#$&* course Phy 121 Here are the remaining ten questions:*********************************************
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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: OK ------------------------------------------------ Self-critique Rating: 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. • 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) 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (type in your solution starting in the next line) (8+3) * 5= First you will add 8 + 3 together because they are in the parenthesis. 8 +3 = 11. Then you will multiply 11 by 5 and it equals 55. 8+3*5= Since there is not an indication of where to start you start by multiplying 3*5 which equals 15. When not indicated in the problem by parenthesis of where to start will always start by multiply and dividing an answer in order from left to right and then go back to the beginning to add and subtract. 8+ 15 = 23 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) 2
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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: OK ------------------------------------------------ Self-critique Rating: 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. • 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) 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (2^4)*3= Solve what is in the parenthesis first. 2^4= 2*2*2*2= 16. 16 * 3= 48. 2^(4*3)=Solve the parenthesis first. 4*3 = 12. 2^12= 2*2*2*2*2*2*2*2*2*2*2*2= 4,096. 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) 2
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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): I need to be more confident in my answers. ********************************************* 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: 3*5-4*3^2= First, you multiply 3*5= 15 and 4*3=12. This turns the problem into 15-12^2=15-12=3^2=3*3= 9. 3*5-(4*3)^2= First, solve the parenthesis. 4*3= 12. 3*5=15. 12^2= 12*12= 144. 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): I did not know to get rid of the exponent first. It has been some time since doing this math and I do like the refresher. If I had remembered this one step I could have gotten the correct answer for the first problem. The second equation I knew how to do. ------------------------------------------------ Self-critique Rating: 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 ********************************************* 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. Y=2(-2)+3. You add the parenthesis to the known x because you will multiply the numbers together. 2*-2= -4. -4+3=-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. x y -2 -1 -1 1 0 3 1 5 2 7 • 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: I chose a liner graph because the line is straight and increasing at a positive slope. 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'. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I did not show enough necessary work. I showed my answer for the first problem and did the rest of the problems in my head based off the first. My answers are correct. ------------------------------------------------ Self-critique Rating:2 ********************************************* 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The graph depicted is an exponential graph. The solutions are symmetrical on each side of the graph. 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. 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I did not go in as depth as I should have. I had the understanding of the graph, just not able to fit it in as many words as I should have been able to do so. ------------------------------------------------ Self-critique Rating:2 ********************************************* 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. On the Y axis 1=5 • 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: The graph is depicting odd power. The graph gradually starts to slop to the right which is positive and then a sharp increase. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Recall that the exponentiation in the expression 2^x + 1 must be done before, not after the addition. When x = 1 we obtain y = 2^1 + 3 = 2 + 3 = 5. When x = 2 we obtain y = 2^2 + 3 = 4 + 3 = 7. When x = 3 we obtain y = 2^3 + 3 = 8 + 3 = 11. When x = 4 we obtain y = 2^4 + 3 = 16 + 3 = 19. x y 1 5 2 7 3 11 4 19 Looking at the numbers in the y column we see that they increase as we go down the column, and that the increases get progressively larger. In fact if we look carefully we see that each increase is double the one before it, with increases of 2, then 4, then 8. When we graph these points we find that the graph rises as we go from left to right, and that it rises faster and faster. From our observations on the table we know that the graph in fact that the rise of the graph doubles with each step we take to the right. The only graph that increases from left to right, getting steeper and steeper with each step, is the graph labeled 'exponential'. