course MTH 163

Here are the remaining ten questions:*********************************************

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.

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

To determine how long it would take to earn $72 at a rate of 8 dollars an hour, I would divide 72 dollars by 8 dollars and come to an answer of 9 hours.

Confidence Assessment: 3

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Self-critique (if necessary): OK

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

I follow the order of operations as such: Powers, Parenthesis, Multiply, Divide, Add, Subtract. To calculate (8+3) * 5, I would first evaluate the expression in parenthesis (8+3) = 11, and then multiply the result by 5, for a final total of 55. In the second equation, I would first multiply 3*5 = 15, and then add 8, for a total of 23.

Confidence Assessment: 3

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Self-critique (if necessary): OK

Self-critique Rating: OK

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:

In the first equation, I would begin by evaluating 2 to the 4th power, which is 16, then multiply by 3, coming to an answer of 48. In the second equation, I would first evaluate the expression in parenthesis, (4*3) = 12, and then calculate the power of 2^12, which is 4096.

Confidence Assessment: 3

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Self-critique (if necessary): OK

Self-critique Rating: OK

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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:

In the first equation, I would begin by evaluations 3 to the 2nd power, which is 9, then multiply, 9 * 4 = 36 and 3 * 5 = 15. The equation would now be 15 - 36 which would evaluate to -21. In the second equation, I would first evaluate the parenthesis 4*3 = 12, then the power 12^2 = 144, then multiply 5*3 = 15, and finally subtract, 15-144 = -129.

Confidence Assessment: 3

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Self-critique (if necessary): OK

Self-critique Rating: OK

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.

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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

0

1

2

*

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:

Evaluating y = 2x + 3, where x = -2 would equate to y = 2(-2) + 3. First multiply 2 * -2 = -4, then add, y = -4 + 3, solution y = -1.

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

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

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

y = 2(2) + 3 = 4 + 3 = 7

x y

-2 -1

-1 1

0 3

1 5

2 7

For the set of coordinates, the graph is a straight line, inclining from the lower left quadrant to the upper right quadrant. This graph most closely resembles the linear graph, as it is basically idential to the one shown.

Confidence Assessment: 3

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Self-critique (if necessary): OK

Self-critique Rating: OK

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Given 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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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

0

1

2

*

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:

Evaluating y = x^2 + 3, where x = -2 would equate to y = (-2)^2 + 3. First evaluate -2^2 = 4, then add, y = 4 + 3, solution y = 7.

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

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

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

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

x y

-2 7

-1 4

0 3

1 4

2 7

For the set of coordinates, the graph resembles a symmetric 'V' or 'U' shape, which most closely resembles the quadratic or parabolic illustration.

Confidence Assessment: 3

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Self-critique (if necessary): OK

Self-critique Rating: OK

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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 = ` + 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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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

1

2

3

4

*

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:

Evaluating y = 2^x + 3, where x = 1 would equate to y = 2^(1) + 3. First evaluate 2^1 = 2, then add, y = 2 + 3, solution y = 5.

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

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

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

x y

1 5

2 7

3 11

4 19

For the set of coordinates, the graph resembles a curve increasing from the lower left to the upper right, which most closely resembles the illustration of the exponential graph, which curves upward in a similar fashion.

Confidence Assessment: 3

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Self-critique (if necessary): OK

Self-critique Rating: OK

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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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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:

Any given positive number divided by 1 will be equal to the original given number.

Confidence Assessment: 3

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Self-critique (if necessary): OK

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:

If you divide a given positive number by a number greater than one, then the result will be less than the original number, as it is being fractioned into smaller parts.

Confidence Assessment: 3

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Self-critique (if necessary): OK

Self-critique Rating: OK

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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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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 a positive number by a number less than 1, you are diving the original number by a decimal or fraction. The result therefore becomes greater than the original number because you splitting the original number up into a smaller fraction, resulting in a larger outcome. This is hard to put into understandable words.

Confidence Assessment: 3

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Self-critique (if necessary): OK

Self-critique Rating: OK

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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.

&#Your work looks good. Let me know if you have any questions. &#