course Mth 174
to stated problem does not match the given solution, you should self-critique per instructions at
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
001. typewriter notation
Question: **** `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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Your solution:
The difference is the presence of the parenthesis in the second problem. These tell you to solve what is inside them first before dividing. The first problem does not have the parenthesis so you divide first before adding and subtracting.
x-2 / x + 4
2-2/ 2-4 = 2 – 1 + 4 = 5
(2-2)/ (2+4) = 0 / 6 = 0
Confidence Assessment:
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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.
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Self-critique (if necessary):
OK
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.
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Your solution:
Again, the difference is the parenthesis which groups different numbers together that must be solved first. In the first problem you solve the exponent where in the second problem you ad x +4.
2 ^ 2 + 4 = 4 +4 = 8
2 ^ (2 + 4)= 2 ^ 6 = 64
Confidence Assessment:
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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.
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Self-critique (if necessary):
I did not show the exponentials fully:
2 ^ 6 = 2*2*2*2*2*2 = 64
Self-critique Rating:3
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?
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Your solution:
Numerator: x-3
Denominator: [ (2x-5)^2 * 3x + 1 ] - 2 + 7x
x=2 :
2-3 / [ (2(2) – 5) ^ 2 * 3(2) + 1] -2 + 7(2)
2-3/ [ -1 ^ 2 * 6 + 1] -2 + 14
2-3 /[ 1 * 6 + 1] -2 + 14
2-3/[6 +1] -2 +14
2-3 / 7 – 2 + 14
2-3/ 7 + 12
-1 / 19
Confidence Assessment: 2
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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.
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Self-critique (if necessary):
x is not part of the numerator because you subtract the fraction by x and there are no parenthesis around the x and 3. also at the end of the problem I should have divided 3/7 first because it comes before addition and subtraction. This would lead to the following expression: 2 - 2 + 14 - 3 / 7 = 14 - 3/7 = 98/7 - 3/7 = 95/7
Self-critique Rating:
Question: **** `q004. Explain, step by step, how you evaluate the expression (x - 5) ^ 2x-1 + 3 / x-2 for x = 4.
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Your solution:
(4 - 5) ^ 2(4)-1 + 3 / 4-2
-1 ^ 8 – 1 + 3 / 4 – 2 First evaluate parenthesis
1- 1 + 3 / 4 – 2 Then exponents
1 – 1 + .75 – 2 Then division
.75 – 2 Then subtraction
= -1.25
Confidence Assessment: 2
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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.
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Self-critique (if necessary):
I multiplied 2 * 4 before evaluating the exponents, so I used -1 ^ 8 instead of -1 ^ 2, which completely changed the solution. The correct answer would have been:
(-1)^2 * 4 - 1 + 3 / 4 - 2
= 1 * 4 - 1 + 3 / 4 - 2 (evaluating exponent before multiplying)
= 4 - 1 + 3/4 - 2
= 1 3/4 = 7 /4
Self-critique Rating: 3
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.
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Your solution:
This site shows different examples of typewriter notation and how to read and type different expressions. It also shows explanations for each expression.
Confidence Assessment: 3
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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
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Self-critique (if necessary):
The first page does not show the pictures of how to read the notations but you can click the word picture under each example to see the explanation. There are 30 examples and it is important to understand how to read these expressions.
Self-critique Rating:3
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.
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Your solution:
It is important to understand how to use order of operations because when a problem is typed, it is written straight out, instead of standard notation that is easier to understand. Understanding order of operations allows me to be able to read the expression correctly and arrive at a correct answer.
&#Good responses. Let me know if you have questions. &#