MAT2500-01/04 17S Test 1 Print Name (Last, First) _________________________________|__
Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use EQUAL SIGNS and arrows when appropriate. Always SIMPLIFY expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation IF appropriate). Indicate where technology is used and what type (Maple, GC). You are encouraged to use technology to check all of your hand results.
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parametrized curve segment 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
is shown in the figure together with 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. Maple will automatically simplify most results to expressions only involving the cosine, or you can ask it to simplify them.
a) Evaluate and simplify 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 , 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, 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| and their values (including 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) at 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.b) Write the parametrized equations of the tangent line through 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. c) Evaluate and simplify LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkmbW92ZXJHRiQ2JS1GIzYoLUkjbWlHRiQ2KFEiYkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJ0Y0RjdGOUY8Rj8tSSNtb0dGJDYvUS0mUmlnaHRBcnJvdztGJ0Y5LyUscGxhY2Vob2xkZXJHRjYvRj1RJ25vcm1hbEYnLyUmZmVuY2VHUSZ1bnNldEYnLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmVuL0ZYRjYtRjE2I1EhRictSShtZmVuY2VkR0YkNiUtRiM2JS1GMTYmUSJ0RidGN0Y5L0Y9USdpdGFsaWNGJ0Y5RkhGOUZILUZDNi5RIj1GJ0Y5RkgvRktGOy9GTkY7L0ZQRjsvRlJGOy9GVEY7L0ZWRjsvRlhGOy9GWlEsMC4yNzc3Nzc4ZW1GJy9GZ25GYXAtRiw2JS1GMTYoUSJyRidGNEY3RjlGPEY/RkJGaG4tRkM2L1EiJ0YnRjdGOUZkb0Zpb0Zqb0ZbcEZccEZdcEZecEZfcC9GWlEsMC4xMTExMTExZW1GJ0ZmbkZcby1GQzYuUSgmdGltZXM7RidGOUZIRmlvRmpvRltwRlxwRl1wRl5wRl9wL0ZaUSwwLjIyMjIyMjJlbUYnL0ZnbkZhcS1GQzYuUSJ+RidGOUZIRmlvRmpvRltwRlxwRl1wRl5wRl9wRllGZm5GY3AtRkM2LkZqcEY5RkhGaW9Gam9GW3BGXHBGXXBGXnBGX3BGW3FGZm5GZnFGXG9GOUZI. Evaluate 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 and 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 .
d) Write the equation of the osculating plane through 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 containing the tangent vector and the second derivative there.e) Evaluate the curvature 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 and its reciprocal, the radius of curvature LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSJ0RicvRjBRJXRydWVGJ0YyL0Y1USdpdGFsaWNGJ0YyRjRGMkY0RjJGNA== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiUtRiM2JS1JJm1mcmFjR0YkNigtRiw2JlEnJiM5NjA7RidGL0YyRjQtRiM2Ji1JI21uR0YkNiVRIjJGJ0YyRjQvRjBRJXRydWVGJ0YyL0Y1USdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGUS8lKWJldmVsbGVkR0YxRjJGNEYyRjRGMkY0 .f) Evaluate and simplify the unit vector 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 in the direction of 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 and then 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.g) Evaluate and simplify the unit normal 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\303\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.h) Evaluate the scalar tangential projection 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 along 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 of the acceleration 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 and its scalar normal projection 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 exactly. i) Write down an integral formula for the length of the curve 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjwtRjk2LlEjLi5GJ0YyRjxGPkZARkJGREZGRkhGSi9GTVEsMC4yMjIyMjIyZW1GJy9GUFEmMC4wZW1GJy1GOTYuUSJ+RidGMkY8Rj5GQEZCRkRGRkZIRkovRk1GZW5GWi1GUjYlUSIyRidGMkY8RmZuLUYsNiZRJyYjOTYwO0YnL0YwRjRGMkY8RjJGPA==. Evaluate it exactly (special function!) and then numerically to 4 decimal places.
