Assin 1

course Mth163

David,I'm a little rusty on this stuff, and find that it's a bit hard with the typwriter notation to know exactly what it is asking for, but I will look over the practice page.

d»Ö‡ûä›˜Ë°žŸÐn€†’³¹VÆ´‚ßà

assignment #001

001. typewriter notation

qa initial problems

01-18-2008

......!!!!!!!!...................................

12:00:45

`q001. Explain the difference between x - 2 / x + 4 and (x - 2) / (x + 4). The evaluate each expression for x = 2.

......!!!!!!!!...................................

RESPONSE -->

for the first equation, division must be done first, for the second, what is in the parenthases must be done first.

1. when x=2 the 2s in the middles are divided first, making the whole equation 2-1+4=5

2. when x=2 the parenthases are completed resulting in 0/6=0

confidence assessment: 2

.................................................

......!!!!!!!!...................................

12:15:18

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.

......!!!!!!!!...................................

RESPONSE -->

I am not sure if this is where I self-critique. If so, I understood the process, and think that I explained it well.

self critique assessment:

.................................................

......!!!!!!!!...................................

12:30:42

`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 -->

#1 would be completed as 2/x+2/4, if x=2 then 2/2+2/4=(because a common denominator has to be found) 4/4+2/4=6/4(then it needs to be reduced) 3/2=1 1/2

#2 the parenthases have to be completed first, and if x=2 then 2^(2+4)= 2^6, then 2 to the 6th power is 64

confidence assessment: 3

.................................................

......!!!!!!!!...................................

12:37:58

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 -->

for the 1st eqation I was treating both x and 4 as if they were in parenthases, which in turn made me divide (I'm not sure why I divided) both by 2. But what I have leared from the response is that only the number next to the power is to be multiplied.

self critique assessment: 2

.................................................

......!!!!!!!!...................................

12:43:59

`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

The denominator is [(2x-5)^2*3x+1]-2+7x

If x=2 then the numerator is -1

and the denominator is 19

so the answer is -1/19

confidence assessment: 2

.................................................

......!!!!!!!!...................................

12:51:23

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 do not undestand this one. Is there a way to go back and see the original question to compare the question and correct answer?

Though it doesn't completely make sense that the x disapears, I can see why. However, where did the -2+7x go? Why does that disapear?

self critique assessment: 2

Nothing disappears. However x isn't part of any numerator, and -2 + 7 x isn't part of any denominator.

A numerator and a denominator occurs when one quantity is divided by another.

In the expression

x - 3 / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x

the only indicated division is 3 / [ (2x-5)^2 * 3x + 1 ].

This is because of two things:

First, the 3 is preceded by the - sign and followed by a division sign. The division must be done before the subtraction. The division is by the bracketed term.
Second, the bracketed term is preceded by the / and followed by the -. The / must be done before the -.

So you have to do 3 / [ (2x-5)^2 * 3x + 1 ] first. This comes out 3/7, as shown in the given solution.

Thus you have

2 - 3/7 - 2 + 14.

As pointed out above the 3/7 is a fraction; terms 2, -2 and 14 are integers.

The additions and subtractions are performed in order. In the given solution we used the fact that as long as the signs are maintained, the order of the terms in a series of additions and subtractions can be changed. So 2 - 3/7 - 2 + 14 is the same as 2 - 2 + 14 - 3/7. This allows us to add the first 3 terms before we deal with the fraction, giving us 14 - 3/7, which as shown give us 95 / 7.

We don't have to change the order of the terms to add these numbers. If we had retained the original order

2 - 3/7 - 2 + 14

we would have needed to put every term over a common denominator, which in this case would be 7. Since 2 = 14/7 and 14 = 98/7 this would have given us

14/7 - 3/7 - 14/7 + 98 / 7 = (14 - 3 - 14 + 98) / 7 = 95 / 7,

which is the same result we obtained previously.

.................................................

......!!!!!!!!...................................

13:59:05

`q004. Explain, step by step, how you evaluate the expression (x - 5) ^ 2x-1 + 3 / x-2 for x = 4.

......!!!!!!!!...................................

RESPONSE -->

1. put (x-5)^2x-1+3 as the numerator, and x-2 as the denominator

2. exponents first so..2*4 then -1 = 7

3. then 4-5=-1

4. -1 to the 7th power is -1

5. -1+3=2

6. 4-2=2

7. 2/2=1

confidence assessment: 3

.................................................

......!!!!!!!!...................................

