Introductory Questions

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course MTH 164

1/11 0100

Question: `q003. If you are earning 8 dollars / hour, how long will it take you to earn $72? The answer may well be obvious, but explain as best you can how you reasoned out your result.YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: (type in your solution starting in the next line)

My Answer: 9 Hours. 8 Dollars per hour times X equals 72 dollars total. 8*X=72, X=72/8, X=9.

confidence rating #$&*:

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Given Solution: Many students simply know, at the level of common sense, that if we divide $72 by $8 / hour we get 9 hours, so 9 hours are required.

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Self-critique (if necessary): OK

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Self-critique Rating: OK

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Question: `q004. Calculate (8 + 3) * 5 and 8 + 3 * 5, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results.

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Your solution:

My Answer: (8+3)*5=55. First do 8 plus 3 because of the parentheses to get 11, Next times your answer by 5 and you get 55. 8+3*5=23. First times 3 by 5 (#### to get 15 since multiplication comes before addition), then add 8 to get 23.

confidence rating #$&*:

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Given Solution: (8 + 3) * 5 and 8 + 3 * 5

To evaluate (8 + 3) * 5, you will first do the calculation in parentheses. 8 + 3 = 11, so

(8 + 3) * 5 = 11 * 5 = 55.

To evaluate 8 + 3 * 5 you have to decide which operation to do first, 8 + 3 or 3 * 5. You should be familiar with the order of operations, which tells you that multiplication precedes addition. The first calculation to do is therefore 3 * 5, which is equal to 15. Thus

8 + 3 * 5 = 8 + 15 = 23

The results are different because the grouping in the first expression dictates that the addition be done first.

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Self-critique (if necessary):OK

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Self-critique Rating: OK

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Question: `q005. Calculate (2^4) * 3 and 2^(4 * 3), indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results. Note that the symbol '^' indicates raising to a power. For example, 4^3 means 4 raised to the third power, which is the same as 4 * 4 * 4 = 64.

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Your solution:

(2^4)*3=48. First do 2 to the power or 4 and it gives you 16, Next times by 3 and you get 48. 2^(4*3)=4,096. First times 4 by 3 since it is in parentheses and you get 12, Next do 2 to the power of 12, (#### or 2*2*2*2*2*2*2*2*2*2*2*2) and it gives you 4,096.

confidence rating #$&*:

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Given Solution:

To evaluate (2^4) * 3 we first evaluate the grouped expression 2^4, which is the fourth power of 2, equal to 2 * 2 * 2 * 2 = 16. So we have

(2^4) * 3 = 16 * 3 = 48.

To evaluate 2^(4 * 3) we first do the operation inside the parentheses, obtaining 4 * 3 = 12. We therefore get

2^(4 * 3) = 2^12 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 4096.

It is easy to multiply by 2, and the powers of 2 are important, so it's appropriate to have asked you to do this problem without using a calculator. Had the exponent been much higher, or had the calculation been, say, 3^12, the calculation would have become tedious and error-prone, and the calculator would have been recommended.

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Question: `q006. Calculate 3 * 5 - 4 * 3 ^ 2 and 3 * 5 - (4 * 3)^2 according to the standard order of operations, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results.

Your solution:

3*5-4*3^2=-21. First do 3 to the power of 2 and get 9, Next times 9 by 4 to get 36, Next times 3 by 5 to get 15, Last take 15 minus 36 and you get Negative 21. 3*5-(4*3)^2=-129. First times 4 by 3 to get 12 since it is in parentheses, Next do 12 squared and get 144, Next times 3 by 5 and get 15, Last take 15 minus 144 to get Negative 129.

confidence rating #$&*:

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Given Solution:

To calculate 3 * 5 - 4 * 3 ^ 2, the first operation is the exponentiation operation ^.

• The two numbers involved in the exponentiation are 3 and 2; the 4 is 'attached' to the 3 by multiplication, and this multiplication can't be done until the exponentiation has been performed.

• The exponentiation operation is therefore 3^2 = 9, and the expression becomes 3 * 5 - 4 * 9.

Evaluating this expression, the multiplications 3 * 5 and 4 * 9 must be performed before the subtraction. 3 * 5 = 15 and 4 * 9 = 36 so we now have

3 * 5 - 4 * 3 ^ 2 = 3 * 5 - 4 * 9 = 15 - 36 = -21.