j) Optional. The center of the osculating circle has position vector: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkmbW92ZXJHRiQ2JS1JI21pR0YkNihRIkNGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjQvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYvUS0mUmlnaHRBcnJvdztGJ0Y3LyUscGxhY2Vob2xkZXJHRjQvRjtRJ25vcm1hbEYnLyUmZmVuY2VHUSZ1bnNldEYnLyUqc2VwYXJhdG9yR0ZKLyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRkovJShsYXJnZW9wR0ZKLyUubW92YWJsZWxpbWl0c0dGSi8lJ2FjY2VudEdGSi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlkvRlZGNC1JKG1mZW5jZWRHRiQ2JS1GIzYlLUYvNiZRInRGJ0Y1RjcvRjtRJ2l0YWxpY0YnRjdGRkY3RkYtRkE2LlEiPUYnRjdGRi9GSUY5L0ZMRjkvRk5GOS9GUEY5L0ZSRjkvRlRGOS9GVkY5L0ZYUSwwLjI3Nzc3NzhlbUYnL0ZlbkZccC1GLDYlLUYvNihRInJGJ0YyRjVGN0Y6Rj1GQEZmbi1GLzYjUSFGJ0Znbi1GQTYuUSIrRidGN0ZGRmRvRmVvRmZvRmdvRmhvRmlvRmpvL0ZYUSwwLjIyMjIyMjJlbUYnL0ZlbkZqcC1GLzYmUScmIzk2MTtGJy9GNkY5RjdGRkZnbi1GQTYuUSJ+RidGN0ZGRmRvRmVvRmZvRmdvRmhvRmlvRmpvRldGWi1GLDYlLUYjNiYtRi82KFEiTkYnRjJGNUY3RjpGPUY1RjdGX29GQEZmbkZnbkY3RkY=. Show that 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.k) Optional. [Just kidding. Ignore this until after the test.] The osculating circle is given by 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. Evaluate this for 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. Then show that this circle just kisses the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn-axis. This is obvious in a plot or from thinking about the relative position of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbW92ZXJHRiQ2JS1JI21pR0YkNihRInJGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjQvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYvUS0mUmlnaHRBcnJvdztGJ0Y3LyUscGxhY2Vob2xkZXJHRjQvRjtRJ25vcm1hbEYnLyUmZmVuY2VHUSZ1bnNldEYnLyUqc2VwYXJhdG9yR0ZKLyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRkovJShsYXJnZW9wR0ZKLyUubW92YWJsZWxpbWl0c0dGSi8lJ2FjY2VudEdGSi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlkvRlZGNC1JKG1mZW5jZWRHRiQ2JS1GIzYlLUkmbWZyYWNHRiQ2KC1GLzYmUSUmcGk7RicvRjZGOUY3RkYtRiM2JC1JI21uR0YkNiVRIjJGJ0Y3RkZGRi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXnAvJSliZXZlbGxlZEdGOS1GLzYjUSFGJ0ZGRjdGRkZjcEZG and 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solutiona) 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 is an elliptic helix.Qys+SSJyRzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkPCw+R0koX3N5c2xpYkdGJTYlLUkkY29zR0YlNiM5JCwkLUkkc2luR0YlRjIiIiNGM0YlRiVGJSEiIjYlLUYkRictLUkiREdGJTYjRiRGJy0tLUkjQEBHRiU2JEY9RjdGPkYnIiIiLUklc3FydEdGJTYjLUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUksVHlwZXNldHRpbmdHNiRGS0YuNiRGO0Y7RkQ+SSZzcGVlZEdGJWYqRidGJUYpRiUtSSlzaW1wbGlmeUdGJTYkLUZGNiMtRkk2JC1GPEYyRmVuSSlzeW1ib2xpY0dGJUYlRiVGJUY4LUZSRic=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Qyc2JS1JInJHNiI2Iy1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YmSS9WZWN0b3JDYWxjdWx1c0c2JUYrL0YtSShTdHVkZW50RzYkRitJKF9zeXNsaWJHRiZGKzYkSSNQaUdGKyMiIiIiIiMtSSVldmFsR0YrNiQtSSVkaWZmR0YrNiQtRiU2I0kieEdGJkZBL0ZBRigtRjo2JC1GPTYlRj9GQUZBRkJGNy1JJXNxcnRHRjI2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYrL0YtSSxUeXBlc2V0dGluZ0dGMjYkRjlGOUY3LUZINiMtRks2JEZDRkM=Qyc+SSJURzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyomLUklZXZhbEdGLzYkLUklZGlmZkdGLzYkLUkickdGJTYjSSJ4R0YlRjwvRjw5JCIiIi1JJnNwZWVkR0YlNiNGPiEiIkYlRiVGJUZDLUYkRidGPy1GJDYjLUkiKkc2JEYvL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiVGLy9GS0koU3R1ZGVudEdGLkYvNiRJI1BpR0YvI0Y/IiIjb) tangent