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I need to be able to explain in more depth. ------------------------------------------------ Self-critique Rating:2 ********************************************* Question: `q010. If you divide a certain positive number by 1, is the result greater than the original number, less than the original number or equal to the original number, or does the answer to this question depend on the original number? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Examples: 1/1=1 2/1=2 3/1=3 The result is always equal to the original number. The original number will always be divided by 1 so the answer will be the resulting original whole number. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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): I understood the question, but I answered in a different manner than the answer, but with the same results. ------------------------------------------------ Self-critique Rating:2 ********************************************* 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: Examples: 10/2=5 9/3=3 16/4=4 If you divide a whole positive number by any number greater than 1 then the result will always be less than the original starting 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 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): I did not go as far in depth as I should have but I did give examples to support my answer. ------------------------------------------------ Self-critique Rating:2 ********************************************* 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: Examples: 10/-1=-10 10/-2=-5 If you divide a certain positive number by any number less than 1 (which will be a negative number) then the resulting outcome will always be less than the original positive number. Any positive number divided by a negative number will always be negative. 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): I did not get the answer correct at all. I must have misunderstood the question being asked and answered a totally separate question all together. #### ------------------------------------------------ Self-critique Rating:0
#$&* course Phy 121 001. typewriter notation
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Given Solution: `aThe order of operations dictates that grouped expressions must be evaluated first, that exponentiation must be done before multiplication or division, which must be done before addition or subtraction. It makes a big difference whether you subtract the 2 from the x or divide the -2 by 4 first. If there are no parentheses you have to divide before you subtract. Substituting 2 for x we get 2 - 2 / 2 + 4 = 2 - 1 + 4 (do multiplications and divisions before additions and subtractions) = 5 (add and subtract in indicated order) If there are parentheses you evaluate the grouped expressions first: (x - 2) / (x + 4) = (2 - 2) / ( 2 + 4 ) = 0 / 6 = 0. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Still need to work on a more descriptive description. ------------------------------------------------ Self-critique Rating:2 ********************************************* Question: `q002. Explain the difference between 2 ^ x + 4 and 2 ^ (x + 4). Then evaluate each expression for x = 2. Note that a ^ b means to raise a to the b power. This process is called exponentiation, and the ^ symbol is used on most calculators, and in most computer algebra systems, to represent exponentiation. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: When there are parentheses in the equation you always solve what is inside of the parenthesis first. When there are no parentheses you will solve to find the exponentiation first and then do what is inside of the parenthesis unless the number connected to the parenthesis is to be solving for the exponentiation. Then, you will solve what is inside the parenthesis first and then solve for the exponentiation. 2^2+4= 2*2+4= 4+4= 8 2^(2+4)= 2^6= 2*2*2*2*2*2=64 confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a2 ^ x + 4 indicates that you are to raise 2 to the x power before adding the 4. 2 ^ (x + 4) indicates that you are to first evaluate x + 4, then raise 2 to this power. If x = 2, then 2 ^ x + 4 = 2 ^ 2 + 4 = 2 * 2 + 4 = 4 + 4 = 8. and 2 ^ (x + 4) = 2 ^ (2 + 4) = 2 ^ 6 = 2*2*2*2*2*2 = 64. