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: 2^5 stands for 2 raised to the fifth power; i.e., 2^5 = 2*2*2*2*2. The result of this calculation is 2^5 = 32. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q002. What does 2^3 * 2^5 mean? Is the result of power of 2? If so, what power of 2 is it? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2^3 * 2^5 means 2 raised to the power of 3 times 2 raised to the power of 5 or you can simplify by saying 2 ^(3+5) or 2 ^ 8 = 256 or 2*2*2*2*2*2*2*2 which shows this is a result of a power of 2. Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 2^3 * 2^5 means (2*2*2) * (2*2*2*2*2). This is the same as 2*2*2*2*2*2*2*2, or 2^8. When we multiply this number out, we obtain 256. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q003. Why do we say that a^b * a^c = a^(b+c)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: as seen in the above answer a(like2) is raised to the different and multiplied by each other. If a like # is raised to two different powers then multiplied it is the same as adding the powers over that number. Ex: a^b * a^c = a^(b+c) Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We saw in the preceding example that 2^3 * 2^5 stood for a product of three 2's, multiply by a product of five 2's. We saw also that the result was identical to a product of eight 2's. This was one instance of the general rule that when we multiply to different powers of the same number, the result is that number raised to the sum of the two powers. One general way to state this rule is to let a stand for the number that is being raised to the different powers, and let b and c stand for those powers. Then we get the statement a^b * a^c = a^(b+c). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q004. What does (2^3)^5 mean? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (2^3)^5 means 2 is raised to the 3 power then that solution is raised to the 5 power. So if 2^3 is 2*2*2 then we would have (2*2*2)(2*2*2)(2*2*2)(2*2*2)(2*2*2), which would actually come out as 2^15 = 32768 Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Since 2^3 stands for 2*2*2, it follows that (2^3)^5 means (2^3)*(2^3)*(2^3)*(2^3)*(2^3) = (2*2*2)*(2*2*2)*(2*2*2)*(2*2*2)*(2*2*2) = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2 = 2^15. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q005. Why do we say that (a^b)^c = a^(b*c)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: As shown in the problem before a^b is raised to the power of c, which will give you a^b c many times, which you can write as a^(b*c) Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We saw in the last example how (2^3)^5 stands for the product of 5 quantities, each equal to the product of three 2's. We saw how this is equivalent to the product of fifteen 2's, and we saw how the fifteen was obtained by multiplying the exponents 3 and 5. In the present question a^b stands for the quantity a multiplied by itself b times. (a^b)^c stands for the quantity a^b multiplied by itself c times, which is equivalent to multiplying a by itself b * c times. Thus we say that (a^b)^c = a^(b * c). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q006. According to the law a^b * a^c = a*(b+c), if we multiply 2^5 by 2^-2 what power of 2 should we get? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: according to the law of exponents if we have 2^5 * 2^-2 then we can say 2^(5-2) which gives us 2^3. Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: To according to the law, 2^5 * 2^-2 = 2^(5 + -2) = 2^(5-2) = 2^3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q007. Since as we saw in the preceding question 2^5 * 2^-2 = 2^3, what therefore must be the value of 2^-2? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: well if 2^5 = 32 and 2^3 = 8 then we would have 32 * 2^-2 = 8 ..2^-2 = 8 / 32 = 1/ 4 Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: One way of seeing this is to note that 2^5 = 32 and 2^3 = 8, so we have 32 * 2^-2 = 8. Dividing both sides by 32 we get 2^-2 = 8 / 32 = 1/4. We can learn something important if we keep the calculation in powers of 2. If 2^5 * 2^-2 = 2^3, then dividing both sides of the equation by 2^5 we obtain 2^-2 = 2^3/2^5, which is equal to 1/2^2. This shows us why 2^-2 = 1/2^2. QUESTIONABLE STUDENT RESPONSE: .25 .25 is of course the value of 2^-2. However I'm not sure you've connected this with the fact that 2^5 * 2^-2 = 2^3, as was the intent of the question. A key word in the given question is 'therefore', which asks you to connect your answer to the fact that 2^5 * 2^-2 = 2^3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q008. Why must we say that 2^-n = 1 / 2^n, where n stands for any counting number? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We cant do a number to the power of a negative #, so we must make it positive by making it a fraction with 1 over that number to it power. Ex: 2^-n must be 1 / 2^n Confidence rating: mostly ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: This is because for any number m, we have 2^m * 2^-n = 2^( m + -n) = 2^(m-n), and we also have 2^m * (1 / 2^n) = 2^m / 2^n = 2^(m-n). So whether we multiply 2^m by 2^-n or by 1 / 2^n we get the same result. This shows that 2^-n and 1 / 2^n are the same. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q009. According to the law a^b * a^c = a*(b+c), if we multiply 2^3 by 2^-3 what power of 2 should we get? Since 2^-3 = 1 / 2^3, what number must we get when we multiply 2^3 by 2^-3? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2^3 * 2^-3 is also written as 2^(3-3) which is 2^0 or 1 because anything to the power of 0 is 1. As is 2^3 * 2^-3 is 2^3 *(1 /2^3) which also equals 1 Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 2^3 * 2^-3 = 2^(3 + -3) = 2^(3-3) = 2^0. Since 2^-3 = 1 / 2^3 it follows that 2^3 * 2^-3 = 2^3 * ( 1 / 2^3) = 1. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q010. Continuing the last question, what therefore should be the value of 2^0? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2^0 is one because anything to the power of 0 is 1. Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Since 2^3 * 2^-3 = 2^0 and also 2^3 * 2^-3 = 1 we see that 2^0 must be 1. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q011. How do we solve the equation x^3 = 12? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: we would have to take 1/3 power of each side: x^(3 * 1/3) = 12^(1/3) x = 12(1/3) = 2.29 approx. Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We solve the equation by taking the 1/3 power of both sides: (x^3)^(1/3) = 12^(1/3), then by the law (a^b)^c = a^(bc) we have x^(3 * 1/3) = 12^(1/3), so that x^1 = 12^(1/3) or just x = 12^(1/3), which we can easily enough evaluate with a calculator. We obtain a result of approximately x = 2.29 . &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q012. How do we solve the equation x^(5/2) = 44? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: we would have to raise both sides to the 2/5 power in order to solve: x^(5/2*2/5) = 44^(2/5) x = 44 ^(2/5) = 4.54 approx. Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: If we take the 2/5 power of both sides we get (x^(5/2))^(2/5) = 44^(2/5) or x^(5/2 * 2/5) = 44^(2/5) or x^1 = 44^(2/5) so that x = 44^(2/5). Evaluating by calculator you should obtain approximately x = 4.54. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: 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. 013. `query 13 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. 013. `query 13 ********************************************* Question: `qQuery last asst before test 1, problem 1.Give your solution to x ^ 3 / 17 = 58 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x^3 / 17 = 58 ..x^3 = 58 * 17 ..x^3 = 986 ..x^(3*1/3) = 986^1/3 .x = 9.95 approx. Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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, pay close attention to order of operation and remember that ( ) change that order. Self-critique rating: ********************************************* Question: `qGive your solution to (3 x) ^ -2 = 19 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (3x) ^ -2 = 19 you would have to take the power of -1/2 on both sides. [(3x)^-2]^(-1/2) = 19^(-1/2) 3x = .229 x = [19^(-1/2)] / 3 = .0765 approx. Confidence rating: mostly ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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 Self-critique rating: ********************************************* Question: `qGive your solution to 4 x ^ -.5 = 7 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 4x^-.5 = 7 first we have to divide both sides by 4: x^-.5 = 7 / 4 then we have to multiply both sides by -2: (x^-.5) ^ -2 = (7/4)^-2 x = (4^2)/(7^2) = 16 / 49 = .327 approx. Confidence rating: mostly ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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 Self-critique rating: ********************************************* Question: `qGive your solution to 14 x ^ (2/3) = 39 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: first we must divide both sides by 14 getting: x^(2/3) = 39/14 then we raise both sides to the power of 3/2 : [x^ (2/3)]^(3/2) = (39/14)^(3/2) ..x = (39/14)^(3/2) = 4.65 approx. Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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: ********************************************* Question: `qGive your solution to 5 ( 3 x / 8) ^ (-3/2) = 9 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: first we must divide both sides by 5 giving you: (3x/8)^(-3/2) = 9 /5 ten raise both sides to the power of -2/3: 3x/8 = (9/5)^(-2/3) then multiply both sides by 8/3: x = (9/5)^(-2/3) * 8/3 x = 1.80 approx. Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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 Self-critique rating: ********************************************* Question: `qQuery problem 2. a(n+1) = a(n) + .5 n, a(0) = 2 What are a(1), a(2), a(3), a(4) and a(5)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: in a(n+1) = a(n) + .5n, a(0) = 2 a(0+1)= a(0) +.5*0 .1 = 0 so a(1) = 2 in a(1+1) = a(1) + .5*1 . if a(1) = 2 then a(2) = 2.5 in a(2+1) = a(2) +.5*2 a(3) = a(2) + 1 ..if a(2) = 2.5 then a(3) = 3.5 in a(3+1) = a(3) +.5*3 ..a(4) = a(3) + 1.5 If a(3) = 3.5 then a(4) = 5 in a(4+1) = a(4) +.5 *4 .a(5) = a(4) + 2 if a(4) = 5 then a(5) = 7 in a(5+1) = a(5) + .5 *5 a(6) = a(5) + 2.5 if a(5) = 7 then a(6) = 9.5 Confidence rating: mostly ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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: ********************************************* Question: `qWhat is your quadratic function and what is its value for n = 4? Does it fit the sequence exactly? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I know the quadratic function is y = ax^2 + b x + c and to graph out three points to have three equations to solve for a, b, and c, but I dont understand what n is how you would check the sequencing. Confidence rating: unsure ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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 0.25x^2 - 0.25x + 2. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique rating: ********************************************* Question: `qQuery problem 3. f(x) = .3 x^2 - 4x + 7, evaluate at x = 0, .4, .8, 1.2, 1.6 and 2.0. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: for f(x) = .3 x^2 4x + 7, x = 0, .4, .8, 1.2, 1.6 f(0) = .3*0^2 -4*0 + 7 = 7 f(.4) = .3*.4^2 -4 * .4 + 7 = 5.448 f(.8) = .3 *.8^2 4 *.8 + 7 = 3.992 f(1.2) = .3 * 1.2^2 4 *1.2 + 7 = 2.632 f(1.6) = .3 * 1.6 ^2 4 * 1.6 + 7 = 1.368 f(2.0) = .3 *2.0^2 4 * 2.0 + 7 = .2 Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): need to give them in (x, y) points and evaluate the difference between the y values and then the difference of sequences which are the same showing that you have a linear function with a constant slope Im not quite sure I understand how you derived at your model y = 2.4 x - 1.6
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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: multiply both sides by 17: x^3 = 986 raise both sides by power of 1/3: x = 9.95 approx. Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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: Confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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): ********************************************* Question: `qGive your solution to 4 x ^ -.5 = 7 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: you need to divide both sides by 4: x^-.5 = 7 / 4 then raise both sides to the power of -2: x = .327 approx. Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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: divide both sides by 14: x^(2/3) = 39/14 raise both sides by the power of 3/2: x = 4.65 approx. Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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: divide both sides by 5: (3x/8) ^(-3/2) = 9/5 then raise both sides to the power of -2/3: 3x/8 = (9/5) ^ (-2/3) then multiply both sides by 8/3: x = 1.8 approx Confidence rating: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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 ********************************************* Question: `qQuery problem 2. a(n+1) = a(n) + .5 n, a(0) = 2 What are a(1), a(2), a(3), a(4) and a(5)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Mr Smith I have already done these problems in the previous part of the quary
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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): ********************************************* Question: `qWhat is your quadratic function and what is its value for n = 4? Does it fit the sequence exactly? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: previously done Confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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 0.25x^2 - 0.25x + 2. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ********************************************* Question: `qQuery problem 3. f(x) = .3 x^2 - 4x + 7, evaluate at x = 0, .4, .8, 1.2, 1.6 and 2.0. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: previously done Confidence 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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ********************************************* Question: `qQuery problem 4. f(x) = a x^2 + b x + c What symbolic expression stands for the average slope between x = h and x = k? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: previously done Confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique rating: "