line at 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 .Ly1JJDwsPkdJKF9zeXNsaWJHNiI2JUkieEdGJkkieUdGJkkiekdGJi1JIitHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YmSS9WZWN0b3JDYWxjdWx1c0c2JUYuL0YwSShTdHVkZW50RzYkRi5GJUYuNiQtSSJyR0YmNiMtSSIqR0YtNiRJI1BpR0YuIyIiIiIiIy1GOzYkSSJzR0YmLUklZXZhbEdGLjYkLUklZGlmZkdGLjYkLUY4NiNGKEYoL0YoRjo=c)QyUtSTJkZWxheUNyb3NzUHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nRzYkRiZJKF9zeXNsaWJHRik2JC1JJWV2YWxHRiY2JC1JJWRpZmZHRiY2JC1JInJHRik2I0kieEdGKUY3L0Y3SSJ0R0YpLUYvNiQtRjI2JUY0RjdGN0Y4IiIiLUkpc2ltcGxpZnlHRis2I0kiJUdGKQ==Qyo+SSJiRzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy1JMmRlbGF5Q3Jvc3NQcm9kdWN0RzYkRi8vSSttb2R1bGVuYW1lR0YlSSxUeXBlc2V0dGluZ0dGLjYkLUklZXZhbEdGLzYkLUklZGlmZkdGLzYkLUkickdGJTYjSSJ4R0YlRkIvRkI5JC1GOjYkLUY9NiVGP0ZCRkJGQ0YlRiVGJSEiIi1GJEYnIiIiLUYtNiMtSSIqRzYkRi8vRjZJL1ZlY3RvckNhbGN1bHVzRzYlRi8vRjZJKFN0dWRlbnRHRi5GLzYkSSIlR0YlKiQtSSVzcXJ0R0YuNiMtSTBkZWxheURvdFByb2R1Y3RHRjQ2JEZXRldGSUZLPkkiQkdGJS1JKHVuYXBwbHlHRi42JEZXRihGSQ==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) osculating plane at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8 as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSNtbkdGJDYlUSIwRidGMi9GNlEnbm9ybWFsRidGMkZBRjJGQUYyRkE= as its 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e) curvature:QyU+SSZrYXBwYUc2ImYqNiNJInRHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQqJi1JJWJtYWdHRiU2IzkkIiIiLUkmc3BlZWRHRiVGNSEiJEkpc3ltYm9saWNHRiVGJUYlRiUhIiItRiRGJw==Qyc+SSRyaG9HNiItSSh1bmFwcGx5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQqJC1JJmthcHBhR0YlNiNJInRHRiUhIiJGMEYxLUYkRi8iIiItSSlzaW1wbGlmeUdGKDYjLUYkNiMtSSIqRzYkRikvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JUYpL0Y9SShTdHVkZW50R0YoRik2JEkjUGlHRikjRjMiIiM=f) binormal.QyctSSJCRzYiNiNJInRHRiUiIiItRiQ2Iy1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JUYuL0YwSShTdHVkZW50RzYkRi5JKF9zeXNsaWJHRiVGLjYkSSNQaUdGLiNGKCIiI0YoLUkpc2ltcGxpZnlHRjU2I0kiJUdGJQ==g) normal evaluated in two ways.QyctSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUkyZGVsYXlDcm9zc1Byb2R1Y3RHNiRGJi9JK21vZHVsZW5hbWVHRihJLFR5cGVzZXR0aW5nR0YlNiQtSSJCR0YoNiNJInRHRigtSSJUR0YoRjMiIiItSSRtYXBHRiY2JUklZXZhbEdGJkkiJUdGKC9GNC1JIipHNiRGJi9GLkkvVmVjdG9yQ2FsY3VsdXNHNiVGJi9GLkkoU3R1ZGVudEdGJUYmNiRJI1BpR0YmI0Y3IiIjRjctRiQ2I0Y8Ignore this.QyktSSVldmFsRyUqcHJvdGVjdGVkRzYkLUklZGlmZkdGJTYkLUkiVEc2IjYjSSJ4R0YsRi4vRi5JInRHRiwiIiItSSlzaW1wbGlmeUc2JEYlSShfc3lzbGliR0YsNiNJIiVHRixGMT5JIk5HRixmKjYjRjBGLDYkSSlvcGVyYXRvckdGLEkmYXJyb3dHRixGLC1GMzYkKiYtRiQ2JEYnL0YuOSRGMS1JJXNxcnRHRjQ2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYlL0krbW9kdWxlbmFtZUdGLEksVHlwZXNldHRpbmdHRjQ2JEZCRkIhIiJJKXN5bWJvbGljR0YsRixGLEYsRlAtRjlGOw==LUkiTkc2IjYjLUkiKkc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiRJL1ZlY3RvckNhbGN1bHVzRzYlRikvRitJKFN0dWRlbnRHNiRGKUkoX3N5c2xpYkdGJEYpNiRJI1BpR0YpIyIiIiIiIw==h) acceleration decomposition.LUklZXZhbEclKnByb3RlY3RlZEc2JC1JJWRpZmZHRiQ2JS1JInJHNiI2I0kieEdGK0YtRi0vRi0tSSIqRzYkRiQvSSttb2R1bGVuYW1lR0YrSS9WZWN0b3JDYWxjdWx1c0c2JUYkL0YzSShTdHVkZW50RzYkRiRJKF9zeXNsaWJHRitGJDYkSSNQaUdGJCMiIiIiIiM=QyUtSSVhbWFnRzYiNiNJInRHRiUiIiItRiQ2Iy1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JUYuL0YwSShTdHVkZW50RzYkRi5JKF9zeXNsaWJHRiVGLjYkSSNQaUdGLiNGKCIiIw==Note