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `q003. What is the numerator of the fraction in the expression x - 3 / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x? What is the denominator? What do you get when you evaluate the expression for x = 2? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The numerator is x-3= -1 The denominator is [(2x-5)^2*3x+1]-2+7x= -16 2-3/[(2(2)-5)^2*3(2)+1]-2+7(2)= 2-3/[(4-5)^2*6+1]-2+14= 2-3/[-1^12+1]-16= 2-3/[-1+1]-16= 2-3/[0]-16= 2-3/-16= -1/-16 confidence rating #$&*:0 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe numerator is 3. x isn't part of the fraction. / indicates division, which must always precede subtraction. Only the 3 is divided by [ (2x-5)^2 * 3x + 1 ] and only [ (2x-5)^2 * 3x + 1 ] divides 3. If we mean (x - 3) / [ (2x-5)^2 * 3x + 1 ] - 2 + 7x we have to write it that way. The preceding comments show that the denominator is [ (2x-5)^2 * 3x + 1 ] Evaluating the expression for x = 2: - 3 / [ (2 * 2 - 5)^2 * 3(2) + 1 ] - 2 + 7*2 = 2 - 3 / [ (4 - 5)^2 * 6 + 1 ] - 2 + 14 = evaluate in parenthese; do multiplications outside parentheses 2 - 3 / [ (-1)^2 * 6 + 1 ] -2 + 14 = add inside parentheses 2 - 3 / [ 1 * 6 + 1 ] - 2 + 14 = exponentiate in bracketed term; 2 - 3 / 7 - 2 + 14 = evaluate in brackets 13 4/7 or 95/7 or about 13.57 add and subtract in order. The details of the calculation 2 - 3 / 7 - 2 + 14: Since multiplication precedes addition or subtraction the 3/7 must be done first, making 3/7 a fraction. Changing the order of the terms we have 2 - 2 + 14 - 3 / 7 = 14 - 3/7 = 98/7 - 3/7 = 95/7. COMMON STUDENT QUESTION: ok, I dont understand why x isnt part of the fraction? And I dont understand why only the brackets are divided by 3..why not the rest of the equation? INSTRUCTOR RESPONSE: Different situations give us different algebraic expressions; the situation dictates the form of the expression. If the above expression was was written otherwise it would be a completely different expression and most likely give you a different result when you substitute. If we intended the numerator to be x - 3 then the expression would be written (x - 3) / [(2x-5)^2 * 3x + 1 ] - 2 + 7x, with the x - 3 grouped. If we intended the numerator to be the entire expression after the / the expression would be written x - 3 / [(2x-5)^2 * 3x + 1 - 2 + 7x ]. STUDENT COMMENT: I wasn't sure if the numerator would be 3 or -3. or is the subtraction sign just that a sign in this case? INSTRUCTOR RESPONSE: In this case you would regard the - sign as an operation to be performed between the value of x and the value of the fraction, rather than as part of the numerator. That is, you would regard x - 3 / [ (2x-5)^2 * 3x + 1 ] as a subtraction of the fraction 3 / [ (2x-5)^2 * 3x + 1 ] from the term x. STUDENT QUESTION: There was another question I had about this problem that wasn’t addressed. At the end when you changed the order of operation from 2 - 2 + 14 - 3/7 = 14 - 3/7 where did the 98/7 - 3/7 come into play before the end solution of 95/7? I must have forgotten how to do this part. INSTRUCTOR RESPONSE: It's not clear how you can get 95/7 without this step. To do the subtraction 14 - 3/7 both terms must be expressed in terms of a common denominator. The most convenient common denominator is 7. So 14 must be expressed with denominator 7. This is accomplished by multiplying 14 by 7 / 7, obtaining 14 * 7 / 7 = 98 / 7. Since 7/7 = 1, we have just multiplied 14 by 1. We chose to use 7 / 7 in order to give us the desired denominator 7. Thus our subtraction is 14 - 3/7 = 98/7 - 3/7 = (98 - 3) / 7 = 95 /7. STUDENT COMMENT It took me a while to think thru this one especially when I got to working with the fraction. Fractions have always been my weak spot. Any tips to make working with fractions a little easier is greatly appreciated. INSTRUCTOR RESPONSE Fractions are seriously undertaught in our schools, so your comment is not unusual. I have to focus my attention on the subject matter of my courses, and while I do address it to a point, I don't have time to do justice to the subject of fractions. In any case , to do so would be redundant on my part, since there are a lot of excellent resources on the Internet. I suggest you search the Web using something like 'review of fractions', and find something appropriate to your needs. You should definitely review the topic, as should 95% of all students entering your course. STUDENT COMMENT I think I am confused on why the Numerator is not the top portion and denominator the bottom portion of the problem. INSTRUCTOR RESPONSE Everything is on one line so there is no top or bottom in the given expression. A numerator and denominator are determined by a division of two expressions. As we know, a denominator divides a numerator. In the given expression the division sign occurs between the 3 and the [ (2x-5)^2 * 3x + 1 ], so 3 is the numerator and [ (2x-5)^2 * 3x + 1 ] is the denominator. x is not divided by the denominator, since the division occurs before the subtraction. For the same reason the -2 + 7x is not involved in the division. So neither the x nor the -2 + 7 x is part of the fractional expression. STUDENT COMMENT Didn’t know that 3 / 7 was 3/7 as a fraction. INSTRUCTOR RESPONSE 3/7 is treated as a fraction because of the order of operations. 