide the -2 by 4 first. If there are no parentheses you have to divide before you subtract. Substituting 2 for x we get 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) / ( 2 + 4 ) = 0 / 6 = 0. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* OK ********************************************* Question: `q002. Explain the difference between 2 ^ x + 4 and 2 ^ (x + 4). Then evaluate each expression for x = 2. Note that a ^ b means to raise a to the b power. This process is called exponentiation, and the ^ symbol is used on most calculators, and in most computer algebra systems, to represent exponentiation. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2 to the x power is to be worked out before you add the 4. 2 ^ 2= 4+4=8 The ( ) indicated that you first solve (x+4) then raise that answer to the 2 ^ (2+4)=6 2^6=64 confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a2 ^ 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* OK, easy problem. ********************************************* Question: `q003. What is the numerator of the fraction in the expression x - 3 / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x? What is the denominator? What do you get when you evaluate the expression for x = 2? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The numerator is 3. This equation can only properly divide into 3. -3/ [2 x2-5)^2x3(2)+1] -2+7x2= 2 - 3 / [(4 - 5) ^2 * 6 + 1 ] - 2 + 14 First do everything inside ( ) Then do everything outside 2 - 3 / [ (-1)^2 * 6 + 1 ] -2 + 14 = add together 2 - 3 / [1 * 6 + 1 ] - 2 + 14 = after evaluating everything inside ( ) 2 - 3 / 7 - 2 + 14 = evaluate brackets 13.57 add and subtract. 2 - 3 / 7 - 2 + 14: confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe 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. COMMON STUDENT QUESTION: ok, I dont understand why x isnt part of the fraction? And I dont understand why only the brackets are divided by 3..why not the rest of the equation? INSTRUCTOR RESPONSE: Different situations give us different algebraic expressions; the situation dictates the form of the expression. If the above expression was was written otherwise it would be a completely different expression and most likely give you a different result when you substitute. If we intended the numerator to be x - 3 then the expression would be written (x - 3) / [(2x-5)^2 * 3x + 1 ] - 2 + 7x, with the x - 3 grouped. If we intended the numerator to be the entire expression after the / the expression would be written x - 3 / [(2x-5)^2 * 3x + 1 - 2 + 7x ]. STUDENT COMMENT: I wasn't sure if the numerator would be 3 or -3. or is the subtraction sign just that a sign in this case? INSTRUCTOR RESPONSE: In this case you would regard the - sign as an operation to be performed between the value of x and the value of the fraction, rather than as part of the numerator. That is, you would regard x - 3 / [ (2x-5)^2 * 3x + 1 ] as a subtraction of the fraction 3 / [ (2x-5)^2 * 3x + 1 ] from the term x. STUDENT QUESTION: There was another question I had about this problem that wasn’t addressed. At the end when you changed the order of operation from 2 – 2 + 14 – 3/7 = 14 – 3/7 where did the 98/7 – 3/7 come into play before the end solution of 95/7? I must have forgotten how to do this part. INSTRUCTOR RESPONSE: It's not clear how you can get 95/7 without this step. To do the subtraction 14 - 3/7 both terms must be expressed in terms of a common denominator. The most convenient common denominator is 7. So 14 must be expressed with denominator 7. This is accomplished by multiplying 14 by 7 / 7, obtaining 14 * 7 / 7 = 98 / 7. Since 7/7 = 1, we have just multiplied 14 by 1. We chose to use 7 / 7 in order to give us the desired denominator 7. Thus our subtraction is 14 - 3/7 = 98/7 - 3/7 = (98 - 3) / 7 = 95 /7. STUDENT COMMENT It took me a while to think thru this one especially when I got to working with the fraction. Fractions have always been my weak spot. Any tips to make working with fractions a little easier is greatly appreciated. INSTRUCTOR RESPONSE Fractions are seriously undertaught in our schools, so your comment is not unusual. I have to focus my attention on the subject matter of my courses, and while I do address it to a point, I don't have time to do justice to the subject of fractions. In any case , to do so would be redundant on my part, since there are a lot of excellent resources on the Internet. I suggest you search the Web using something like 'review of fractions', and find something appropriate to your needs. You should definitely review the topic, as should 95% of all students entering your course. STUDENT COMMENT I think I am confused on why the Numerator is not the top portion and denominator the bottom portion of the problem. INSTRUCTOR RESPONSE Everything is on one line so there is no top or bottom in the given expression. A numerator and denominator are determined by a division of two expressions. As we know, a denominator divides a numerator. In the given expression the division sign occurs between the 3 and the [ (2x-5)^2 * 3x + 1 ], so 3 is the numerator and [ (2x-5)^2 * 3x + 1 ] is the denominator. x is not divided by the denominator, since the division occurs before the subtraction. For the same reason the -2 + 7x is not involved in the division. So neither the x nor the -2 + 7 x is part of the fractional expression. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* The brackets thru me off a lot. The student question helped me a little. I just get confused when I see brackets and parenthesis together. I feel like I am doing okay on the problem then I have to start over because the number(s) I got cannot be right. ********************************************* Question: `q004. Explain, step by step, how you evaluate the expression (x - 5) ^ 2x-1 + 3 / x-2 for x = 4. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: First you have to consider order of operations. First, you do everything inside ( )… (4-5)= -1 -1^2 x 4-1 + ¾-2 1x4-1+ ¾ -2 4-1 + ¾ -2 =7/4 confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aWe get (4-5)^2 * 4 - 1 + 3 / 4 - 2 = (-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. STUDENT QUESTION: if it's read (-1)^8 it would be 1 or would you apply the sign afterward even if it is grouped and it be a -1? INSTRUCTOR RESPONSE: The 8th power won't occur in this problem, of course, but you ask a good question. -1^8 would require raising 1 to the 8th power, then applying the negative sign, and the result would be -1. (-1)^8 would be the 8th power of -1, which as you see would be 1. STUDENT COMMENT: I think it would be easier to visualize what your trying to raise to the exponent if you actually put parenthesis around the 2, that part seems to get tricky on the computer. INSTRUCTOR RESPONSE: The expression was intentionally written to be misleading and make the point that, to avoid ambiguity, order of operations apply strictly, no matter what the expression looks like. Normally, for clarity, the parentheses would be included. They aren't necessary, but when helpful it's a good idea to include them. You can, of course, have too many parentheses in an expression, making it harder than necessary to sort out. In practice we try to strike a balance. The original expression was (x - 5) ^ 2x-1 + 3 / x-2 White spaces make no difference in how an expression is evaluated, but they can help show the structure; e.g., (x - 5)^2 * x - 1 + 3 / x -2 is a visual improvement over the original. The * between the 2 and the x is not strictly necessary, but is also helpful. ((((x - 5) ^ 2)) * x) - 1 + (3 / x) - 2 verges on having too many parentheses at the beginning; it does help clarify the 3 / x. STUDENT COMMENT Although I read through your explanation and do see the point you are making, that 2x is actually 2 * x, I still think that (-1) should be raised to 2x rather than 2. Kaking the answer -11/4, not 7/4. INSTRUCTOR RESPONSE When the expression (x - 5) ^ 2x-1 + 3 / x-2 is copied and pasted into a computer algebra system it is translated as This notation is universal and unambiguous. Any deviation from strict interpretation (which does occur among some authors and among manufacturers of some calculators) tends to result in ambiguity and confusion. STUDENT COMMENT While I do understand what you are trying to relate, I will continue to make these mistakes on more than one occasion and will not penalize myself for not rewriting years of mathematics because of a syntax issue in an online class. INSTRUCTOR RESPONSE I don't penalize errors in typed notation when the intent is clear (though I will sometimes point out these errors), and when you take your tests you'll be writing them out by hand and this won't be an issue. However this is not a syntax issue in an online class. This is the order of operations, as it has been since algebra was developed hundreds of years ago, and it's completely consistent with the mathematics you appear to know (quite well). As stated here, if you use the wrong syntax in any computer algebra system, your expression will not be interpreted correctly. For this reason alone you need to understand the notation. For this and other valid reasons you need to understand how the order of operations are represented in 'linear' fashion (i.e., 'typewriter notation') and to correctly interpret expressions written in this notatation. Any mathematics that has been learned correctly is completely consistent with the order of operations and with the notation used in this course. If the mathematics you've learned was inconsistent with the order of operations (and I don't believe this is so in your case, but it is with many students), then you would need to adjust your thinking. Fortunately this is very easy to do. Interpret expressions literally, assume nothing, and everything works out. You will also find that the notation quickly becomes easy to read and use, and that it expands your comprehension of all mathematical notation. STUDENT COMMENT I used -1^(2*4). I didn't realize that was doing multiplication before exponents. All of this typewriter notations seems ambiguous to me but I think that had I seen the expression in standard notation I would probably have made the same mistake in this instance. If I were writing this expression I would probably use a parenthesis or * to show the necessary separation. INSTRUCTOR RESPONSE Parentheses, even when they aren't strictly necessary, are often useful to clarify the expression. An parentheses, even when not necessary, are part of the order of operations. Spacing is not part of the order of operations. An expression has the same meaning even if all spaces are removed. However as long as an expression is correctly formed, spacing as well as parentheses can certainly be used to make it more readable. I don't go to any trouble in this exercise to make the expressions readable, since my goal here is to make the point about order of operations, which give an expression its unambiguous meaning. However in most of the documents you will be working with, I do make an effort to clarify the meanings of expressions through their formatting, often using unnecessary parentheses and spacing to help clarify meaning. Certainly I encourage you to do the same. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* This is a tricky question. I forgot that a negative number in ( ) would change to a positive number, but after reading this student question and answer I understand it. ********************************************* Question: `q005. At the link http://www.vhcc.edu/dsmith/genInfo/introductory problems/typewriter_notation_examples_with_links.htm (copy this path into the Address box of your Internet browser; alternatively use the path http://vhmthphy.vhcc.edu/ > General Information > Startup and Orientation (either scroll to bottom of page or click on Links to Supplemental Sites) > typewriter notation examples and you will find a page containing a number of additional exercises and/or examples of typewriter notation.Locate this site, click on a few of the links, and describe what you see there. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I see numerous equations that involve system of operations. Along with the problems are pictures that represent the problem in a different way. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aYou should see a brief set of instructions and over 30 numbered examples. If you click on the word Picture you will see the standard-notation format of the expression. The link entitled Examples and Pictures, located in the initial instructions, shows all the examples and pictures without requiring you to click on the links. There is also a file which includes explanations. The instructions include a note indicating that Liberal Arts Mathematics students don't need a deep understanding of the notation, Mth 173-4 and University Physics students need a very good understanding, while students in other courses should understand the notation and should understand the more basic simplifications. There is also a link to a page with pictures only, to provide the opportunity to translated standard notation into typewriter notation. end program &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* A little confusing. I would rate these problems a 2. The standard way to express the problems confuses me. I will understand them I just have to think about them for awhile. ********************************************* Question: `q006 Standard mathematics notation is easier to look at; it's easier to see the meaning of the expressions. However it's very important to understand order of operations, and students do get used to this way of doing it. You should of course write everything out in standard notation when you work it on paper. It is likely that you will at some point use a computer algebra system, and when you do you will probably have to enter expressions using a keyboard, so it is well worth the trouble to get used to this notation. Indicate your understanding of why it is important to understand this notation. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Although standard notation can be confusing it is important to write it this way. One, It is the right way. Two, you can get errors and a totally different answer if you do not use standard notation. confidence rating #$&* I am pretty confidence with this set of problems. With all math I just have to pay extra close attention to all the symbols and know which operations to do first. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ "
course Mth 164 6/3/10 9:37pm Question: `q001. If you are earning money at the rate of 8 dollars / hour and work for 4 hours, how much money do you make during this time? Answer in such a way as to explain your reasoning as fully as possible. A solution to this problem appears several lines below, but enter your own solution before you look at the given solution.YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
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Given Solution: 8 dollars / hour means '8 dollars per hour', indicating that for every hour you work you earn 8 dollars. If you work for 4 hours, then if you earn 8 dollars for every one of those hours you earn 4 * 8 dollars = 32 dollars. 3 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): (If you believe your solution matches the given solution then just type in 'OK'. Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.) OK ------------------------------------------------ Self-critique rating #$&* (If you believe your solution matches the given solution then just type in 'OK'. Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3. • 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation. • 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand • 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand • 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase) ********************************************* Question: `q002. If you work 12 hours and earn $168, then at what rate, in dollars / hour, were you making money? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (type in your solution starting in the next line) 168/12 = 14 dollars an hour 168 dollars made total in 12 hours means that you would divide the numbers to figure out how much money was made for each of those hours. confidence rating #$&* (Type in a number from 0 to 3, indicating your level of confidence in your solution. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution 2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution 1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing 0 means that you're pretty sure you didn't get anything right) 3
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Given Solution: $168 earned in 12 hours implies that $168 / 12 = $14 were made per hour, so the rate is $14 / hour. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): (If you believe your solution matches the given solution then just type in 'OK'. Otherwise explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.) ok ------------------------------------------------ Self-critique rating #$&* (If you believe your solution matches the given solution then just type in 'OK'. Otherwise evaluate the quality of your self-critique, using a number between 0 and 3. 3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation. 2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand 1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand 0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)"