course Mth 164 ӌݒTlOn`ݴ͏assignment #003
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17:08:40 Goals for this Assignment include but are not limited to the following: 1. Construct a table of the values of y = sin(x) for a complete cycle of this function, with x equal to multiples of pi/6 or pi/4, and using the table construct a graph of one cycle of y = sin(x ). 2. Given a function y = sin(theta) with theta given as a function of x, construct a table of the values of y = sin(theta) for a complete cycle of this function with theta equal to multiples of pi/6 or pi/4, then determine the x value corresponding to each value of theta. Using a table of y vs. x construct a graph of one cycle of y = sin(theta) in terms of the given function theta of x, clearly labeling the x axis for each quarter-cycle of the function. 3. Interpret the function and graph corresponding to Goal 2 in terms of angular motion on a unit circle. Click once more on Next Question/Answer for a note on Previous Assignments.
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RESPONSE --> self critique assessment:
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17:08:43 Previous Assignments: Be sure you have completed Assignment 1 as instructed under the Assts link on the homepage at 164.106.222.236 and submitted the result of the Query and q_a_ from that Assignment.
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RESPONSE --> self critique assessment:
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lxxߎ assignment #004 004. Extending to circle of radius 3 Precalculus II 02-06-2007
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17:09:03 Goals for this Assignment include but are not limited to the following: 1. Construct a table of the values of y = A sin(x) for a given value of A, extending for a complete cycle of this function, with x equal to multiples of pi/6 or pi/4, and using the table construct a graph of one cycle of y = A sin(x ). 2. Given a function y = A sin(theta) with theta given as a function of x, construct a table of the values of y = A sin(theta) for a complete cycle of this function with theta equal to multiples of pi/6 or pi/4, then determine the x value corresponding to each value of theta. Using a table of y vs. x construct a graph of one cycle of y = A sin(theta) in terms of the given function theta of x, clearly labeling the x axis for each quarter-cycle of the function. 3. Interpret the function and graph corresponding to Goal 2 in terms of angular motion on a unit circle. Click once more on Next Question/Answer for a note on Previous Assignments.
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RESPONSE --> self critique assessment:
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17:09:05 Previous Assignments: Be sure you have completed Assignment 2 as instructed under the Assts link on the homepage at 164.106.222.236 and submitted the result of the Query and q_a_ from that Assignment.
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RESPONSE --> self critique assessment:
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17:11:10 `q001. Now we have a circle of radius 3. An angular position of 1 radian again corresponds to an arc displacement equal to the radius of the circle. Which point on the circle in the picture corresponds to the angular position of 1 radian?
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RESPONSE --> The circle of radius three with an angular position of 1 radian corresponds to the arc displacement of 3 since the arc displacement of 1 radian is equal to that of the radius of the circle. confidence assessment: 2
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17:11:58 The distance along the arc will be equal to the radius at point b. So the angular position of one radian occurs at point b. We see that when the circle is scaled up by a factor of 3, the radius becomes 3 times as great so that the necessary displacement along the arc becomes 3 times as great. Note that the 1-radian angle therefore makes the same angle as for a circle of radius 1. The radius of the circle doesn't matter.
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RESPONSE --> self critique assessment: 3
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17:14:01 `q002. On the circle of radius 3 what arc distance will correspond to an angle of pi/6?
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RESPONSE --> The arc distance to the angle of pi/6 would be 3pi/6 since one radian angle = 3 arc displacement. confidence assessment: 2
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17:14:40 On a circle of radius 1 the arc distance pi/6 corresponds to an arc displacement of pi/6 units. When the circle is scaled up to radius 3 the arc distance will become three times as great, scaling up to 3 * pi/6 = pi/2 units.
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RESPONSE --> self critique assessment: .
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17:16:59 `q003. If the red ant is moving along a circle of radius 3 at a speed of 2 units per second, then what is its angular velocity--i.e., its the rate in radians / second at which its angular position changes?
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RESPONSE --> If an ant moves at 2 units per second then the angular velocity would be 2 units per second also. confidence assessment: 1
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17:18:43 Since 3 units corresponds to one radian, 2 units corresponds to 2/3 radian, and 2 units per second will correspond to 2/3 radian/second.
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RESPONSE --> The ant makes a move of 2 units per second which is 2/3 the amount total of one radian, which corresponds to 3 units, therefore, the ant is moving at 2/3 units per second. self critique assessment: 2
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17:19:46 `q004. If the red ant is moving along at angular velocity 5 radians/second on a circle of radius 3, what is its speed?
