course Mth 271 ¹ØÓ¼Þ·ìüªÍ{{‘Ò¸ŒÂeŠ…{ôÇ¥àassignment #001
......!!!!!!!!...................................
22:06:10 `q001. Explain the difference between x - 2 / x + 4 and (x - 2) / (x + 4). The evaluate each expression for x = 2.
......!!!!!!!!...................................
RESPONSE --> The difference between x-2/x+4 and [x-2] / [x+4] is the order of opperations changes the result. In the first equation you must divide 2 /x then subtract that from x and then add 4. In the second equation, order of opperations calls to do x - 2 and x + 4 separate because they are in parenthesis then divide the answer of those. So if x = 2 the first equation will be 5 and the second will be 0/6 or 0. confidence assessment: 3
.................................................
‹ë¦“véÃóÙí¾ƒÏózÛÍÆÉØ¿Î assignment #001 001. typewriter notation qa initial problems 01-11-2007
......!!!!!!!!...................................
22:07:52 `q001. Explain the difference between x - 2 / x + 4 and (x - 2) / (x + 4). The evaluate each expression for x = 2.
......!!!!!!!!...................................
RESPONSE --> The difference between x-2/x+4 and [x-2] / [x+4] is the order of operations changes the result. In the first equation you must divide 2 /x then subtract that from x and then add 4. In the second equation, order of operations calls to do x - 2 and x + 4 separate because they are in parenthesis then divide the answer of those. So if x = 2 the first equation will be 5 and the second will be 0/6 or 0. confidence assessment: 3
.................................................
......!!!!!!!!...................................
22:09:02 `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.
......!!!!!!!!...................................
RESPONSE --> I answered everything for this problem. confidence assessment: 3
.................................................
......!!!!!!!!...................................
22:09:33 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.
......!!!!!!!!...................................
RESPONSE --> Ok self critique assessment: 3
.................................................
......!!!!!!!!...................................
22:13:30 `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?
......!!!!!!!!...................................
RESPONSE --> The numerator is x - 3 and the denominator is [[2x-5]^2*3x+1]-2+7x after evaluating the expression my answer was -1/19 confidence assessment: 3
.................................................
......!!!!!!!!...................................
22:17:27 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 ].
......!!!!!!!!...................................
RESPONSE --> I understand what I did wrong in the problem. I did not pay close enough attention to order of operations. I assumed that x -3 was together instead of realizing there were no parentheses which caused me to get the numerator wrong. The same goes for the denominator. That also messed up my ending number because I divided 2-3 by the denominator insteead of just -3 then using the 2. self critique assessment: 2
.................................................
......!!!!!!!!...................................
22:22:36 `q004. Explain, step by step, how you evaluate the expression (x - 5) ^ 2x-1 + 3 / x-2 for x = 4.
......!!!!!!!!...................................
RESPONSE --> [4-5]^ 2(4)-1 + 3 / 4 - 2. Parentheses must be dealt with first, so 4-5 = -1. Then exponents so after evaluating 2*4-1 the answer is 7. [-1]^ 7 is -1. then division is next. 3/4 is .75 So -1 +.75 - 2 = -2.25 confidence assessment: 2
.................................................
......!!!!!!!!...................................
22:26:32 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.
......!!!!!!!!...................................
RESPONSE --> I understand what I did wrong, again. I realize that the [x-5]^2 is separate and that it was not meant to be ^2*x-1. I need to be careful where there aren't parentheses. self critique assessment: 1
.................................................
......!!!!!!!!...................................
22:27:32 *&*& 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.
......!!!!!!!!...................................
RESPONSE --> I realize this is very important. I almost have it down, but I get confused with some of the equations. I need to make sure that if it was meant to be together, there would have been a parenthesis basically. self critique assessment: 2
................................................."