#$&* course MTH 163 Jan. 22 11:04 pm 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
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Given Solution: Many students simply know, at the level of common sense, that if we divide $72 by $8 / hour we get 9 hours, so 9 hours are required. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): If you are sure your solution matches the given solution, and/or are sure you completely understand the given solution, then just type in 'OK'. Otherwise you should include a self-critique. In your self-critique you should explain in your own words how your solution differs from the given solution, and demonstrate what you did not originally understand but now understand about the problem and its solution. Note that your instructor scans your document for questions and indications that you are having difficulty, usually beginning with your self-critique. If no self-critique is present, your instructor assumes you understand the solution to your satisfaction and do not need additional information or assistance. If you do not fully understand the given solution, and/or if you still have questions after reading and taking notes on the given solution, you should self-critique in the manner described in the preceding paragraph. Insert your 'OK' or your self-critique, as appropriate, starting in the next line: ------------------------------------------------ Self-critique Rating: OK Your self-critique rating should be entered on the line above, after the colon at the end of the prompt. ********************************************* Question: `q004. Calculate (8 + 3) * 5 and 8 + 3 * 5, indicating the order of your steps. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: First solve the parenthesis: (8+3)=11 then add 11*5 which equals 55. Multiplication comes first, so multiply 3*5 = 15 and then add 15 + 8 = 23 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& 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: ------------------------------------------------ Self-critique Rating: OK ********************************************* 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: First evaluate parenthesis, 2 to the 4th which equals 16, so 16*3 = 48. First evaluate 4*3 which equals 12, so 2 to the 12th power equals 4096. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: To evaluate (2^4) * 3 we first evaluate the grouped expression 2^4, which is the fourth power of 2, equal to 2 * 2 * 2 * 2 = 16. So we have (2^4) * 3 = 16 * 3 = 48. To evaluate 2^(4 * 3) we first do the operation inside the parentheses, obtaining 4 * 3 = 12. We therefore get 2^(4 * 3) = 2^12 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 4096. It is easy to multiply by 2, and the powers of 2 are important, so it's appropriate to have asked you to do this problem without using a calculator. Had the exponent been much higher, or had the calculation been, say, 3^12, the calculation would have become tedious and error-prone, and the calculator would have been recommended. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ********************************************* Question: `q006. Calculate 3 * 5 - 4 * 3 ^ 2 and 3 * 5 - (4 * 3)^2 according to the standard order of operations, indicating the order of your steps. Explain, as best you can, the reasons for the difference in your results. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: First do exponents, so 3 to the 2nd power equals 9, so 3*5-4*9 . Do multiplication next, so 3*5=15 and 4*9=36, so 15-36=-21. First do parenthesis, so 4*3=12. Next do 12 to the 2nd power=144. Then multiply 3*5=15 so, 15-144=-129. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: ********************************************* 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 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: when x = -2, y = -1, when x=-1, y= 1, when x= 0 y=3, when x=1 y=5, when x=2 y=7. I get this by plugging in x and evaluating the equation by equaling both sides. My graph had a straight line, making it 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. 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'. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: ********************************************* 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: when x=-2 y= 7, when x=-1 y=4, when x= 0 y=3, when x=1 y=4, when x=2 y=7. I solved the equation for x by plugging in -2,-1,0,1,2 when x was. My graph made a U shape, making it quadratic. 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 = ` + 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. 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique Rating: ********************************************* 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: when I plug in the x value substitution, and solve for y I get: x=1 y=5 x=2 y=7 x=3 y=11 x=4 y=19 My graph went from bottom and rose steeper, making it an exponential. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Recall that the exponentiation in the expression 2^x + 1 must be done before, not after the addition. When x = 1 we obtain y = 2^1 + 3 = 2 + 3 = 5. When x = 2 we obtain y = 2^2 + 3 = 4 + 3 = 7. When x = 3 we obtain y = 2^3 + 3 = 8 + 3 = 11. When x = 4 we obtain y = 2^4 + 3 = 16 + 3 = 19. Looking at the numbers in the y column we see that they increase as we go down the column, and that the increases get progressively larger. In fact if we look carefully we see that each increase is double the one before it, with increases of 2, then 4, then 8. When we graph these points we find that the graph rises as we go from left to right, and that it rises faster and faster. From our observations on the table we know that the graph in fact that the rise of the graph doubles with each step we take to the right. The only graph that increases from left to right, getting steeper and steeper with each step, is the graph labeled 'exponential'. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q010. If you divide a certain positive number by 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If you divide anything by one, it equals itself. No matter what. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: ********************************************* 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: If you have a whole, say 30 and divide it by a small number it will be more (30/3=10). but, if you divide it by a large number it will be smaller (30/10=3) 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): This is a tricky question, when I really think about it, it all made sense. ------------------------------------------------ Self-critique 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: The larger the number you divide something into, the number is smaller but when dividing by a small number, it will equal a larger number. 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Another tricky question! ------------------------------------------------ Self-critique 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 In my work, I did not do the charts to the graph questions exactly like you did. When I concentrated on yours, it did make a lot of sense and seemed easier. "