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RESPONSE --> The ant would be moving at an angular velocity of 5/3 radians per second. confidence assessment: 2
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17:21:14 Each radian on a circle of radius 3 corresponds to 3 units of distance. Therefore 5 radians corresponds to 5 * 3 = 15 units of distance and 5 radians/second corresponds to a speed of 15 units per second.
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RESPONSE --> One radian equals 3 units of distance on the arc, on a circle of radius 3. So if the ant has traveled 5 units of distance, then to find the radians involved, multiply this by three, to get the speed of 15 units per second. self critique assessment: 2
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17:34:37 `q005. Figure 17 shows a circle of radius 3 superimposed on a grid with .3 unit between gridmarks in both x and y directions. Verify that this grid does indeed correspond to a circle of radius 3. Estimate the y coordinate of each of the points whose angular positions correspond to 0, pi/6, pi/3, pi/2, 2 pi/3, 5 pi/6, pi, 7 pi/6, 4 pi/3, 3 pi/2, 5 pi/3, 11 pi/6 and 2 pi.
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RESPONSE --> 10 spaces each of .3 = 3, verified. 0 = 0 pi/6 = 1.5 pi/3 = 2.6 pi/2 = 3 2pi/3 =2.6 5pi/6 = 1.5 pi = 0 7pi/6 = -1.5 4pi/3 = -2.6 3pi/2 = 3 5pi/3 = -2.6 11pi/6 = -1.5 2 pi = 0 confidence assessment: 2
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17:34:58 The angular positions of the points coinciding with the positive and negative x axes all have y coordinate 0; these angles include 0, pi and 2 pi. At angular position pi/2 the y coordinate is equal to the radius 3 of the circle; at 3 pi/2 the y coordinate is -3. At angular position pi/6 the point on the circle appears to be close to (2.7,1.5); the x coordinate is actually a bit less than 2.7, perhaps 2.6, so perhaps the coordinates of the point are (2.6, 1.5). Any estimate close to these would be reasonable. The y coordinate of the pi/6 point is therefore 1.5. The coordinates of the pi/3 point are (1.5, .87), just the reverse of those of the pi/6 point; so the y coordinate of the pi/3 point is approximately 2.6. The 2 pi/3 point will also have y coordinate approximately 2.6, while the 4 pi/3 and 5 pi/3 points will have y coordinates approximately -2.6. The 5 pi/6 point will have y coordinate 1.5, while the 7 pi/6 and 11 pi/6 points will have y coordinate -1.5.
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RESPONSE --> self critique assessment: 3
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17:36:49 `q006. The y coordinates of the unit-circle positions 0, pi/6, pi/3, pi/2, 2 pi/3, 5 pi/6, pi, 7 pi/6, 4 pi/3, 3 pi/2, 5 pi/3, 11 pi/6 and 2 pi are 0, .5, .87, 1, .87, .5, 0, -.5, -.87, -1, -.87, -.5, 0. What should be the corresponding y coordinates of the points lying at these angular positions on the circle of radius 3? Are these coordinates consistent with those you obtained in the preceding problem?
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RESPONSE --> The y coordinates of the unit circle graph should be 1/3 the amount of the circle of radius 3, so all the y values of the unit circle, when multiplied by 3 should equal that of the 3 radius circle, 0 * 3 = 0, .5 * 3 = 1.5, .87 * 3 = 2.61, 1*3=3, everything checks out. confidence assessment: 3
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17:36:58 On a radius 3 circle the y coordinates would each be 3 times as great. The coordinates would therefore be obtained by multiplying the values 0, .5, .87, 1, .87, .5, 0, -.5, -.87, -1, -.87, -.5, 0 each by 3, obtaining 0, 1.5, 2.61, 3, 2.61, 1.5, 0, -1.5, -2.61, -3, -2.61, -1.5, 0. These values should be close, within .1 or so, of the estimates you made for this circle in the preceding problem.
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RESPONSE --> self critique assessment: 3
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17:39:11 `q007. The exact y coordinates of the unit-circle positions 0, pi/6, pi/3, pi/2, 2 pi/3, 5 pi/6, pi, 7 pi/6, 4 pi/3, 3 pi/2, 5 pi/3, 11 pi/6 and 2 pi are 0, .5, sqrt(3) / 2, 1, sqrt(3) / 2, .5, 0, -.5, -sqrt(3) / 2, -1, -sqrt(3) / 2, -.5, 0. What should be the corresponding y coordinates of the points lying at these angular positions on the circle of radius 3? Are these coordinates consistent with those you obtained in the preceding problem?