that the tangential acceleration equals the speed for this curve.QyUtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUkwZGVsYXlEb3RQcm9kdWN0RzYkRiYvSSttb2R1bGVuYW1lR0YoSSxUeXBlc2V0dGluZ0dGJTYkLUklZXZhbEdGJjYkLUklZGlmZkdGJjYlLUkickdGKDYjSSJ4R0YoRjpGOi9GOkkidEdGKC1JIlRHRig2I0Y8IiIiLUkmc3BlZWRHRihGPw==LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYlL0krbW9kdWxlbmFtZUdGJ0ksVHlwZXNldHRpbmdHRiQ2JC1JJWV2YWxHRiU2JC1JJWRpZmZHRiU2JS1JInJHRic2I0kieEdGJ0Y5RjkvRjlJInRHRictSSJOR0YnNiNGOw==Qyo+SSR0YXVHNiItSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiVGKS9GK0koU3R1ZGVudEc2JEYpSShfc3lzbGliR0YlRik2JEkjUGlHRikjIiIiIiIjISIiLUkiVEdGJTYjRiRGNT5JI2FUR0YlLUkwZGVsYXlEb3RQcm9kdWN0RzYkRikvRitJLFR5cGVzZXR0aW5nR0YwNiQtSSVldmFsR0YpNiQtSSVkaWZmR0YpNiUtSSJyR0YlNiNJInhHRiVGTEZML0ZMRiRGOEY1PkkjYU5HRiUtRj42JEZDLUkiTkdGJUY6RjU=Pythagorean check:QyU2JCokSSNhVEc2IiIiIyokSSNhTkdGJkYnIiIiLy1JIitHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YmSS9WZWN0b3JDYWxjdWx1c0c2JUYvL0YxSShTdHVkZW50RzYkRi9JKF9zeXNsaWJHRiZGL0YjKiQtSSVhbWFnR0YmNiMtSSIqR0YuNiRJI1BpR0YvI0YqRidGJw==To get an idea of the shape of the projection rectangle, here is the rectangle whose sides are the two scalar projections and whose diagonal is the acceleration vector length.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hmm.i) Length of curve is integral of speed.QyctSSZzcGVlZEc2IjYjSSJ0R0YlIiIiLUkkaW50R0YlNiRGIy9GJzsiIiEtSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiVGMi9GNEkoU3R1ZGVudEc2JEYySShfc3lzbGliR0YlRjI2JCIiI0kjUGlHRjJGKC9JIkxHRiUtSSZldmFsZkdGMjYjSSIlR0Ylj) osculating circle center.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k) osculating circle and intersection with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8.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Maple only gives one of the two solutions seen in the plot, symmetric about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GLDYmUScmIzk2MDtGJy9GMEY0RjJGPEYyRjw=.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The reference angle compared to the horizontal axis on the unit circle 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 to find the coordinates of these points of intersection.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JSFHmaking the diagram and 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JSFHHere we include the part of the tangent line.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 allows us to zoom in on the point of contact, by centering the plot box on the contact point.It does not take much more effort (easy for you to say, bob) to put in all the vectors of the projection process as well as the TNB triad.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiMEYnRjItRjY2LVEiOkYnRjJGOUY7Rj1GP0ZBRkNGRUZHRkovJStleGVjdXRhYmxlR0YxRjI=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JSFHpledgeWhen you have completed the exam, please read and sign the dr bob integrity pledge and hand this test sheet on top of your answer sheets as a cover page, with the first test page facing up:"During this examination, all work has been my own. I have not accessed any of the class web pages or any other sites during the exam. I give my word that I have not resorted to any ethically questionable means of improving my grade or anyone else's on this examination and that