3 must be divided by 7 before any other operation is applied to either number, and 3 divided by 7 is the fraction 3/7. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): A lot of helpful tips and information in this response. I guess I have forgotten how to math all together. Did not realize that I would need to back track as far as to how to relearn how to handle fractions. ------------------------------------------------ Self-critique Rating:0 ********************************************* Question: `q004. Explain, step by step, how you evaluate the expression (x - 5) ^ 2x-1 + 3 / x-2 for x = 4. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (x-5)^2x-1+3/x-2= (4-5)^2*4-1+3/4-2= -1^2*4-1+3/4-2= 1*4-1+3/4-2= 4-1+3/4-2= 4-1-2+3/4= 1+3/4= 4/4+3/4= 1/4 confidence rating #$&*:1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aWe get (4-5)^2 * 4 - 1 + 3 / 4 - 2 = (-1)^2 * 4 - 1 + 3 / 4 - 2 evaluating the term in parentheses = 1 * 4 - 1 + 3 / 4 - 2 exponentiating (2 is the exponent, which is applied to -1 rather than multiplying the 2 by 4 = 4 - 1 + 3/4 - 2 noting that 3/4 is a fraction and adding and subtracting in order we get = 1 3/4 = 7 /4 (Note that we could group the expression as 4 - 1 - 2 + 3/4 = 1 + 3/4 = 1 3/4 = 7/4). COMMON ERROR: (4 - 5) ^ 2*4 - 1 + 3 / 4 - 2 = -1 ^ 2*4 - 1 + 3 / 4-2 = -1 ^ 8 -1 + 3 / 4 - 2. INSTRUCTOR COMMENTS: There are two errors here. In the second step you can't multiply 2 * 4 because you have (-1)^2, which must be done first. Exponentiation precedes multiplication. Also it isn't quite correct to write -1^2*4 at the beginning of the second step. If you were supposed to multiply 2 * 4 the expression would be (-1)^(2 * 4). Note also that the -1 needs to be grouped because the entire expression (-1) is taken to the power. -1^8 would be -1 because you would raise 1 to the power 8 before applying the - sign, which is effectively a multiplication by -1. STUDENT QUESTION: if it's read (-1)^8 it would be 1 or would you apply the sign afterward even if it is grouped and it be a -1? INSTRUCTOR RESPONSE: The 8th power won't occur in this problem, of course, but you ask a good question. -1^8 would require raising 1 to the 8th power, then applying the negative sign, and the result would be -1. (-1)^8 would be the 8th power of -1, which as you see would be 1. STUDENT COMMENT: I think it would be easier to visualize what your trying to raise to the exponent if you actually put parenthesis around the 2, that part seems to get tricky on the computer. INSTRUCTOR RESPONSE: The expression was intentionally written to be misleading and make the point that, to avoid ambiguity, order of operations apply strictly, no matter what the expression looks like. Normally, for clarity, the parentheses would be included. They aren't necessary, but when helpful it's a good idea to include them. You can, of course, have too many parentheses in an expression, making it harder than necessary to sort out. In practice we try to strike a balance. The original expression was (x - 5) ^ 2x-1 + 3 / x-2 White spaces make no difference in how an expression is evaluated, but they can help show the structure; e.g., (x - 5)^2 * x - 1 + 3 / x -2 is a visual improvement over the original. The * between the 2 and the x is not strictly necessary, but is also helpful. ((((x - 5) ^ 2)) * x) - 1 + (3 / x) - 2 verges on having too many parentheses at the beginning; it does help clarify the 3 / x. STUDENT COMMENT Although I read through your explanation and do see the point you are making, that 2x is actually 2 * x, I still think that (-1) should be raised to 2x rather than 2. Kaking the answer -11/4, not 7/4. INSTRUCTOR RESPONSE When the expression (x - 5) ^ 2x-1 + 3 / x-2 is copied and pasted into a computer algebra system it is translated as This notation is universal and unambiguous. Any deviation from strict interpretation (which does occur among some authors and among manufacturers of some calculators) tends to result in ambiguity and confusion. STUDENT COMMENT While I do understand what you are trying to relate, I will continue to make these mistakes on more than one occasion and will not penalize myself for not