To calculate 3 * 5 - (4 * 3)^2 we first do the operation in parentheses, obtaining 4 * 3 = 12. Then we apply the exponentiation to get 12 ^2 = 144. Finally we multiply 3 * 5 to get 15. Putting this all together we get

3 * 5 - (4 * 3)^2 =

3 * 5 - 12^2 =

3 * 5 - 144 =

15 - 144 =

-129.

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Question: `q007. Let y = 2 x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result.

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Your Solution:

My Answer: -1. If x equals Negative 2 then negative 2 times 2 equals negative 4 plus 3 equals negative 1.

Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table.

My Answer: X=-2, Y=2(-2)+3, Y=-1

X=-1, Y=2(-1)+3, Y=1

X=0, Y=2(0)+3, Y=3

X=1, Y=2(1)+3, Y=5

X=2, Y=2(2)+3, Y=7

x y

-2 -1

-1 1

0 3

1 5

2 7

My Answer: My graph best represents a linear graph. I chose this because it goes from the bottom left to the top right in a straight line (#### Which looks most similar to linear).

confidence rating #$&*:

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Question: `q008. Let y = x^2 + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result.

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Your Solution:

My Answer: 7. If x equals negative 2 then negative 2 squared equals 4 since squaring a negative number gives you a positive number, then add 3 to get 7.

Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table.

My Answer: X=-2, Y=(-2)^2+3, Y=4+3, Y=7

X=-1, Y=(-1)^2+3, Y=1+3, Y=4

X=0, Y=(0)^2+3, Y=0+3, Y=3

X=1, Y=(1)^2+3, Y=1+3, Y=4

X=2, Y=(2)^2+3, Y=4+3, Y=7

x y

-2 7

-1 4

0 3

1 4

2 7

My Answer: The graph I chose to best represent this equation would be Quadratic or Parabolic since it starts from the top left, comes down to 3 when X equals 0, then goes back up to the top right, and it would be a curved line (#### and equal points left or right of the Y axis).

confidence rating #$&*:

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Question: `q009. Let y = 2 ^ x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

Evaluate y for x = 1. What is your result? In your solution explain the steps you took to get this result.

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Your Solution:

My Answer:5. If X equals 1 then 2 to the power of 1 equals 2, then add 3 to get 5.

Evaluate y for x values 2, 3 and 4. Write out a copy of the table below. In your solution give the y values you obtained in your table.

My Answer: X=1, Y=2^1+3, Y=2+3, Y=5

X=2, Y=2^2+3, Y=4+3, Y=7

X=3, Y=2^3+3, Y=8+3, Y=11

X=4, Y=2^4+3, Y=16+3, Y=19

x y

1 5

2 7

3 11

4 19

My Answer: The graph I would chose to best represent this equation would be Exponential since at X equals 1, Y equals 5 and as X gets bigger, Y increases faster which would make the line curve up sharply.

confidence rating #$&*:

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Question: `q010. If you divide a certain positive number by 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

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Your solution:

My Answer: The result would be the same as the original number because any number divided by one equals the original number.

confidence rating #$&*:

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Question: `q011. If you divide a certain positive number by a number greater than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

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Your solution:

My Answer: The answer will always be smaller. As long as the number you are dividing by is greater than one then the answer has to be smaller than the original since a number divided by one is equal to the number.

confidence rating #$&*:

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Question: `q012. If you divide a certain positive number by a positive number less than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

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Your solution:

My Answer: The answer will always be larger. As long as the number you are dividing by is less than one, then the answer has to be larger since a number divided by one is equal to the number.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Question: `q013. Students often get the basic answers to nearly all, or even all these questions, correct. Your instructor has however never seen anyone who addressed all the subtleties in the given solutions in their self-critiques, and it is very common for a student to have given no self-critiques. It is very likely that there is something in the given solutions that is not expressed in your solution.

This doesn't mean that you did a bad job. If you got most of the 'answers' right, you did fine.

However, in order to better understand the process, you are asked here to go back and find something in one of the given solutions that you did not address in your solution, and insert a self-critique. You should choose something that isn't trivial to you--something you're not 100% sure you understand.

If you can't find anything, you can indicate this below, and the instructor will point out something and request a response (the instructor will select something reasonable, but will then expect a very good and complete response). However it will probably be less work for you if you find something yourself.

