Query 13

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Given Solution: `a** The solution to x^3 / 17 = 58 is found by first multiplying both sides by 17 to get x^3 = 58 * 17 then taking the 1/3 power of both sides, obtaining (x^3)^(1/3) = (58 * 17)^(1/3) or x = 9.95, approx.. COMMON ERROR: If you interpret the equation as x^(3/17) = 58 you will get solution x = 58^(17/3) = 9834643694. However this is not the solution to the given equation To interpret x ^ 3 / 17 you have to follow the order of operations. This means that x is first cubed (exponentiation precedes multiplication or division) then divided by 17. If you introduce the grouping x^(3/17) you are changing the meaning of the expression, causing 3 to be divided by 17 before exponentiation. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating:ok

@& In your solution you quoted a rule as opposed to giving an explanation (the given solution didn't give a great explanation either).

An explanation might go something like this:

a^b represents a string of a's multiplied together. The length of that string is b.

a^c represents a string of a's multiplied together. The length of that string is c.

The result is a string of multiplied a's, of length b + c. This string can be represented by the expression a^(b + c). *@

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Given Solution: `a** (3x)^-2 = 19 is solved by taking the -1/2 power of both sides, or the negative of the result: ((3x)^-2)^(-1/2)) = 19^(-1/2) gives us 3x = 19^(-1/2) so that x = [ 19^(-1/2) ] / 3 = .0765 or -.0765. ** STUDENT QUESTION: I'm not sure how I was supposed to arrive at two solutions. INSTRUCTOR RESPONSE If an even power of a number has a given value, so does that power of the negative of that number. It's fairly easy to understand why. For example: 5^2 = 25 and (-5)^2 = 25 3^4 = 81 and (-3)^4 = 81 2^(-4) = 1/16 and (-2)^(-4) = 1/16. So when solving an equation of the form (c)^n = d, where n is an even integer (whether positive or negative) and d is a positive quantity, there are two solutions: c = d^(1/n) and c = - d^(1/n). (note that - d^(1/n) is the negative of d^(1/n), not (-d)^(1/n), which would not be defined). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Ok ------------------------------------------------ Self-critique rating:ok

@& Good.

Note on terminology:

You want to call this the 'cube root', not the 'cubed root'. Nothing is being cubed (i.e., raised to the 3d power) when you take that root, and the use of 'cubed' could invoke the idea that maybe something is being cubed. *@

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Given Solution: `a** to solve the equation we first multiply both sides by 1/4 to get x ^ -.5 = 7 / 4. Then we raise both sides to the -2 power: (x^-.5)^-2 = (7/4)^-2 so x = .327 approx ** STUDENT QUESTION Why do we multiply by ¼? INSTRUCTOR RESPONSE 4 is not part of the expression which is raised to the -.5 power. ^ precedes *. Since 4 x ^ -.5 means 4 * x ^ -.5, we do x^-.5 before we multiply by 4. So it's easier to multiply both sides by 1/4, so that the left-hand side is just x^-.5, without the confusion of the 4. Your solution illustrates why this is the best idea: Quoting from your solution 4x^-.5 = 7 raise both sides by -1/.5 (4x^-.5)^-1/.5 = 7^-(1/.5) (this step is valid, even if it's not the best way to proceed) 4x = 0.020 (this step is in error) The error on that last step is that (4 x ^-.5) ^ (-1 / .5) is not equal to 4x. (4 x ^-.5) ^ (-1 / .5) = 4 ^ (-1 / .5) * (x^-.5) ^ (1 / (-.5) ) = 4^(-1/.5) * x &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating:ok

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Given Solution: `a** We first multiply both sides by 1/14 to get x^(2/3) = 39/14. The we raise both sides to the 3/2 power to get x = (39/14)^(3/2) = 4.65. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Ok ------------------------------------------------ Self-critique rating:ok

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Given Solution: `a** multiplying both sides by 1/5 we get (3x/8)^(-3/2) = 9/5. Raising both sides to the -2/3 power we have 3x / 8 = (9/5)^(-2/3). Multiplying both sides by 8/3 we obtain x = 8/3 * (9/5)^(-2/3) = 1.80 ** STUDENT QUESTION Still confused on the steps after we get down to 9/5. INSTRUCTOR COMMENT We take the -2/3 power of the -3/2 power: (3x/8)^(-3/2) = 9/5. Raising both sides to the -3/2 power we have ((3x / 8)^(-3/2) ) ^ (-2/3) = (9/5)^(-2/3), so that (by the laws of exponents) (3x / 8)^(-3/2 * (-2/3) ) = (9/5)^(-2/3). Since -3/2 * -2/3 = 1 so we have (3x / 8)^1 = (9/5)^(-2/3), that is 3 x / 8 = (9/5)^(-2/3). Multiplying both sides by 8/3 we complete the solution. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating:ok