14:08:13

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 was assuming 2*4-1 was the entire exponent. I'm not sure how to avoid getting this confused in the future.

self critique assessment: 1

The order of operations dicates the following:

Exponentiation precedes multiplication and division, which precedes addition and subtraction.
Expressions inside signs of grouping must be done first, with the innermost grouping taking precedence.
A sequence of multiplications and divisions is performed in the order specified.
A sequence of additions and subtractions is performed in the order specified.

This order is unambiguous. Any properly formed expression can be evaluated in this manner. (an expression like 7 * - / + 4 ^ can't be evaluated because it isn't well-formed, doesn't make any sense at all).

The expression given in this problem is well-formed.

In the fragment

^ 2x

the x is multiplied by the factor that precedes it, so the fragment means the same as

^ 2 * x.

The 2 is preceded by ^ and followed by . The order of operations dicates that you have to do the ^ before the .

So whatever factor precedes the ^ 2 must be raised to the power 2 before the multiplication.

The factor preceding the ^ 2 is (x - 5). This means that we must do (x - 5) ^ 2 before doing the * x.

Similarly the - 1 is not to be done until the * x is completed.

If you were intended to do 2 * 4 - 1 first, then the expression would have to read

(x - 5) ^ (2x-1) + 3 / x-2.

It might 'look like' that's what is 'really' meant by the expression, but the order of operations 'trumps' everything else. If a grouping is intended, then it has to be written out, not implied.

Different people will 'see' different things when they interpret an expression. Mathematical expressions need to be evaluated by uambiguous rules, not by potentially inconsistent visual interpretations.

THE EXPRESSION, WHILE UNAMBIGUOUS, COULD HAVE BEEN WRITTEN MORE CLEARLY:

That said, in practice the expression (x - 5) ^ 2x-1 + 3 / x-2 could be written more clearly. For example the 2x really does look like it belongs together, even though by the order of operations is very clear that the 2 is 'connected' to the (x - 5) by the ^, and not to the x. However, to 'help out' the reader, the form

(x - 5) ^ 2 * x - 1 + 3 / x - 2

would be better, with consistent spacing between the numbers and the operators so that no grouping appears to be implied.

It might also help to explicitly group the first factor, writing

( (x - 5) ^ 2 ) * x - 1 + 3 / x - 2 .

It would also be wise to help the reader by grouping the 3 / x, avoiding the error of treating the x - 2 as a denominator:

( (x - 5) ^ 2 ) * x - 1 + (3 / x) - 2 .

None of this is necessary. This expression has exactly the same meaning as the original. However, this expression is easier to interpret than the original, and this form will be less prone to error.

Additional grouping is possible. However if too many parentheses are nested, the expression can actually lose some of its clarity. So there is a balance to be found between making the expression easier to read by adding groupings, and relying on the user's knowledge of the order of operations. The additional grouping shown below is unnecessary; each reader can decide whether the groupings are helpful or needless:

Some users might tend to group the x - 1, even though the x is clearly to be multiplied by ( (x-5)^2 ) before the - 1 is performed. To avoid this possibility we can write the expression as

( ( (x - 5) ^ 2 ) * x ) - 1 + ( 3 / x ) - 2 .

You can decide for yourself whether or not this is prefereable to the preceding form

( (x - 5) ^ 2 ) * x - 1 + (3 / x) - 2 .

However, to reemphasize, while some of these forms might be easier to read, every form given here is identical to the original expression

(x - 5) ^ 2x-1 + 3 / x-2.

.................................................

......!!!!!!!!...................................

14:11:37

*&*& 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 -->

standard notation is easiest to use, and should be worked out on paper, however, it is imporant to get used to doing order of operations on the computer.

self critique assessment: 3

.................................................

......!!!!!!!!...................................

14:19:11

`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.

......!!!!!!!!...................................

RESPONSE -->

On each of the sites, I see links and descriptions. At the bottom of the pages I see examples

confidence assessment:

.................................................

......!!!!!!!!...................................

14:19:59

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,

......!!!!!!!!...................................

RESPONSE -->

self critique assessment:

.................................................

......!!!!!!!!...................................

14:20:35

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.

......!!!!!!!!...................................

RESPONSE -->

self critique assessment: 3

.................................................

......!!!!!!!!...................................

14:20:46

end program

......!!!!!!!!...................................

RESPONSE -->

self critique assessment: 3

.................................................

I've inserted fairly extensive notes.

Look back over everything and be sure you understand it. If you have questions, copy the given question, your solution, the given solution, your self-critique and my comments, and insert your answers as appropriate. Indicate your insertions by &&&& so I can find them easily.