course Phy 201 §×¤Ć‡˙™Ő·Đďë–J˙˘Űą‡čůŹ¨•assignment #001
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09:46:48 `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 --> In the question without parenthesis, the division, according to the order of operations, must be done first. In order to do this, though, one needs to multiply both the top and the bottom of the equation by x-4 (which is the reverse sign seen in the denominator). After multiplying the terms together and adding in two, the answer comes out to be 0 (0/-12). The latter with the parenthesis, however, indicate that the parenthesis should be done first, which makes the task much more simple. The answer, after filling in 2, comes out ot be -2/3 (simplified). confidence assessment: 1
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11:20:05 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 --> I realize now that on the equation with no parenthesis it would have been much easier (and obviously far more correct) to have substituted initially instead of rationalizing the denominator. In the equation with parenthesis, clearly I didn't do my math correctly, but I am familiar with such order of operations: always do anything in parenthesis first before finishing out the equation, thus making the subtraction, this time, come before the division. self critique assessment: 2
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11:23:39 `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 first equation (without parenthesis) means that the two is raised to the x and THEN four is added; the second, however, indicates that four is added to the x, THEN the two is raised to the sum of the four and the x. The answer to the first equation, the one without parenthesis is 8. 2 squared is four, and then four added to four is 8. The answer to the second equation is 64. Four added to two is six, then two raised to the sixth power is 64. confidence assessment: 3
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11:24:14 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 understood this exactly as it is written here, and that is, basically, what I wrote myself. self critique assessment: 3
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11:28:55 `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. The answer is -1/19. confidence assessment: 3
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11:31:36 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 --> Wow. I really missed that one. I understand, now, that unless the x and the 3 are connected by parenthesis, then both cannot be considered part of the denominator. Had I realized that (it was inexcusable that I didn't), I would have been able to solve the rest of the order of operations.
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11:44:58 `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 --> 1. Since x-5 is in parenthesis, that part can go ahead and be evaluated as -1, which I get from subtracting 5 from 4. 2. 2x is considered basically one term, and it is the term to which -1 is raised. 2 x 4 = 8, so this step involves taking -1 and raising it to 8. This answer is 1. 3. The next step is realizing that the -1 after the 2x is NOT a part of the exponent. So, after figuring out that -1^8 = 1, I also have to subtract the 1 after the 2x from the 1 that resulted from the exponential equation, leaving zero for the first part of the equation. 4. Because x-2 is not joined by parenthesis, the next step is to divide 3/x, which = 3/4 after filling in four. 5. Then, 3/4 - 2 (again, because there was no initial parenthesis joining x-2, we have to take the two from the product of 3/4) = -1.25. 6. So, taking the 0 that resulted from the first part of the equation added to the -1.25 indicates that the final answer to the equation is -1.25. confidence assessment: 1
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11:49:08 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 did everything correctly EXCEPT that I mutiplied the 2 by 4 before using the 2 as an exponent. I did not realize that even side by side terms (where one is a number, the other a variable) without parenthesis can't be added together. I do understand this better now, though not perfectly. I hope there are more practice problems! self critique assessment: 2
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11:50:43 *&*& 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 --> Although a lot of mathematics is conducted in standard math notation, in this class and possibly in the future we will be required to type in this typewriter-style notation; thus, it is important to learn it and, with time, it should more easily. self critique assessment: 2
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