3-13Questions

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course Phy 232

5/29 8

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)

The rate the individual is earning is given at 8 dollars per hour. Furthermore, we divide the rate by the total amount of money earned, $72. Therefore, $72/8 dollars per hour equals 9 hours. This works because the dollars cancel out and the hours is moved to the numerator through the division process.

confidence rating #$&*:

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Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

2 indicates that you believe you addressed most of the discrepancies between the given solution and your solution but are unsure of some aspects of the situation; you would at this point consider including a question or a statement of what you're not sure you understand

1 indicates that you believe you understand the overall idea of the solution but have not been able to address the specifics of the discrepancies between your solution and the given solution; in this case you would normally include a question or a statement of what you're not sure you understand

0 indicates that you don't understand the given solution, and/or can't make a reasonable judgement about whether or not your solution is correct; in this case you would be expected to address the given solution phrase-by-phrase and state what you do and do not understand about each phrase)

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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): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

'OK'

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Self-critique Rating:3

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

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

These problems are easily answered using order of operations.

For (8+3)*5, the first step is to do what is inside the parenthesis. So, 8+3=11. Then, the next step would to be to multiply to get 11*5=55.

For 8+3*5, the first step would to be to multiply the 5 and 3. After that step, the individual is left with 8+15=23.

As stated earlier, the reason for the differences was order of operations and in this case the use of parenthesis. The parenthesis makes it so the addition is the first step for the first question where as the second question does the multiplication first.

confidence rating #$&*:

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Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

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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): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'.

Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique.

If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance.

Insert your 'OK' or your self-critique, as appropriate, starting in the next line:

'OK'

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Self-critique Rating:'OK'

Your self-critique rating should be entered on the line above, after the colon at the end of the prompt.

Your self-critique rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

(If you believe your solution matches the given solution then just type in 'OK'.

Otherwise evaluate the quality of your self-critique by typing in a number between 0 and 3.

3 indicates that you believe you have addressed all discrepancies between the given solution and your solution, in such a way as to demonstrate your complete understanding of the situation.

In subsequent problems the detailed instructions that accompanied the first four problems are missing. We assume you will know to follow the same instructions in answering the remaining questions.

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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:

Once again, significant knowledge of the order of operations is needed to solve these problems.

Both of these problems begin with the numbers within the parenthesis.

So, first calculate (2^4) which is 16. Then, multiply 16*3 to get 48.

The second problem starts by multiplying 4*3 which is equal to 12. Then, do 2^12. After a little calculating, one would come to the answer of 4096.

confidence rating #$&*:

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

Your Confidence Rating should be entered on the line above, after the colon at the end of the prompt.

Your Confidence Rating is a number from 0 to 3, which is to indicate your level of confidence in your solution.

3 means you are at least 90% confident of your solution, or that you are confident you got at least 90% of the solution

2 means that you are more that 50% confident of your solution, or that you are confident you got at least 50% of the solution

1 means that you think you probably got at least some of the solution correct but don't think you got the whole thing

0 means that you're pretty sure you didn't get anything right)

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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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Self-critique (if necessary):3. My solution could of been more thorough in the exponential process. You did a nice job of explaining why a

number to a power equals a certain number. That is something I had failed to do and believe it shows in my less thorugh solution.

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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.

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Your solution:

The first problem starts with doing the 3^2. Which as shown earlier through your solution process, equals 3*3=9.Then, do the multiplication steps of the problem.

This includes 3*5 which equals 15 and 4*9 which equals 36. As a result, we are left with 15-36 which equals -21.

The second problem starts by doing the math within the parenthesis, which is 4*3=12. Then we are left with 3*5-12^2. As we know, the exponential step comes next which is 12*12=144.

Lastly, we do 3*5=15 and subtract 15 by 144. As a result, we are left with -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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Self-critique (if necessary):

Even though we both took a different approach to explaining the problem, I believe I do an adequate job in my method, However, I believe I will use your method next time because it is easier to follow and understand.

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Self-critique Rating:3

In the next three problems, the graphs will be of one of the basic shapes listed below. You will be asked to construct graphs for three simple functions, and determine which of the depicted graphs each of your graphs most closely resembles. At this point you won't be expected to know these terms or these graph shapes; if at some point in your course you are expected to know these things, they will be presented at that point.

Linear:

Quadratic or parabolic:

Exponential:

Odd power:

Fractional positive power:

Even negative power:

partial graph of polynomial of degree 3

more extensive graph of polynomial of degree 3

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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.

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.

x y

-2

-1

Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

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Your solution:

To find the coordinates, you plug the x-values into the formula y=2x+3.

x y

-2 -1 y=2(-2)+3=-1

-1 1 y=2(-1)+3=1

0 3 y=2(0)+3=3

1 5 y=2(1)+3=5

2 7 y=2(2)+3=7

My graph shows as a straight line going through the coordinate points given above. Since my graph is a straight line, the graph is best described as linear.

confidence rating #$&*:

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Given Solution:

Two slightly different explanations are give below, one by a student and one by the instructor. Neither format is inherently better than the other.