rewriting years of mathematics because of a syntax issue in an online class. INSTRUCTOR RESPONSE I don't penalize errors in typed notation when the intent is clear (though I will sometimes point out these errors), and when you take your tests you'll be writing them out by hand and this won't be an issue. However this is not a syntax issue in an online class. This is the order of operations, as it has been since algebra was developed hundreds of years ago, and it's completely consistent with the mathematics you appear to know (quite well). As stated here, if you use the wrong syntax in any computer algebra system, your expression will not be interpreted correctly. For this reason alone you need to understand the notation. For this and other valid reasons you need to understand how the order of operations are represented in 'linear' fashion (i.e., 'typewriter notation') and to correctly interpret expressions written in this notatation. Any mathematics that has been learned correctly is completely consistent with the order of operations and with the notation used in this course. If the mathematics you've learned was inconsistent with the order of operations (and I don't believe this is so in your case, but it is with many students), then you would need to adjust your thinking. Fortunately this is very easy to do. Interpret expressions literally, assume nothing, and everything works out. You will also find that the notation quickly becomes easy to read and use, and that it expands your comprehension of all mathematical notation. STUDENT COMMENT I used -1^(2*4). I didn't realize that was doing multiplication before exponents. All of this typewriter notations seems ambiguous to me but I think that had I seen the expression in standard notation I would probably have made the same mistake in this instance. If I were writing this expression I would probably use a parenthesis or * to show the necessary separation. INSTRUCTOR RESPONSE Parentheses, even when they aren't strictly necessary, are often useful to clarify the expression. An parentheses, even when not necessary, are part of the order of operations. Spacing is not part of the order of operations. An expression has the same meaning even if all spaces are removed. However as long as an expression is correctly formed, spacing as well as parentheses can certainly be used to make it more readable. I don't go to any trouble in this exercise to make the expressions readable, since my goal here is to make the point about order of operations, which give an expression its unambiguous meaning. However in most of the documents you will be working with, I do make an effort to clarify the meanings of expressions through their formatting, often using unnecessary parentheses and spacing to help clarify meaning. Certainly I encourage you to do the same. STUDENT QUESTION I didn’t separate the ¾ as a stand alone fraction, I am confused about why you don’t treat it as an equation that the denominator isn’t treated as a denominator. INSTRUCTOR RESPONSE Your work was good throughout most of this problem. You did forget to copy down a -1 in one of the early steps, but otherwise followed the order of operations correctly until nearly the last step. However near the end you said that 4+3/4-2=7/2. You appear to have performed the addition 4 + 3 and the subtraction 4 - 2 before dividing. However the division has to be done first. The division sign is between the 3 and the 4, so the division is 3/4, and that gives you the fraction 3/4. Therefore the expression 4+3/4-2 tells you to 'add 3/4 to 4 then subtract 2'. When actually writing this out we would probably include parentheses. That wasn't done here, as it would have defeated the point being made about order of operations, but for clarity we might have written 4 + (3/4) - 2. The parentheses are not necessary around the 3/4, since the order of operations is sufficient to unambiguously define the result, but they do make the expression easier to read and reduce the likelihood of error. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I worked through the problem without any major issues until the end. I subtracted instead of adding and I do not know why. I wasn’t paying attention. ------------------------------------------------ Self-critique Rating:2 ********************************************* Question: `q005. Evaluate the expression x^3x+2/x-1 for x = 2, according to the order of operations. Show all your steps. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: X^3x+2/x-1= 2^3*2+2/2-1= 8*2+2/2-1= 16+2/2-1= 16-1+2/2= 15+2/2= 15+1= 16 confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-critique Rating: Where’s the answer? I would like to think I answered this correctly.