I have not discussed this exam with anyone other than my instructor, nor will I until after the exam period is terminated for all participants."Signature: Date: TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzMxNjIyMzI2WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQtSSRjb3NHNiQlKnByb3RlY3RlZEclKF9zeXNsaWJHNiMlInRHLCQtSSRzaW5HNiRGLEYtRi4iIiNGL0YlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY1MTAwMzkwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJC1JJHNpbkc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IyUidEchIiIsJC1JJGNvc0c2JEYtRi5GLyIiIyIiIkYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY1MTAwOTkwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJC1JJGNvc0c2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IyUidEchIiIsJC1JJHNpbkc2JEYtRi5GLyEiIyIiIUYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY1MDUyOTE4WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQiIiEiIiMsJCUjUGlHIyIiIkYqRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY1MDUzMjc4WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQhIiIiIiEiIiJGJQ==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TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY1MDYwNjMwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQlInhHJSJ5RyUiekdGJQ==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TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY1MDc3NDk0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJC1JJHNpbkc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IyUidEciIiMsJC1JJGNvc0c2JEYtRi5GLyEiIkYxRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY1MDc4NTc0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJComLCYqJC1JJGNvc0c2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IyUidEciIiMhIiQiIikiIiIjISIiRjQtSSRzaW5HNiRGMEYxRjJGN0Y0LCQqJkYrRjhGLUY3RjksJCokRitGOEY0RiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzY1MDc1MDc4WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJC1JJHNpbkc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IyUidEciIiMsJC1JJGNvc0c2JEYtRi5GLyEiIkYxRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzQxNTYxODc4WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJComLCYqJC1JJGNvc0c2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IyUidEciIiMhIiQiIikiIiIjISIiRjQtSSRzaW5HNiRGMEYxRjJGN0Y0LCQqJkYrRjhGLUY3RjksJCokRitGOEY0RiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzQxNTYxOTk4WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJCokIiIpIyIiIiIiIyNGLSIiJSIiIUYpRiU=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TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzQxNTU1OTY2WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQiIiEhIiJGKUYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzQxMzUxMTY2WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJiooLCYqJC1JJGNvc0c2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IyUidEciIiMiIiRGNCIiIiMhIiRGNC1JJHNpbkc2JEYwRjFGMkY0Ri1GNkY4KiZGKyMhIiJGNEYtRjZGPiwmKihGK0Y3Ri1GNEY5RjYiIicqJkYrRj1GOUY2ISIjLCQqKEYrRjdGLUY2RjlGNkY1RiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzQxMzQzMjMwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJComLUkkY29zRzYkJSpwcm90ZWN0ZWRHJShfc3lzbGliRzYjJSJ0RyIiIiwmKiRGKyIiIyIiJEY1RjIjISIkRjUhIiYsJComLUkkc2luRzYkRi5GL0YwRjJGM0Y3ISIlLCQqKEYzRjdGK0YyRjxGMkY2RiU=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