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Given Solution: `a** Substituting n = 0 we get a(0+1) = a(0) + .5 * 0 which we simplify to get a(1) = a(0). Substituting a(0) = 2 from the given information we get a(1) = 2. Substituting n = 1 we get a(1+1) = a(1) + .5 * 1 which we simplify to get a(2) = a(1) + .5. Substituting a(1) = 2 from the previous step we get a(2) = 2.5. Substituting n = 2 we get a(2+1) = a(2) + .5 * 2 which we simplify to get a(3) = a(2) + 1. Substituting a(2) = 2.5 from the previous step we get a(3) = 2.5 + 1 = 3.5. Substituting n = 3 we get a(3+1) = a(3) + .5 * 3 which we simplify to get a(4) = a(3) + 1.5. Substituting a(3) = 3.5 from the previous step we get a(4) = 3.5 + 1.5 = 5. Substituting n = 4 we get a(4+1) = a(4) + .5 * 4 which we simplify to get a(5) = a(4) + 2. Substituting a(4) = 5 from the previous step we get a(5) = 5 + 2 = 7. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating:ok

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Given Solution: `a** Using points (1,2), (3,3.5) and (7,5) we substitute into the form y = a x^2 + b x + c to obtain the three equations 2 = a * 1^2 + b * 1 + c 3.5 = a * 3^2 + b * 3 + c 7 = a * 5^2 + b * 5 + c. Solving the resulting system for a, b and c we obtain a = .25, b = -.25 and c = 2, giving us the equation 0.25•x^2 - 0.25•x + 2. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating:

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Given Solution: `a** We obtain the points (0, 7) (4, 5.448) (.8, 3.992) (1.2, 2.632) (1.6, 1.368) (2, .2) y values are 7, 5.448, 3.992, 2.632, 1.368, 0.2. Differences are 7-5.448 = -1.552, 3.992 - 2.632 = -1.456, etc. The sequence of differences is -1.552, -1.456, -1.36, -1.264, -1.168. The rate of change of the original sequence is proportional to this sequence of differences. The differences of the sequence of differences (i.e., the second differences) are .096, .096, .096, .096, .096.. These differences are constant, meaning that the sequence of differences is linear.. This constant sequence is proportional to the rate of change of the sequence of differences. The differences are associated with the midpoints of the intervals over which they occur. Therefore the difference -1.552, which occurs between x = 0 and x = .4, is associated with x = .2; the difference -1.456 occuring between x = .4 and x = .8 is associated with x = .6, etc.. The table of differences vs. midpoints is } 0.2, -1.552 .6, -1.456 1, -1.36 1.4, -1.264 1.8, -1.168 This table yields a graph whose slope is easily found to be constant at .24, with y intercept -1.6. The function that models these differences is therefore y = 2.4 x - 1.6. ** STUDENT QUESTION: I’m not quite sure I understand how you derived at your model y = 2.4 x - 1.6 INSTRUCTOR RESPONSE: For example the rise between the first and second point is .096 and the run is .4, making the slope .096 / .4 = .24. You get the same slope for any pair of points (you can easily see that the x values change by the same amount each time, and if you calculate the changes in the y values you get .096 for each change, leading fairly quickly to the conclusion that the slopes are all the same). Between 0 and 0.2 the 'rise' would therefore be 0.2 * .24 = .048. This would put the x = 0 point .48 units lower than the x = .2 point, so that at the x = 0 point the y value is -1.552 - .048 = -1.6. The line is therefore y = 2.4 x - 1.6. STUDENT QUESTION: I think I may have misunderstood the directions for this one. I believe the problem statement asked to graph the avg. rate of change vs the midpoints, which I assumed to be the slopes vs the midpoints, not the difference between x values (which would just be the run between each point?). So my midpoint coordinates and the resulting model do not match up to the given solution. INSTRUCTOR RESPONSE: Your solution was fine. There is however a fine distinction in the wording of the question. The statement referred to a 'quadratic sequence', which isn't the same thing as a 'quadratic function'. A quadratic sequence is a sequence of numbers which can be obtained by evaluating a quadratic function. Once the numbers are obtained, the function is effectively discarded, and the sequence stands on its own. The function yields the sequence in the manner you outline: f(x) = 0.6x - 4 f(0.2) = 0.6(0.2) - 4 = -3.88 f(0.6) = 0.6(0.6) - 4 = -3.64 f(1.0) = 0.6(1.0) - 4 = -3.4 f(1.4) = 0.6(1.4) - 4 = -3.16 f(1.8) = 0.6(1.8) - 4 = -2.92 The original function f(x) is at this point forgotten. The sequence is -3.88, -3.64, -3.4, -3.16, -2.92, ... A sequence is numbered by integers, so that these values could now be referred to as a(1), a(2), a(3), a(4) and a(5). Since a sequence is numbered by consecutive integers, the rates at which a sequence changes are just the differences in consecutive members of the sequence. You might question why we want to forget the origins of the sequence. The reason is that the sequence is just simpler, and has certain properties and behaviors that are better understood if we treat it as such. Having done so we can relate our results for the sequence back to the original function and its rate-of-change behavior. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating:ok

