course Mth163 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: If we substitute n = 1 into a(n) = a(n-1) + 2^n we get a(1) = a(1-1) + 2^1 or, since 1-1 = 0 and 2^1 = 2 a(1) = a(0) + 2. Since we are given a(0) = 3 we now have a(1) = 3 + 2 = 5. If we substitute n = 2 into a(n) = a(n-1) + 2^n we get a(2) = a(2-1) + 2^2 or, since 2-1 = 1 and 2^2 = 4 a(2) = a(1) + 4. Since we are given a(1) = 5 we now have a(2) = 5 + 4 = 9. If we substitute n = 3 into a(n) = a(n-1) + 2^n we get a(3) = a(3-1) + 2^3 or, since 3-1 = 2 and 2^3 = 8 a(3) = a(2) + 8. Since we are given a(2) = 9 we now have a(3) = 9 + 8 = 17. If we substitute n = 4 into a(n) = a(n-1) + 2^n we get a(4) = a(4-1) + 2^4 or, since 4-1 = 3 and 2^4 = 16 a(4) = a(3) + 16. Since we are given a(3) = 16 we now have a(4) = 17 + 16 = 33. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q002. If a(n) = 2 * a(n-1) + n with a(0) = 3, then what are the values of a(1), a(2), a(3) and a(4)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: a(1) = 2 * a(1-1) + 1….a(1) = 2 *a(0)+1….if a(0) = 3 then a(1) = 2*3+1=7 a(2)= 2*a(2-1)+2…a(2) = 2*a(1)+2…if a(1)=7 then a(2) = 2*7 +2=16 a(3) = 2*a(3-1)+3….a(3) = 2*a(2) +3…if a(2) =16 then a(3) = 2*16 +3 = 35 a(4) = 2* a(4-1) +4…a(4) = 2*a(3) +4…if a(3) = 35 then a(4) = 2*35 +4 = 74 Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: If we substitute n = 1 into a(n) = 2 * a(n-1) + n we get a(1) = 2 * a(1-1) + 1 or since 1-1 = 0 a(1) = 2 * a(0) + 1. Since we know that a(0) = 3 we have a(1) = 2 * 3 + 1 = 7. If we substitute n = 1 into a(n) = 2 * a(n-1) + n we get a(2) = 2 * a(2-1) + 2 or since 2-1 = 1 a(2) = 2 * a(1) + 2. Since we know that a(0) = 3 we have a(2) = 2 * 7 + 2 = 16. If we substitute n = 1 into a(n) = 2 * a(n-1) + n we get a(3) = 2 * a(3-1) + 3 or since 3-1 = 2 a(3) = 2 * a(2) + 3. Since we know that a(0) = 3 we have a(3) = 2 * 16 + 3 = 35. If we substitute n = 1 into a(n) = 2 * a(n-1) + n we get a(4) = 2 * a(4-1) + 4 or since 4-1 = 3 a(4) = 2 * a(3) + 4. Since we know that a(0) = 3 we have a(4) = 2 * 35 + 4 = 74. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q003. What are the average slopes of the graph of y = x^2 + x - 2 between the x = 1 and x= 3 points, between the x = 3 and x = 5 points, between the x = 5 and x = 7 points, and between the x = 7 and x = 9 points? What is the pattern of this sequence of slopes? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: using y = x^2 + x – 2 to solve y values if x = 1,3,5,7,9 y = 1^2 +1 – 2 = 0 (1,0) y = 3^2 +3 -2 = 10 (3,10) y = 5^2 + 5 – 2 = 28 (5,28) y = 7^2 + 7 – 2 = 54 (7,54) y = 9^2 + 9 – 2 = 88 (9,88) slope is found as rise / run: between x = 1 and x = 3: (10 – 0)/(3 -1) = a slope of 5 between x = 3 and x = 5: (28 – 10)/(5 – 3) = a slope of 9 between x = 5 and x = 7: (54 – 28)/(7 – 5) = a slope of 13 between x = 7 and x = 9: (88 – 54)/ (9 – 7) = a slope of 17 the pattern is that the run is always 2. Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: At x = 1, 3, 5 , 7 and 9 we find by substituting that y = 0, 10, 28, 54 and 88. The x = 1, 3, 5, 7 and 9 points are therefore (1,0), (3,10), (5,28), (7,54) and (9,88). The run from one point to the next is always 2. The rises are respectively 10, 18, 26 and 34. The slopes are therefore slope between x = 1 and x = 3: slope = rise / run = 10 / 2 = 6. slope between x = 3 and x = 5: slope = rise / run = 18 / 2 = 9. slope between x = 5 and x = 7: slope = rise / run = 26 / 2 = 13. slope between x = 7 and x = 9: slope = rise / run = 34 / 2 = 17. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q004. If a solid stone sphere 4 inches in diameter weighs 3 pounds, then what would be the weight of a solid stone sphere 2 feet in diameter? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: A solid stone sphere weight is proportional to that of a cube volume. w for v The volume of sphere is proportional to cubes diameter d for x Giving you a problem of: w = k d^3 3 = k * 4^3….k = 3 / 4^3…..k = 3 / 64 giving you a function of w = 3/64 * d^3 In a diameter of 2 feet what would be the weight: Converting 2 ft to inches – 24 inch and substituting into function to solve for weight: w = 3/64 * 24^3….w = 648 as you total weight Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The volume of a sphere is proportional to the cube of its diameters, and weight is directly proportional to volume so we have the proportionality w = k d^3, where w and d stand for weight and diameter and k is the proportionality constant. Substituting the known weight and diameter we get 3 = k * 4^3, where we understand that the weight is in pounds and the diameter in inches. This gives us 3 = 64 k so that k = 3 / 64. Our proportionality equation is now w = 3/64 * d^3. So when the diameter is 2 feet, we first recall that diameter must be in inches and say that d = 24, which we then substitute to obtain w = 3/64 * 24^3. A simple calculation gives us the final weight w = 748. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I have all the steps the same as your except the last step and I’m still getting 648 as my weight.
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Given Solution: The ratios 36/12, 54/18 and 72/24 of the corresponding sides are all the same and all equal to 3, so the dimensions of the sides of the second box are 3 times those of the first. Since the thickness of the cardboard is the same on both boxes, only the dimensions of the rectangular sides change. The only thing that matters, therefore, is the surface area of the box. The proportionality is therefore of the form w = k x^2, where w is the weight of the box and x stands any linear dimension. It follows that w2 / w1 = (x2 / x1)^2. Since as we just saw x2 / x1 = 3, we see that w2 / w1 = 3^2 = 9. Since w1 = 22 oz, we write this as w2 / 22 oz = 9. Multiplying both sides by 22 oz we see that w2 = 22 oz * 9 = 198 oz. If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm. Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution. 014. `query 14 ********************************************* Question: `qQuery two examples and a picture ...explain the statement 'the rate of change of a quadratic function changes at a constant rate' YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: say it takes two times to reach a zero consonance you would have a 2 degree polynomial and the 2 increase by the? amount as the first increases by the ? amount at a constant rate Confidence rating: unsure ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** We can calculate the rates of change of a quadratic function based on a series of consecutive intervals of constant length. We find that these rates change from interval to interval, and always by the same amount. Since the rates of change always change by the same amount, they are changing at a constant rate. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Calculated by a series of consecutive intervals. length : increase of rates from interval to interval is constant and changes by same amount. Self-critique rating: ********************************************* Question: `qexplain how to get the first few members of a sequence from its recurrence relation YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: say you have a(n) with n being a set of numbers like 1,2,3 and so on, we would substitute n into the given equation to solve and then the next number will be the same except we will use the solved first number of a(n) to add or subtract to get the next solved number of a(n) and continue onto the next using the previous until we are finished. Confidence rating: positive if I didn’t sound confussing ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** We let n be the first integer for which the value a(n) is not given, and we substitute this integer into the recurrence relation to evaluate a(n) for this 'new' integer, using values of a(n) for previous integers. If this is not possible then we have not been given enough information to evaluate the sequence. We then substitute the next integer and use values of a(n) for previous integers. We continue this process as long as necessary to get the results we need. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique rating: "