#$&* course Mth 164 12:07 pm1/30/15 Question: `q001. In the preceding assignment we saw how to model the sine function using a circle of radius 1.
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Given Solution: The distance along the arc will be equal to the radius at point b. So the angular position of one radian occurs at point b. We see that when the circle is scaled up by a factor of 3, the radius becomes 3 times as great so that the necessary displacement along the arc becomes 3 times as great. Note that the 1-radian angle therefore makes the same angle as for a circle of radius 1. The radius of the circle doesn't affect the picture; the radius simply determines the scale at which the picture is interpreted. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q002. On the circle of radius 3 what arc distance will correspond to an angle of pi/6? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Since this radius of 3 instead of a radius of 1 I would suppose these distances would be 3 times as far so pi/6*3 =3pi/6 which when reduced gives us pi/2 confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aOn 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I see how upscaling the circle effects the arc distance as well. ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I’m not sure but I believe it may be 1/6 radians per sec. confidence rating #$&*: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aSince 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I have this in my notes and I’ve reviewed the text section and I don’t understand why this is. I may be making this more difficult than it is. ------------------------------------------------ Self-critique Rating: 0
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Given Solution: `aEach 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I think I am starting to see the concept here. The circle has a radius of 3 and there are 3 units around the outside of this circle. This means when moving 2 units per second around the circle we have 2/3 radians per second and when moving 15 units per second this is 5 radians per second. I believe I was overlooking a simple concept. ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I would reason that these points would be 3 times as far as the original y values in our studies of a circle with radius 1. 0 y = 0 pi/6 y = 1.5 pi/3 y = 3 Sqrt3 / 2 = 2.598 pi/2 y = 3 2 pi/3 y = 2.598 5 pi/6 y = 1.5 Pi y = 0 7 pi/6 y = -1.5 4 pi/3 y = -2.598 3 pi/2 y = -3 5 pi/3 y = -2.598 11 pi/6 y = -1.5 2 pi. Y = 0 Looking at the circle graph given I can see that these seem to be the correct points. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Angular position radius = 1 radius = 3 0 y = 0 y = 0 pi/6 y = 0.5 y = 1.5 pi/3 y = 0.87 y = 3 Sqrt3 / 2 = 2.598 pi/2 y = 1 y = 3 2 pi/3 y = 0.87 y = 2.598 5 pi/6 y = 0.5 y = 1.5 Pi y = 0 y = 0 7 pi/6 y = -0.5 y = -1.5 4 pi/3 y = -0.87 y = -2.598 3 pi/2 y = -1 y = -3 5 pi/3 y = -0.87 y = -2.598 11 pi/6 y = -0.5 y = -1.5 2 pi. y = 0 y = 0 Yes these are consistent to my previous solution. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aOn a radius 3 circle the y coordinates would each be 3 times as great. The coordinates would therefore be obtained by multiplying the values 0, .5, .87, 1, .87, .5, 0, -.5, -.87, -1, -.87, -.5, 0 each by 3, obtaining 0, 1.5, 2.61, 3, 2.61, 1.5, 0, -1.5, -2.61, -3, -2.61, -1.5, 0. These values should be close, within .1 or so, of the estimates you made for this circle in the preceding problem. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q007. The exact y coordinates of the unit-circle positions 0, pi/6, pi/3, pi/2, 2 pi/3, 5 pi/6, pi, 7 pi/6, 4 pi/3, 3 pi/2, 5 pi/3, 11 pi/6 and 2 pi are 0, .5, sqrt(3) / 2, 1, sqrt(3) / 2, .5, 0, -.5, -sqrt(3) / 2, -1, -sqrt(3) / 2, -.5, 0. • What should be the corresponding y coordinates of the points lying at these angular positions on the circle of radius 3? • Are these coordinates consistent with those you obtained in the preceding problem? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 0 y = 0 pi/6 y = 1.5 pi/3 y = 3 Sqrt3 / 2 pi/2 y = 3 2 pi/3 y = 3 Sqrt3 / 2 5 pi/6 y = 1.5 Pi y = 0 7 pi/6 y = -1.5 4 pi/3 y = -3 Sqrt3 / 2 3 pi/2 y = -3 5 pi/3 y = -3 Sqrt3 / 2 11 pi/6 y = -1.5 2 pi. Y = 0 Yes these are consistent to my previous solution. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q008. Sketch a graph of the y coordinate obtained for a circle of radius 3 in the preceding problem vs. the anglular position theta. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: This graph gave me the points: (0, 0), (pi/6, 1.5), (pi/3, 2.6), (pi/2, 3), (2pi/3, 2.6), (5pi/6, 1.5), (pi, 0)…… The highest point was at (pi/2, 3) and the lowest point was at (3pi/2, -3) The zeros were at (0, 0), (pi, 0) and (2pi, 0) confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aYour graph should be as shown in Figure 54. This graph as the same description as a graph of y = sin(theta) vs. theta, except that the slopes are all 3 times as great and the maximum and minimum values are 3 and -3, instead of 1 and -1. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q009. If the red ant starts on the circle of radius 3, at position pi/3 radians, and proceeds at pi/3 radians per second then what will be its angular position after 1, 2, 3, 4, 5 and 6 seconds? What will be the y coordinates at these points? Make a table and sketch a graph of the y coordinate vs. the time t. Describe the graph of y position vs. clock time. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Starting at Pi/3 (y = 2.6): 1 second 2pi/3 y = 2.6 2 seconds 3pi/3 y = 0 3 seconds 4pi/3 y = -2.6 4 seconds 5pi/3 y = -2.6 5 seconds 6pi/3 y = 0 6 seconds 7pi/3 y = 2.6 This graph peaks at 3 points starting at (pi/3, 2.6) and then (2pi/3, 2.6) and (7pi/3, 2.6) The lowest points are at (4pi/3, -2.6) and (5pi/3, -2.6). The only other points lie at the x axis (pi, 0) and (2pi, 0) confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): My graph did only contain the listed points so I missed the peak of 3. I thought this was incorrect because of the appearance of the graph. ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: `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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Starting at Pi/3 (y = 2.6): 1 second 2pi/3 y = 2.6 2 seconds 3pi/3 y = 0 3 seconds 4pi/3 y = -2.6 4 seconds 5pi/3 y = -2.6 5 seconds 6pi/3 y = 0 6 seconds 7pi/3 y = 2.6 This graph peaks at 3 points starting at (pi/3, 2.6) and then (2pi/3, 2.6) and (7pi/3, 2.6) The lowest points are at (4pi/3, -2.6) and (5pi/3, -2.6). The only other points lie at the x axis (pi, 0) and (2pi, 0) confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): My graph did only contain the listed points so I missed the peak of 3. I thought this was incorrect because of the appearance of the graph. ------------------------------------------------ Self-critique Rating: 3 #*&! ********************************************* Question: `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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Starting at Pi/3 (y = 2.6): 1 second 2pi/3 y = 2.6 2 seconds 3pi/3 y = 0 3 seconds 4pi/3 y = -2.6 4 seconds 5pi/3 y = -2.6 5 seconds 6pi/3 y = 0 6 seconds 7pi/3 y = 2.6 This graph peaks at 3 points starting at (pi/3, 2.6) and then (2pi/3, 2.6) and (7pi/3, 2.6) The lowest points are at (4pi/3, -2.6) and (5pi/3, -2.6). The only other points lie at the x axis (pi, 0) and (2pi, 0) confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): My graph did only contain the listed points so I missed the peak of 3. I thought this was incorrect because of the appearance of the graph. ------------------------------------------------ Self-critique Rating: 3 #*&!#*&!