course Phy 121 ÿy”‹ Žïøêȯ’µß‚®¸äøõœúŸ¿Ù‰assignment #001
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11:45:27 `q001. Explain the difference between x - 2 / x + 4 and (x - 2) / (x + 4). The evaluate each expression for x = 2.
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RESPONSE --> One must first consider the order of operations: Parentheses Exponents Multiplication Division Addition Subtration In the first equation, one would first divide 2 by whatever x is and get a solution. That answer would then be added to 4, and that answer would then be subtracted from x. In the second equation, one would first work with the equations in parentheses [(x+4) and then (x-2)]. Once answers have been achieved for each equation, respectively, the resulting value of (x-2) would be divided by the resulting value for (x-4) to arrive at an answer. This reasoning is derived from the order of operations. x - 2/x + 4 = 2 - 2/2 + 4 = 2 - 1 + 4 = 2 - 5 = -3 (x - 2)/(x + 4) = (2 - 2)/(2 + 4) = (0)/(6) = 0 confidence assessment: 2
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11:49:11 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 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) / ( 4 - 2) = 0 / 2 = 0.
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RESPONSE --> In the =2 - 1 + 4 step, why would you subtract before you add? According to the order of operations additions should occur before subtractions. self critique assessment: 2
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12:01:40 `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 --> In the first equation, 2 is raised to the power of x. The resulting quantity is then added to 4. In the second equation, 2 is raised to the power of x+4. This means that the value of x will be plugged in to the equation x+4. The resulting quantity will be the power that 2 is raised to. 2 ^ x + 4 = (2^2) + 4 = 4 + 4 = 8 2 ^ (x +4) = 2 ^ (2 + 4) = 2 ^ 6 = 64 confidence assessment: 2
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12:02:21 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 achieved the correct answers on all parts of the question. self critique assessment: 3
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12:17:01 `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 is x - 3. The denominator is [(2x - 5)^2 * 3x + 1] - 2 + 7x. I base this conclusion on where the division symbol is placed and on how the numbers are grouped. [(2x - 5)^2 * 3x + 1] is grouped together and the order of operations says it is to be completed first and the - 2 + 7x should be figured into the result of the previously mentioned grouping. 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 * 7] +12 2 - 3 / [1 * 7] + 12 2 - 3 / 7 + 12 2 - .428 + 12 1.572 + 12 = 13.572 confidence assessment: 2
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12:26:15 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. 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 ].
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RESPONSE --> Based on the instructor response, I now understand which the numerator and which is the denominator and why this is so. Since, I am pretty sure my numerical response is close to that of the correct one I will say that there is usually more than one way to work a problem and get the right answer; so, to my mind I don't necessarily understand all aspects of this explanation. For example, I do understand that how things are notated and grouped affects how substitution will change the answer. However, I still do not understand why x is not considered as part of the numerator. self critique assessment: 2
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12:39:33 `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 --> First, go ahead and plug in 4 for x: (4 - 5) ^ 2(4) - 1 + 3 / (4) - 2 Follow the order of operations. First deal with parentheses: (-1) ^ 8 - 1 + 3 / (4) - 2 Next deal with exponents. 2x is the power to which (x - 5 ) is raised. 1 - 1 + 3 / (4) - 2 Continue to follow the order by doing division: 1 - 1 + .75 - 2 Then subtract, add, and subtract again: 0 + .75 - 2 = .75 - 2 = -1.25 confidence assessment: 2
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12:40:49 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 --> After reading this explanation, I see where I made my mistake. self critique assessment: 3
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12:43:58 *&*& Standard mathematics notation is easier to see. On the other hand 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 have to enter expressions through a typewriter, so it is well worth the trouble to get used to this notation. Indicate your understanding of the necessity to understand this notation.
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RESPONSE --> Despite the fact that standard mathematics notation is easier to see, students will often use the order of operations so it is important to get used to this way of doing it. It is best to write out everything on paper in standard notation before attempting to put it in a computer notated answer. self critique assessment: 2
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12:47:39 `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.
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RESPONSE --> The various links show examples of the different types of notation (typewriter notation converted to standard notation and vice versa) and how to convert one type of notation to the other, as well as detailed instructions on how to do this and understand it. confidence assessment: 2
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12:49:55 You 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,
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RESPONSE --> I went to the links and clicked on the word Picture. I also went to the link entitled Examples and Pictures located in the initial instructions. self critique assessment:
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12:50:52 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.
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RESPONSE --> I went to and studied the mentioned link. self critique assessment: 2
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12:51:10 end program
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RESPONSE --> end of exercise self critique assessment: 2
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