Your response should be inserted at the appropriate place in this document, and should be indicated by preceding it with ####.

As an answer to this question, include a copy of whatever you inserted above, or an indication that you can't find anything.

your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv

Question 4: (#### to get 15 since multiplication comes before addition)

Question 5: (#### or 2*2*2*2*2*2*2*2*2*2*2*2)

Question 7: (#### Which looks most similar to linear).

Question 8: (#### and equal points left or right of the Y axis).

"

Self-critique (if necessary):

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Self-critique rating:

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Question: `q005. Calculate (2^4) * 3 and 2^(4 * 3), indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results. Note that the symbol '^' indicates raising to a power. For example, 4^3 means 4 raised to the third power, which is the same as 4 * 4 * 4 = 64.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

(2^4)*3=48. First do 2 to the power or 4 and it gives you 16, Next times by 3 and you get 48. 2^(4*3)=4,096. First times 4 by 3 since it is in parentheses and you get 12, Next do 2 to the power of 12, (#### or 2*2*2*2*2*2*2*2*2*2*2*2) and it gives you 4,096.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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

Given Solution:

To evaluate (2^4) * 3 we first evaluate the grouped expression 2^4, which is the fourth power of 2, equal to 2 * 2 * 2 * 2 = 16. So we have

(2^4) * 3 = 16 * 3 = 48.

To evaluate 2^(4 * 3) we first do the operation inside the parentheses, obtaining 4 * 3 = 12. We therefore get

2^(4 * 3) = 2^12 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 4096.

It is easy to multiply by 2, and the powers of 2 are important, so it's appropriate to have asked you to do this problem without using a calculator. Had the exponent been much higher, or had the calculation been, say, 3^12, the calculation would have become tedious and error-prone, and the calculator would have been recommended.

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Question: `q006. Calculate 3 * 5 - 4 * 3 ^ 2 and 3 * 5 - (4 * 3)^2 according to the standard order of operations, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results.

Your solution:

3*5-4*3^2=-21. First do 3 to the power of 2 and get 9, Next times 9 by 4 to get 36, Next times 3 by 5 to get 15, Last take 15 minus 36 and you get Negative 21. 3*5-(4*3)^2=-129. First times 4 by 3 to get 12 since it is in parentheses, Next do 12 squared and get 144, Next times 3 by 5 and get 15, Last take 15 minus 144 to get Negative 129.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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

Given Solution:

To calculate 3 * 5 - 4 * 3 ^ 2, the first operation is the exponentiation operation ^.

• The two numbers involved in the exponentiation are 3 and 2; the 4 is 'attached' to the 3 by multiplication, and this multiplication can't be done until the exponentiation has been performed.

• The exponentiation operation is therefore 3^2 = 9, and the expression becomes 3 * 5 - 4 * 9.

Evaluating this expression, the multiplications 3 * 5 and 4 * 9 must be performed before the subtraction. 3 * 5 = 15 and 4 * 9 = 36 so we now have

3 * 5 - 4 * 3 ^ 2 = 3 * 5 - 4 * 9 = 15 - 36 = -21.

To calculate 3 * 5 - (4 * 3)^2 we first do the operation in parentheses, obtaining 4 * 3 = 12. Then we apply the exponentiation to get 12 ^2 = 144. Finally we multiply 3 * 5 to get 15. Putting this all together we get

3 * 5 - (4 * 3)^2 =

3 * 5 - 12^2 =

3 * 5 - 144 =

15 - 144 =

-129.

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Question: `q007. Let y = 2 x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your Solution:

My Answer: -1. If x equals Negative 2 then negative 2 times 2 equals negative 4 plus 3 equals negative 1.

Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table.

My Answer: X=-2, Y=2(-2)+3, Y=-1

X=-1, Y=2(-1)+3, Y=1

X=0, Y=2(0)+3, Y=3

X=1, Y=2(1)+3, Y=5

X=2, Y=2(2)+3, Y=7

x y

-2 -1

-1 1

0 3

1 5

2 7

My Answer: My graph best represents a linear graph. I chose this because it goes from the bottom left to the top right in a straight line (#### Which looks most similar to linear).

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Question: `q008. Let y = x^2 + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

Evaluate y for x = -2. What is your result? In your solution explain the steps you took to get this result.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your Solution:

My Answer: 7. If x equals negative 2 then negative 2 squared equals 4 since squaring a negative number gives you a positive number, then add 3 to get 7.