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RESPONSE --> The corresponding y coordinates of the points lying at the angular positions of 0, pi/6, pi/3, pi/2, 2 pi/3, 5 pi/6, pi, 7 pi/6, 4 pi/3, 3 pi/2, 5 pi/3, 11 pi/6 and 2 pi on the circle of radius 3 should be 0, 3/2, 3sqrt(3)/2, 3, 3sqrt(3)/2, 3/2, 0, -3/2, -3sqrt(3)/2, -3, -3sqrt(3)/2, -3/2, and 0. these are still consistent with the ones of the preceding problem. confidence assessment: 3
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17:39:26 On a radius 3 circle the y coordinates would each be 3 times as great. The coordinates would therefore be obtained by multiplying the values 0, .5, sqrt(3) / 2, 1, sqrt(3) / 2, .5, 0, -.5, -sqrt(3) / 2, -1, -sqrt(3) / 2, -.5, 0 each by 3, obtaining 0, 1.5, 3 sqrt(3) / 2, 3, 3 sqrt(3) / 2, 1.5, 0, -3 sqrt(3) / 2, -3 sqrt(3) / 2, -3, -2.61, -1.5, 0.
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RESPONSE --> self critique assessment: 3
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17:42:35 `q008. Sketch a graph of the y coordinate obtained for a circle of radius 3 in the preceding problem vs. the anglular position theta.
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RESPONSE --> Looks like unit cirlce except all values are multiplied by three. confidence assessment: 3
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17:44:17 Your graph should be as shown in Figure 54. This graph as the same description as a graph of y = sin(theta) vs. theta, except that the slopes are all 3 times as great and the maximum and minimum values are 3 and -3, instead of 1 and -1.
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RESPONSE --> I was mistaken and did a graph of the circle itself, however, the maximum y value is 3, minimum is -3 because this is the radius of the circle, and the x intercept is at pi and 2 pi, etc. self critique assessment: 2
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12:03:10 `q009. If the red ant starts on the circle of radius 3, at position pi/3 radians, and proceeds at pi/3 radians per second then what will be its angular position after 1, 2, 3, 4, 5 and 6 seconds? What will be the y coordinates at these points? Make a table and sketch a graph of the y coordinate vs. the time t. Describe the graph of y position vs. clock time.
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RESPONSE --> time, angular position, y coordinate 0 sec, pi/3, 3sqrt(3)/2 1 sec, 2pi/3, 3sqrt(3)/2 2 sec, pi, 0 3 sec, 4pi/3, -3sqrt(3)/2 4 sec, 5pi/3, -3sqrt(3)/2 5 sec, 2pi, 0 6 sec, pi/3, 3sqrt(3)/2 The graph has y value of 3sqrt(3)/2 and y value of - 3sqrt(3)/2 with the clock time being marked off for six seconds. The graph begins at y = 3sqrt(3)/2, increases at a decreasing rate and reaches a peak somewhere past y = 3sqrt(3)/2, possibly 1/2, then begins to decrease at an increasing rate, passing through 3sqrt(3)/2 at one second, and the x axis at 2 sec, where the graph begins to decrease at a decreasing rate, reaches the bottom peak some where around - 1/2 i would guess, y = - 3sqrt(3)/2 is passed through at 3 and 4 seconds, the graph increases at an increasing rate up to the x axis at 5 seconds and stops at y = 3sqrt(3)/2 at 6 seconds. confidence assessment: 3
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12:06:11 The angular positions at t = 1, 2, 3, 4, 5 and 6 are 2 pi/3, pi, 4 pi/3, 5 pi/3, 2 pi and 7 pi/3. The corresponding y coordinates are 3 * sqrt(3) / 2, 0, -3 * sqrt(3) / 2, -3 * sqrt(3) / 2, 0 and 3 * sqrt(3) / 2. If you just graph the corresponding points you will miss the fact that the graph also passes through y coordinates 3 and -3; from what you have seen about these functions in should be clear why this happens, and it should be clear that to make the graph accurate you must show this behavior. See these points plotted in red in Figure 45, with the t = 0, 2, 4, 6 values of theta indicated on the graph. The graph therefore runs through its complete cycle between t = 0 and t = 6, starting at the point (0, 3 * sqrt(3) / 2), or approximately (0, 2.6), reaching its peak value of 3 between this point and (1, 3 * sqrt(3) / 2), or approximately (1, 2.6), then reaching the x axis at t = 3 as indicated by the point (2, 0) before descending to (3, -3 * sqrt(3) / 2) or approximately (3, -2.6), then through a low point where y = -3 before again rising to (4, -3 * sqrt(3) / 2) then to (5, 0) and completing its cycle at (6, 3 * sqrt(3) / 2). This graph is shown in Figure 86.