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Given Solution: `aYou should see a brief set of instructions and over 30 numbered examples. If you click on the word Picture you will see the standard-notation format of the expression. The link entitled Examples and Pictures, located in the initial instructions, shows all the examples and pictures without requiring you to click on the links. There is also a file which includes explanations. The instructions include a note indicating that Liberal Arts Mathematics students don't need a deep understanding of the notation, Mth 173-4 and University Physics students need a very good understanding, while students in other courses should understand the notation and should understand the more basic simplifications. There is also a link to a page with pictures only, to provide the opportunity to translated standard notation into typewriter notation. end program STUDENT COMMENT (not quite correct) I see a collection of typewriter problems, after looking at some of them I see that the slash mark is to create a fraction rather than to denote division. INSTRUCTOR CORRECTION A fraction is a division of the numerator by the denominator. The slash mark indicates division, which can often be denoted by a fraction. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): A fraction is division. ------------------------------------------------ Self-critique Rating:2 ********************************************* Question: `q007. Standard mathematics notation is easier to look at; it's easier to see the meaning of the expressions. However it's very important to understand order of operations, and students do get used to this way of doing it. You should of course write everything out in standard notation when you work it on paper. It is likely that you will at some point use a computer algebra system, and when you do you will probably have to enter expressions using a keyboard, so it is well worth the trouble to get used to this notation. As one example take a minute and go to Wolfram Alpha at http://www.wolframalpha.com/. If this link doesn't work just search the Web for 'Wolfram Alpha'. When the page comes up, you can simply copy the expression x - 3 / (2x + 4) into the box. Think about what you would get were you to evaluate this expression, then click on the = sign. Repeat the process with each of the following expressions. Be sure you think in each case about what expression you would expect to see. (x - 3) / (2 x + 4) x - 2 / 3 (x - 2) / 3 (x+2) ^ 2x (x+2) ^ (2x) (x - 3) / 3x (x - 3) / (3 x) x - 3 / 3x Do these expressions act the way most people would expect, or do they act in the way dictated by the order of operations? Indicate your understanding of why it is important to understand this notation. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The expressions are what I am used to seeing in school. The web page makes it easier to see the equivalent to what I am used to learning. I think doing the math problems on the computer are easier because you can see everything and it is not as messy. I t is important to understand both ways of seeing a problem for future reference. confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q007. Standard mathematics notation is easier to look at; it's easier to see the meaning of the expressions. However it's very important to understand order of operations, and students do get used to this way of doing it. You should of course write everything out in standard notation when you work it on paper. It is likely that you will at some point use a computer algebra system, and when you do you will probably have to enter expressions using a keyboard, so it is well worth the trouble to get used to this notation. As one example take a minute and go to Wolfram Alpha at http://www.wolframalpha.com/. If this link doesn't work just search the Web for 'Wolfram Alpha'. When the page comes up, you can simply copy the expression x - 3 / (2x + 4) into the box. Think about what you would get were you to evaluate this expression, then click on the = sign. Repeat the process with each of the following expressions. Be sure you think in each case about what expression you would expect to see. (x - 3) / (2 x + 4) x - 2 / 3 (x - 2) / 3 (x+2) ^ 2x (x+2) ^ (2x) (x - 3) / 3x (x - 3) / (3 x) x - 3 / 3x Do these expressions act the way most people would expect, or do they act in the way dictated by the order of operations? Indicate your understanding of why it is important to understand this notation. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The expressions are what I am used to seeing in school. The web page makes it easier to see the equivalent to what I am used to learning. I think doing the math problems on the computer are easier because you can see everything and it is not as messy. I t is important to understand both ways of seeing a problem for future reference. confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!