Evaluate y for x values -1, 0, 1 and 2. Write out a copy of the table below. In your solution give the y values you obtained in your table.

My Answer: X=-2, Y=(-2)^2+3, Y=4+3, Y=7

X=-1, Y=(-1)^2+3, Y=1+3, Y=4

X=0, Y=(0)^2+3, Y=0+3, Y=3

X=1, Y=(1)^2+3, Y=1+3, Y=4

X=2, Y=(2)^2+3, Y=4+3, Y=7

x y

-2 7

-1 4

0 3

1 4

2 7

My Answer: The graph I chose to best represent this equation would be Quadratic or Parabolic since it starts from the top left, comes down to 3 when X equals 0, then goes back up to the top right, and it would be a curved line (#### and equal points left or right of the Y axis).

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Question: `q009. Let y = 2 ^ x + 3. (Note: Liberal Arts Mathematics students are encouraged to do this problem, but are not required to do it).

Evaluate y for x = 1. What is your result? In your solution explain the steps you took to get this result.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your Solution:

My Answer:5. If X equals 1 then 2 to the power of 1 equals 2, then add 3 to get 5.

Evaluate y for x values 2, 3 and 4. Write out a copy of the table below. In your solution give the y values you obtained in your table.

My Answer: X=1, Y=2^1+3, Y=2+3, Y=5

X=2, Y=2^2+3, Y=4+3, Y=7

X=3, Y=2^3+3, Y=8+3, Y=11

X=4, Y=2^4+3, Y=16+3, Y=19

x y

1 5

2 7

3 11

4 19

My Answer: The graph I would chose to best represent this equation would be Exponential since at X equals 1, Y equals 5 and as X gets bigger, Y increases faster which would make the line curve up sharply.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Question: `q010. If you divide a certain positive number by 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

My Answer: The result would be the same as the original number because any number divided by one equals the original number.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Question: `q011. If you divide a certain positive number by a number greater than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

My Answer: The answer will always be smaller. As long as the number you are dividing by is greater than one then the answer has to be smaller than the original since a number divided by one is equal to the number.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q012. If you divide a certain positive number by a positive number less than 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

My Answer: The answer will always be larger. As long as the number you are dividing by is less than one, then the answer has to be larger since a number divided by one is equal to the number.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q013. Students often get the basic answers to nearly all, or even all these questions, correct. Your instructor has however never seen anyone who addressed all the subtleties in the given solutions in their self-critiques, and it is very common for a student to have given no self-critiques. It is very likely that there is something in the given solutions that is not expressed in your solution.

This doesn't mean that you did a bad job. If you got most of the 'answers' right, you did fine.

However, in order to better understand the process, you are asked here to go back and find something in one of the given solutions that you did not address in your solution, and insert a self-critique. You should choose something that isn't trivial to you--something you're not 100% sure you understand.

If you can't find anything, you can indicate this below, and the instructor will point out something and request a response (the instructor will select something reasonable, but will then expect a very good and complete response). However it will probably be less work for you if you find something yourself.

Your response should be inserted at the appropriate place in this document, and should be indicated by preceding it with ####.

As an answer to this question, include a copy of whatever you inserted above, or an indication that you can't find anything.

your answer: vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv

Question 4: (#### to get 15 since multiplication comes before addition)

Question 5: (#### or 2*2*2*2*2*2*2*2*2*2*2*2)

Question 7: (#### Which looks most similar to linear).

Question 8: (#### and equal points left or right of the Y axis).

"

@& Instructions request that you insert your responses into a copy of the original web document. You appear to have eliminated some of the text from the original document.

The original text includes certain strings of characters that allow me to efficiently isolate your insertions from the rest of the text, allowing me to review much more student work, much more accurately, than would otherwise be possible.

These and other characters are also used as triggers in collecting databases of student responses, and in some instances the lack of these characters can cause your document not to post at all.

If so requested below, you should resubmit this document, and insert your responses into a complete, unaltered copy of the original document. If you are not specifically requested to do so on this document, it won't be necessary, but you should follow this practice on all future submissions.

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*@

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You are welcome, if you wish, to insert your answers into a complete copy of the document and resubmit it, but if you understand everything this won't be necessary.*@