GOOD SOLUTION BY STUDENT:

First we need to complete the table. I have added a column to the right of the table to show the calculation of �y� when we us the �x� values as given.

x y Calculation: If y = 2x + 3

-2 -1 If x = -2, then y = 2(-2)+3 = -4+3 = -1

-1 1 If x= -1, then y = 2(-1)+3 = -2+3 = 1

0 3 If x= 0, then y = 2(0)+3 = 0+3 = 3

1 5 If x= 1, then y = 2(1)+3 = 2+3 = 5

2 7 If x= 2, then y = 2(2)+3 = 4+3 = 7

Once an answer has been determined, the �y� value can be filled in. Now we have both the �x� and �y� values and we can begin our graph. The charted values continue on a straight line representing a linear function as shown above.

INSTRUCTOR'S SOLUTION:

We easily evaluate the expression:

When x = -2, we get y = 2 x + 3 = 2 * (-2) + 3 = -4 + 3 = -1.

When x = -1, we get y = 2 x + 3 = 2 * (-1) + 3 = -2 + 3 = 1.

When x = 0, we get y = 2 x + 3 = 2 * (0) + 3 = 0 + 3 = 3.

When x = 1, we get y = 2 x + 3 = 2 * (1) + 3 = 2 + 3 = 5.

When x = 2, we get y = 2 x + 3 = 2 * (2) + 3 = 4 + 3 = 7.

Filling in the table we have

x y

-2 -1

-1 1

0 3

1 5

2 7

When we graph these points we find that they lie along a straight line.

Only one of the depicted graphs consists of a straight line, and we conclude that the appropriate graph is the one labeled 'linear'.

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Self-critique (if necessary):'OK'

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Self-critique Rating:'OK'

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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.

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.

x y

-2

-1

Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

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Your solution:

To find the coordinates, you plug the x-values into the formula y=x^2 + 3.

x y

-2 7 y=(-2)^2 + 3= 7

-1 4 y=(-1)^2 + 3= 4

0 3 y=(0)^2 + 3= 3

1 4 y=(1)^2 + 3= 4

2 7 y=(2)^2 + 3= 7

After plotting the points one will notice that the graph has a very well known shape. This shape is a parabola. This is defined based on its symmetric shape reflected about the y-axis.

The lowest point of the graph is at the coordinates (0,3). This is where the graph dips and could be thought of as the start of the symmetric shape.From the spot, the graph is easily made using the points already discovered.

Therefore, the graph best resembles the one named quadratic or parabolic.

confidence rating #$&*:

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Given Solution:

Evaluating y = x^2 + 3 at the five points:

If x = -2 then we obtain y = x^2 + 3 = (-2)^2 + 3 = 4 + 3 = 7.

If x = -1 then we obtain y = x^2 + 3 = (-1)^2 + 3 = 1 + 3 = 4.

If x = 0 then we obtain y = x^2 + 3 = (0)^2 + 3 = 0 + 3 = 3.

If x = 1 then we obtain y = x^2 + 3 = (1)^2 + 3 = 1 + 3 = 4.

If x = 2 then we obtain y = x^2 + 3 = (2)^2 + 3 = 4 + 3 = 7.

The table becomes

x y

-2 7

-1 4

0 3

1 4

2 7

We note that there is a symmetry to the y values. The lowest y value is 3, and whether we move up or down the y column from the value 3, we find the same numbers (i.e., if we move 1 space up from the value 3 the y value is 4, and if we move one space down we again encounter 4; if we move two spaces in either direction from the value 3, we find the value 7).

A graph of y vs. x has its lowest point at (0, 3).

If we move from this point, 1 unit to the right our graph rises 1 unit, to (1, 4), and if we move 1 unit to the left of our 'low point' the graph rises 1 unit, to (-1, 4).

If we move 2 units to the right or the left from our 'low point', the graph rises 4 units, to (2, 7) on the right, and to (-2, 7) on the left.

Thus as we move from our 'low point' the graph rises up, becoming increasingly steep, and the behavior is the same whether we move to the left or right of our 'low point'. This reflects the symmetry we observed in the table. So our graph will have a right-left symmetry.

Two of the depicted graphs curve upward away from the 'low point'. One is the graph labeled 'quadratic or parabolic'. The other is the graph labeled 'partial graph of degree 3 polynomial'.

If we look closely at these graphs, we find that only the first has the right-left symmetry, so the appropriate graph is the 'quadratic or parabolic' graph.

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Self-critique (if necessary):

I believe my solution was good, but lacked some of the details that the given solution had. Some of the aspects my solution lacked, was how the increase in the graph was not the same throughout but increased as the the x values got farther from the y axis.

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Self-critique Rating:3

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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.

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.

x y

Sketch a graph of y vs. x on a set of coordinate axes resembling the one shown below. You may of course adjust the scale of the x or the y axis to best depict the shape of your graph.