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Given Solution: `a** The average slope is rise / run = [ f(k) - f(h) ] / (k - h) = ( a k^2 + b k + c - ( a h^2 + b h + c) ) / ( k - h). We simplify this to get ave slope = ( a ( k^2 - h^2) + b ( k - h) ) / ( k - h), which we write as ave slope = ( a ( k-h) ( k+h) ) + b ( k - h) ) / (k - h). k - h is a factor of the numerator so we have the final form ave slope = a ( k + h) + b. ** ********************************************* Question: `qQuery last asst before test 1, problem 1.Give your solution to x ^ 3 / 17 = 58 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 9.953----- I did the same problem in Assignment 13 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution: `a** The solution to x^3 / 17 = 58 is found by first multiplying both sides by 17 to get x^3 = 58 * 17 then taking the 1/3 power of both sides, obtaining (x^3)^(1/3) = (58 * 17)^(1/3) or x = 9.95, approx.. COMMON ERROR: If you interpret the equation as x^(3/17) = 58 you will get solution x = 58^(17/3) = 9834643694. However this is not the solution to the given equation To interpret x ^ 3 / 17 you have to follow the order of operations. This means that x is first cubed (exponentiation precedes multiplication or division) then divided by 17. If you introduce the grouping x^(3/17) you are changing the meaning of the expression, causing 3 to be divided by 17 before exponentiation. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ********************************************* Question: `qGive your solution to (3 x) ^ -2 = 19 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: -.0764 and .0764 confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution: `a** (3x)^-2 = 19 is solved by taking the -1/2 power of both sides, or the negative of the result: ((3x)^-2)^(-1/2)) = 19^(-1/2) gives us 3x = 19^(-1/2) so that x = [ 19^(-1/2) ] / 3 = .0765 or -.0765. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ********************************************* Question: `qGive your solution to 4 x ^ -.5 = 7 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 16/49 ---- question completed in assignment 13 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution: `a** to solve the equation we first multiply both sides by 1/4 to get x ^ -.5 = 7 / 4. Then we raise both sides to the -2 power: (x^-.5)^-2 = (7/4)^-2 so x = .327 approx ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ********************************************* Question: `qGive your solution to 14 x ^ (2/3) = 39 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 4.6494 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution: `a** We first multiply both sides by 1/14 to get x^(2/3) = 39/14. The we raise both sides to the 3/2 power to get x = (39/14)^(3/2) = 4.65. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ********************************************* Question: `qGive your solution to 5 ( 3 x / 8) ^ (-3/2) = 9 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 1.802 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution: `a** multiplying both sides by 1/5 we get (3x/8)^(-3/2) = 9/5. Raising both sides to the -2/3 power we have 3x / 8 = (9/5)^(-2/3). Multiplying both sides by 8/3 we obtain x = 8/3 * (9/5)^(-2/3) = 1.80 ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok"" "

Query 13

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@&

In your solution you quoted a rule as opposed to giving an explanation (the given solution didn't give a great explanation either).

An explanation might go something like this:

a^b represents a string of a's multiplied together. The length of that string is b.

a^c represents a string of a's multiplied together. The length of that string is c.

The result is a string of multiplied a's, of length b + c. This string can be represented by the expression a^(b + c).

*@

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@&

Good.

Note on terminology:

You want to call this the 'cube root', not the 'cubed root'. Nothing is being cubed (i.e., raised to the 3d power) when you take that root, and the use of 'cubed' could invoke the idea that maybe something is being cubed.

*@

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&#This looks good. See my notes. Let me know if you have any questions. &#