course Phy 201
This is for phy 201 and 202
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.*********************************************
Your solution: (type in your solution starting in the next line)
Using the rate of $8/hour, you can divide $72 by $8 to get the number of hours it would take to achieve this payment.
$72/ ($8/hr) = 9 hours
Dividing $72 by $8/hr cancels out the $’s and leaves only hours
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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
Self-critique Rating: OK
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)
(8+3) * 5= 55
First you add 8 and 3 giving 11, then you multiply by 5 giving 55. The rules regarding order of operations requires that you start with things in parentheses and then multiply
8+3*5= 23
First you multiply 3 and 5 giving 15, then you add 8 giving 23
Order of operations requires that you start with multiplication then add.
The differences result from the differences in order of operations brought on by the parentheses.
Confidence Assessment: 3
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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
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.
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:
Grouping again dictates this answer, the larger answer in the second equation results because 4 and 3 are multiplied first.
(2^4) * 3 = 48
Start with 2^4 which equals 16, then multiply by 3 which equals 48
2^(4*3)= 4096
Start my multiplying 4 by 3 which equals 12, then compute 2^12 which equals 2096
Confidence Assessment:
3
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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, I could have better explained the order of operations and the exponential equation.
Self Critique Rating: OK
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:
3*5-4*3^2=
First you raise 3 to the 2nd power which is 9 giving, 3*5-4*9. Then you multiply each component (3*5 and 4*9). This gives the equation 15-36, which equals -21.
The answer is -21.
For the second equation, 3*5- (4*3)^2 you start with the parentheses, leading to 3*5- 12^2, then you do the exponent, leading to 3*5-144, then you multiply giving 15-144 and a final answer of
-129.
The answer is -129.
The difference between these two equations is a result of grouping, the parentheses in the second equation cause that group to be multiplied first while in the first equation, the exponent is done first.
Confidence Assessment:
3
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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):
OK
Self-critique Rating:
OK
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.
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Your solution:
y= 2x + 3
Solving for x=-2 gives 2(-2) +3 which equals -4 +3 which equals -1.
The answer to the first bullet is -1
The first step is plugging -2 in for x and then solving based on order of operations. This requires the multiplication of 2(-2) first followed by the addition of 3.
For x= -2, -1, 0, 1, 2
X= -2 y=-1, x= -1 y= 1, x=0 y=3, x= 1 y=5, x=2 y=7
The graph is a straight line meaning it is a linear graph. It has a positive slope of 2. The y intercept is 3 and the x intercept seems to be about -1.5. There is only one x intercept.
Confidence Assessment: 3
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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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Self-critique (if necessary):
OK
Self-critique Rating: OK
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.
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Your solution:
Y= x^2 + 3
When x=-2; y=-2^2 + 3 = 7. (-2)(-2)=4+3=7. Then answer for x=-2 is 7.
When x= -2, y=7. When x=-1, (-1)(-1)=1+3= 4, so when x=-1, y=4. When x=0, (0)(0)+3=3, so y=3. When x=1, (1)(1) + 3, so when x=1, y=4. When x=2, (2)(2)=4 +3= 7.
The negative and positive integers equal the same do to the exponent of 2. A negative times a negative is a positive and a positive by a positive is also a positive. Due to these equal values, the graph has an upward, bowl like shape, or is concave up. This graph is a parabola. The Y intercept is 3 and there are no x intercepts. The point at (0,3) is the minimum of this graph and there are no maximums.
Confidence Assessment: 3
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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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Self-critique (if necessary):
3, I could have discussed the symmetry of the graph and compared it to the other graphs more closely.
Self-critique Rating: 3
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.
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Your solution:
y = 2 ^ x + 3
Solve for x= . First you take 2^1 which is equal to 2. Then you add this number (2) to 3 which gives a solution of 5. When x=1, y=5.
When x=2, First you take 2^2 which equals 4 and then add this 4 to 3 which equals 7.
When x=3, first you take 2^3, which equals 8 and add it to 3 which equals 11.
When x=4, first you take 2^4, which equals 16 and add 3 which equals 19.
The graph intersects the y axis at 4. This is because 2^0 is equal to 1 and adding 3= 4. The graph increases in an exponential fashion to the right. The slope increases dramatically as you move to the right and decreases to 0 as you move to the left. There is no symmetry. This information indicates the graph is exponential.
Confidence Assessment: 3
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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
Self-critique Rating:
OK
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:
The result is always equal to the original number, any integer divided by 1 is equal to itself.
Confidence Assessment: 3
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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
Self-critique Rating:
OK
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:
The answer will always be less than the original number. Taking any number and dividing by a number greater than one will lead to a splitting of the value and a smaller number.
Confidence Assessment: 3
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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):
OK
Self-critique Rating:
OK
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:
Dividing by a number smaller than one always results in a greater number. Division by a fraction is the same as multiplication by the inverse of the fraction. Because of this the number will always be greater.
Confidence Assessment: 3
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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):
I could have made my explanation a bit more clear. It may not make perfect sense to an outside reader.
Self-critique Rating:
3
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