In your solution, describe your graph in words, and indicate which of the graphs depicted previously your graph most resembles. Explain why you chose the graph you did.

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Your solution:

x y

1 5 y = 2 ^ 1 + 3 = 2+3 = 5

2 7 y = 2 ^ 2 + 3 = 4+3 = 7

3 11 y = 2 ^ 3 + 3 = 8+3 = 11

4 19 y = 2 ^ 4 + 3 = 16+3 = 19

After graphing the coordinates, it is obvious that the graph is listed as exponential. This is because the graph's y values exponentially change as the x values stay constant.

To describe this, all one needs to do is look at the values. The difference between the y values gets larger as the x value increase. This is shown as the first difference in y values is 7-5=2 and the second increases to 11-7=4.

This idea is defined with the word exponential because the y values get exponentially bigger as the x-value increases.

confidence rating #$&*:

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Given Solution:

Recall that the exponentiation in the expression 2^x + 1 must be done before, not after the addition.

When x = 1 we obtain y = 2^1 + 3 = 2 + 3 = 5.

When x = 2 we obtain y = 2^2 + 3 = 4 + 3 = 7.

When x = 3 we obtain y = 2^3 + 3 = 8 + 3 = 11.

When x = 4 we obtain y = 2^4 + 3 = 16 + 3 = 19.

x y

1 5

2 7

3 11

4 19

Looking at the numbers in the y column we see that they increase as we go down the column, and that the increases get progressively larger. In fact if we look carefully we see that each increase is double the one before it, with increases of 2, then 4, then 8.

When we graph these points we find that the graph rises as we go from left to right, and that it rises faster and faster. From our observations on the table we know that the graph in fact that the rise of the graph doubles with each step we take to the right.

The only graph that increases from left to right, getting steeper and steeper with each step, is the graph labeled 'exponential'.

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Self-critique (if necessary):'OK'

I believe I could of better described the graph and how it rises faster as the graph goes from left to right. I also could of better explained the graphing techniques and how the graph looked overall. ####

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Self-critique Rating:'OK'

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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:

If you divide a positive number by 1, your result will be equal to the original number. This is shown by 15/1=15. The number stays the same when dividing by 1.

confidence rating #$&*:

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Given Solution: If you divide any number by 1, the result is the same as the original number. Doesn't matter what the original number is, if you divide it by 1, you don't change it.

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Self-critique (if necessary):'OK'

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Self-critique Rating:'OK'

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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:

When dividing a positive number by anumber greater than 1, you get a number smaller than the original number. This is shown by 4/2=2. You can better understand this by thinking it of spliting 4 into 2 equal parts. Therefore, the number is obviously going to be smaller than before.

This is concept can be used with all numbers and the example should not be thought of as the only possible example.

confidence rating #$&*:

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Given Solution: If you split something up into equal parts, the more parts you have, the less will be in each one. Dividing a positive number by another number is similar. The bigger the number you divide by, the less you get.

Now if you divide a positive number by 1, the result is the same as your original number. So if you divide the positive number by a number greater than 1, what you get has to be smaller than the original number. Again it doesn't matter what the original number is, as long as it's positive.

Students will often reason from examples. For instance, the following reasoning might be offered:

OK, let's say the original number is 36. Let's divide 36 be a few numbers and see what happens:

36/2 = 18. Now 3 is bigger than 2, and

36 / 3 = 12. The quotient got smaller. Now 4 is bigger than 3, and

36 / 4 = 9. The quotient got smaller again. Let's skip 5 because it doesn't divide evenly into 36.

36 / 6 = 4. Again we divided by a larger number and the quotient was smaller.

I'm convinced.

That is a pretty convincing argument, mainly because it is so consistent with our previous experience. In that sense it's a good argument. It's also useful, giving us a concrete example of how dividing by bigger and bigger numbers gives us smaller and smaller results.

However specific examples, however convincing and however useful, don't actually prove anything. The argument given at the beginning of this solution is general, and applies to all positive numbers, not just the specific positive number chosen here.

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Self-critique (if necessary):

Good, but I think I could of better described the concept rather than giving an example.

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Self-critique 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:

If you divide any positive number by a positive number less than 1, you get a greater number than the original value. A good way to think of this is the smaller the number you divide by the greater your result will be. This idea is pretty much the exact opposite when the number is greater than one.

confidence rating #$&*:

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Given Solution: If you split something up into equal parts, the more parts you have, the less will be in each one. Dividing a positive number by some other number is similar. The bigger the number you divide by, the less you get. The smaller the number you divide by, the more you get.

Now if you divide a positive number by 1, the result is the same as your original number. So if you divide the positive number by a positive number less than 1, what you get has to be larger than the original number. Again it doesn't matter what the original number is, as long as it's positive.

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Self-critique (if necessary):'OK'

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Self-critique Rating:'OK'

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Your solution:

confidence rating #$&*:

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Self-critique (if necessary):'OK'

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Self-critique Rating:'OK'

#*&!

3-13Questions

&#Good responses. Let me know if you have questions. &#