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20:43:13 `q001. Explain the difference between x - 2 / x + 4 and (x - 2) / (x + 4).
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RESPONSE --> I think the difference is that in the first problem, to get the solution, you start at the beginning and work it straight through and in the second problem, you solve what is in the parenthesis first and then divide two solutions. So I think the difference is the way you solve the problem and the answer might also be different.
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20:43:58 The 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 2 or divide the -2 by 4 first. If there are no parentheses you have to divide before you subtract: 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) / ( 4 - 2) = 0 / 2 = 0.......!!!!!!!!...................................
RESPONSE --> OK
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20:53:47 `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.......!!!!!!!!...................................
RESPONSE --> The difference is that in the first problem, 2 is raised to the power of x plus 4 and in the second problem you will add x plus 4 and then find the power of 2 and that number.
Solution 1: 2^ x + 4= x = 2 2^ 2+4= 4+4=8 Solution 2: 2^ (x+4)= x=2 2^ (6)=64.................................................
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20:54:03 2 ^ 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.......!!!!!!!!...................................
RESPONSE --> OK
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21:03:55 `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?
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RESPONSE --> The numerator is: -1 The denominator is: -2
x-3 / [ (2x-5)^2 * 3x + 1] -2 +7x -1 / [ (4-5)^2 * 6 +1] -2 +14 -1 / [-2 * 7] -2 +14 -1 / [-14] - 2 + 14 -1 / -2.................................................
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21:06:04 The 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.......!!!!!!!!...................................
RESPONSE --> OK, I really got this one wrong.
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21:18:38 `q004. Explain, step by step, how you evaluate the expression (x - 5) ^ 2x-1 + 3 / x-2 for x = 4.
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RESPONSE --> Step 1: Simplify to: (4-5) ^ 2 (4) -1 + 3/ 4-2
Step 2: Solve what is in the ()'s 1 (4) -1 + 3 / 4 - 2 Step 3: Do the multipliation 4 - 1 + .75 -2 Step 4: Simplify further to 3 + -1.25 Step 5: Solution = 1.75.................................................
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21:25:47 We get
(4-5)^2 * 4 - 1 + 3 / 1 - 4 = (-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.......!!!!!!!!...................................
RESPONSE --> I think I understand the solution you have given here. I think that my main problem is knowing the order of operations. I went back and reworked this problem on paper based on your solution and I do understand how you came about the solution.
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§ôòòÕŸË“çxà{ÆÀÌî¢ý– Student Name: assignment #002 002. Describing Graphs
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21:47:22 `q001. You will frequently need to describe the graphs you have constructed in this course. This exercise is designed to get you used to some of the terminology we use to describe graphs. Please complete this exercise and email your work to the instructor.
Problem 1. We make a table for y = 2x + 7 as follows: We construct two columns, and label the first column 'x' and the second 'y'. Put the numbers -3, -2, -1, -, 1, 2, 3 in the 'x' column. We substitute -3 into the expression and get y = 2(-3) + 7 = 1. We substitute -2 and get y = 2(-2) + 7 = 3. Substituting the remaining numbers we get y values 5, 7, 9, 11 and 13. These numbers go into the second column, each next to the x value from which it was obtained. We then graph these points on a set of x-y coordinate axes. Noting that these points lie on a straight line, we then construct the line through the points. Now make a table for and graph the function y = 3x - 4. Identify the intercepts of the graph, i.e., the points where the graph goes through the x and the y axes.......!!!!!!!!...................................
RESPONSE --> Ok, the coordinates I got were: -3, -13; -2, -10; -1, -7; 0, -4; 1, -1; 2,2; 3,5. I graphed them out and they all made a straight line except for the coordinate 0, -4. I'm not sure if this is right or if it wrong, I'm really not sure why. I remember a little about graphing, but there may be a little something that I'm forgetting.
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21:50:04 The graph goes through the x axis when y = 0 and through the y axis when x = 0.
The x-intercept is therefore when 0 = 3x - 4, so 4 = 3x and x = 4/3. The y-intercept is when y = 3 * 0 - 4 = -4. Thus the x intercept is at (4/3, 0) and the y intercept is at (0, -4). Your graph should confirm this.......!!!!!!!!...................................
RESPONSE --> OK. I think I got this one.
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21:53:10 `q002. Does the steepness of the graph in the preceding exercise (of the function y = 3x - 4) change? If so describe how it changes.
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RESPONSE --> I don't think the steepness changes compared to what I have graphed. I'm not exactly sure what this question is asking.
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21:53:46 The graph forms a straight line with no change in steepness.
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RESPONSE --> Ok, so I was right. I did understand it the right way. My graph also formed a straight line.
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21:59:12 `q003. What is the slope of the graph of the preceding two exercises (the function ia y = 3x - 4;slope is rise / run between two points of the graph)?
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RESPONSE --> I do not remember how to find the slope of the graph. I know it is pretty simple, but I don't remember exactly how to get it.
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22:00:32 Between any two points of the graph rise / run = 3.
For example, when x = 2 we have y = 3 * 2 - 4 = 2 and when x = 8 we have y = 3 * 8 - 4 = 20. Between these points the rise is 20 - 2 = 18 and the run is 8 - 2 = 6 so the slope is rise / run = 18 / 6 = 3. Note that 3 is the coefficient of x in y = 3x - 4. Note the following for reference in subsequent problems: The graph of this function is a straight line. The graph increases as we move from left to right. We therefore say that the graph is increasing, and that it is increasing at constant rate because the steepness of a straight line doesn't change.......!!!!!!!!...................................