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RESPONSE --> The y value i was referring to as possibly 1/2 is of course 3, I was thinking of pi/6 on the unit circle which still would even have to be translated to pi/2, but the maximum y value of a circle with radius 3 is 3 and the minimum is -3. self critique assessment: 2
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12:19:15 `q010. Make a table for the graph of y = 3 sin(pi/4 * t + pi/3), using theta = 0, pi/4, pi/2, etc., and plot the corresponding graph. Describe your graph.
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RESPONSE --> confidence assessment:
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12:27:21 Using columns for t, theta and sin(theta) as we have done before, and an additional column for 3 * sin(theta) we obtain the following initial table: t theta = pi/4 t + pi/3 sin(theta) 3 * sin(theta) 0 0 pi/4 .71 pi/2 0 3 pi/4 .71 pi 0 5 pi/4 -.71 3 pi/2 -1 7 pi/4 -.71 2 pi 0 0 The solution to pi/4 t + pi/3 = theta is obtained by first adding -pi/3 to both sides to obtain pi/4 t = theta - pi/3, then multiplying both sides by 4 / pi to obtain t = 4 / pi * theta - 4 / pi * pi/3, and finally simplifying to get t = 4 / pi * theta - 4/3. Substituting in the given values of theta we obtain t values -4/3, -1/3, 2/3, 5/3, 8/3, 11/3, 14/3, 17/3, 20/3. We also multiply the values of sin(theta) by 3 to get the values of 3 sin(theta): t theta = pi/4 t + pi/3 sin(theta) 3 * sin(theta) -4/3 0 0 0 -1/3 pi/4 .71 2.1 2/3 pi/2 0 0 5/3 3 pi/4 .71 2.1 8/3 pi 0 0 11/3 5 pi/4 -.71 -2.1 14/3 3 pi/2 -1 -1 17/3 7 pi/4 -.71 -2.1 20/3 2 pi 0 0 Since theta = pi/4 t + pi/3, if we graph the final column vs. the first we have the graph of y = 3 sin(pi/4 t + pi/3) vs. t. This graph is shown in Figure 19, with red dots indicating points corresponding to rows of the table. The graph is as in the figure (fig 3 sin(pi/4 * t + pi/3)), with a standard cycle running from t = -4/3 to t = 20/3. During this cycle y goes from 0 to 3 to 0 to -3 to 0. The duration of the cycle is 20/3 - (-4/3) = 24/3.
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RESPONSE --> I accidently hit enter and the box came up to evaluate your confidence and when hitting cancel, it moved on to the next question. I arrived at mostly the same answer execpt theta pi/2, that sin would be 1 and multiplied by 3 equalling 3 instead of 0 and 0 i imagine. self critique assessment: 3
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12:27:37 Complete Assignment 3, including Class Notes, text problems and Web-based problems as specified on the Assts page. When you have completed the entire assignment run the Query program. Submit SEND files from Query and q_a_.
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RESPONSE --> self critique assessment:
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}TwOΞ]ҳ assignment #003 003. Precalculus II 02-07-2007
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12:45:08 query problem 5.5.42 transformations to graph 3 cos x + 3
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RESPONSE --> confidence assessment:
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12:45:45 explain how you use transformations to construct the graph.
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RESPONSE --> The final graph is that of the cos x graph with a vertical stretch of 3 and a vertical shift of 3 also. confidence assessment: 3
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12:46:21 Describe the resulting graph by given its period and the maximum and minimum y values
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RESPONSE --> The period is still 2pi as the normal cosine graph. However, because of the vertical stretch and shift, the maximum y value is 6 and the minimum is 0. confidence assessment: 3
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12:46:28 query problem 5.5.54 transformations to graph 4 tan(.5 x)
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RESPONSE --> confidence assessment:
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12:46:56 explain how you use transformations to construct the graph.
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RESPONSE --> The graph is that of the tangent graph only with a vertical shift of 4 and a period adjustment to 2pi. confidence assessment: 3
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12:47:26 describe the graph by giving the locations of its vertical asymptotes
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RESPONSE --> Vertical asymptotes occur at, ...-3pi, -pi, pi, 3pi... confidence assessment: 3
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12:47:30 Comm on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> confidence assessment:
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