RESPONSE --> OK.
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22:05:40 `q004. Make a table of y vs. x for y = x^2. Graph y = x^2 between x = 0 and x = 3.
Would you say that the graph is increasing or decreasing? Does the steepness of the graph change and if so, how? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?......!!!!!!!!...................................
RESPONSE --> I would say that the graph is increasing. I think the steepness stays consistant.
I would say that this graph is increasing at an increasing rate..................................................
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22:06:11 Graph points include (0,0), (1,1), (2,4) and (3,9). The y values are 0, 1, 4 and 9, which increase as we move from left to right.
The increases between these points are 1, 3 and 5, so the graph not only increases, it increases at an increasing rate.......!!!!!!!!...................................
RESPONSE --> Great! I got this one right!
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22:09:56 `q005. Make a table of y vs. x for y = x^2. Graph y = x^2 between x = -3 and x = 0.
Would you say that the graph is increasing or decreasing? Does the steepness of the graph change and if so, how? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?......!!!!!!!!...................................
RESPONSE --> I think this graph is decreasing.
I looks like the steepness stays consistant with what I graphed out. I would say that this graph is decreasing at an increasing rate..................................................
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22:12:38 From left to right the graph is decreasing (points (-3,9), (-2,4), (-1,1), (0,0) show y values 9, 4, 1, 0 as we move from left to right ). The magnitudes of the changes in x from 9 to 4 to 1 to 0 decrease, so the steepness is decreasing.
Thus the graph is decreasing, but more and more slowly. We therefore say that the graph is decreasing at a decreasing rate.......!!!!!!!!...................................
RESPONSE --> I see where I made my mistake. I actually graphed out wrong. I graphed on the negative side of the y axis instead of the positive side. I regraphed and I see how the steepness decreases and how it is decreasing at a decreasing rate.
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22:17:31 `q006. Make a table of y vs. x for y = `sqrt(x). [note: `sqrt(x) means 'the square root of x']. Graph y = `sqrt(x) between x = 0 and x = 3.
Would you say that the graph is increasing or decreasing? Does the steepness of the graph change and if so, how? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?......!!!!!!!!...................................
RESPONSE --> I would say that the graph is increasing.
The steepness stays consistant. The graph is increasing at a decreasing rate..................................................
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22:18:13 If you use x values 0, 1, 2, 3, 4 you will obtain graph points (0,0), (1,1), (2,1.414), (3. 1.732), (4,2). The y value changes by less and less for every succeeding x value. Thus the steepness of the graph is decreasing.
The graph would be increasing at a decreasing rate.{}{} If the graph respresents the profile of a hill, the hill starts out very steep but gets easier and easier to climb. You are still climbing but you go up by less with each step, so the rate of increase is decreasing. {}{}If your graph doesn't look like this then you probably are not using a consistent scale for at least one of the axes. If your graph isn't as desribed take another look at your plot and make a note in your response indicating any difficulties.......!!!!!!!!...................................
RESPONSE --> OK. My graph looks as described.
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22:25:35 `q007. Make a table of y vs. x for y = 5 * 2^(-x). Graph y = 5 * 2^(-x) between x = 0 and x = 3.
Would you say that the graph is increasing or decreasing? Does the steepness of the graph change and if so, how? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?......!!!!!!!!...................................
RESPONSE --> I don't remember how to get 2^(-x). The best that I can guess is that the graph is decreasing.
I don't know if the steepness changes. I think that it is decreasing at an increasing rate. I think my problem started when I couldn't remember how to solve 2^(-x)..................................................
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22:27:09 ** From basic algebra recall that a^(-b) = 1 / (a^b).
So, for example: 2^-2 = 1 / (2^2) = 1/4, so 5 * 2^-2 = 5 * 1/4 = 5/4. 5* 2^-3 = 5 * (1 / 2^3) = 5 * 1/8 = 5/8. Etc. The decimal equivalents of the values for x = 0 to x = 3 will be 5, 2.5, 1.25, .625. These values decrease, but by less and less each time. The graph is therefore decreasing at a decreasing rate. **......!!!!!!!!...................................
RESPONSE --> Ok. I think this is still a little confusing. I don't remember doing problems like this one.
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22:29:07 `q008. Suppose you stand still in front of a driveway. A car starts out next to you and moves away from you, traveling faster and faster.
If y represents the distance from you to the car and t represents the time in seconds since the car started out, would a graph of y vs. t be increasing or decreasing? Would you say that the graph is increasing at an increasing rate, increasing at a constant rate, increasing at a decreasing rate, decreasing at an decreasing rate, decreasing at a constant rate, or decreasing at a decreasing rate?......!!!!!!!!...................................
RESPONSE --> I think you would graph it to be increasing.
I think that it is increasing at an increasing rate..................................................
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22:29:21 ** The speed of the car increases so it goes further each second. On a graph of distance vs. clock time there would be a greater change in distance with each second, which would cause a greater slope with each subsequent second. The graph would therefore be increasing at an increasing rate. **
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RESPONSE --> OK.
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‰ä×~r…ÊÚF™ß°ÝîwÚÏ‹¨¶†Y‚TùóÑ‚™ Student Name: assignment #004 004. Liberal Arts Mathematics