JSFHMAT2500-02 08S Quiz 3 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 arrows and equal signs when appropriate. Always simplify expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation). Indicate where technology is used and what type (Maple, GC).Given the vector-valued function 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 for its natural domain 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:
a) Evaluate 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 and remember to simplify your results (no credit for unidentified expressions).
b) Evaluate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2MEYrLUYjNiVGKy1JJm1vdmVyR0YkNiUtRiw2J1EickYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGOy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYuUS0mUmlnaHRBcnJvdztGJy8lLHBsYWNlaG9sZGVyR0Y7L0Y/USdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGTi8lKXN0cmV0Y2h5R0Y7LyUqc3ltbWV0cmljR0ZOLyUobGFyZ2VvcEdGTi8lLm1vdmFibGVsaW1pdHNHRk4vJSdhY2NlbnRHRk4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Znbi8lJ2FjY2VudEdGO0ZKLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtbkdGJDYkUSIxRidGSkZKRkotRkU2LVEiLEYnRkovRk1RJmZhbHNlRicvRlBGOy9GUkZpby9GVEZpby9GVkZpby9GWEZpby9GWkZpb0Zlbi9GaW5RLDAuMzMzMzMzM2VtRidGMS1GRTYtUSInRidGSkZoby9GUEZpb0ZbcEZccEZdcEZecEZfcC9GZm5RLDAuMTExMTExMWVtRidGaG5GXG9GZW8tRkU2LVEifkYnRkpGaG9GZXBGW3BGXHBGXXBGXnBGX3BGZW5GaG5GMUZicEZicEZcb0ZKRitGK0ZK, 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 and remember to simplify your results (no credit for unidentified expressions).
c) Evaluate the exact angle \316\270 in radians between 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 and 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 and a single decimal place approximation in degrees.
d) Evaluate the vector 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 which is the vector projection of 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 orthogonal to 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.e) Optional: Make a suggestive hand diagram illustrating the various vectors derived above at 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.solutiona) We introduce the position vectors and the two relevant difference 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QyU+SSJyRzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkPCw+R0YlNiU5JCokRi8iIiMqJEYvIiIkRiVGJUYlISIiLUYkRic=a)QyctSSJyRzYiNiNJInRHRiUiIiItSSVldmFsRyUqcHJvdGVjdGVkRzYkLUklZGlmZkdGKzYkLUYkNiNJInhHRiVGMi9GMkYnRigtRio2JC1GLjYlRjBGMkYyRjM=b)QyctSSJyRzYiNiMiIiJGJy1JJWV2YWxHJSpwcm90ZWN0ZWRHNiQtSSVkaWZmR0YqNiQtRiQ2I0kieEdGJUYxL0YxRidGJy1GKTYkLUYtNiVGL0YxRjFGMg==a) again and b)QyY+SSZzcGVlZEc2Ii1JKXNpbXBsaWZ5R0YlNiQtSSVzcXJ0R0YlNiMtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSSxUeXBlc2V0dGluZ0c2JEYvSShfc3lzbGliR0YlNiQtSSVldmFsR0YvNiQtSSVkaWZmR0YvNiQtSSJyR0YlNiNJInhHRiVGPy9GP0kidEdGJUY2SSlzeW1ib2xpY0dGJSIiIj5JIlRHRiUtRic2JC1JIipHNiRGLy9GMUkvVmVjdG9yQ2FsY3VsdXNHNiVGL0YvL0YxSShTdHVkZW50R0Y0NiRGNiokRiQhIiJGQkZDQyc+SSNUMEc2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRJIlRHRiUvSSJ0R0YlIiIiRi0tSSVzdWJzR0YoNiRGK0kmc3BlZWRHRiVGLS1JJmV2YWxmR0YoNiNJIiVHRiU=c)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d)QyctSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiRJI1QwR0YpLUklZXZhbEdGJjYkLUklZGlmZkdGJjYlLUkickdGKTYjSSJ4R0YpRjhGOC9GOCIiIkY6PkkmQTBwYXJHRiktSSIqRzYkRiYvRihJL1ZlY3RvckNhbGN1bHVzRzYlRiZGJi9GKEkoU3R1ZGVudEdGLDYkSSIlR0YpRi5GOj5JJ0EwcGVycEdGKS1JIitHRj82JEYvLUkiLUdGPzYjRkY=QyUtSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMtSTBkZWxheURvdFByb2R1Y3RHNiRGJi9JK21vZHVsZW5hbWVHRihJLFR5cGVzZXR0aW5nR0YlNiRJJ0EwcGVycEdGKEYxIiIiLUkmZXZhbGZHRiY2I0kiJUdGKA==e) Here is the 3d plot:QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJnBsb3RzR0YlISIiQyQ+SSljdXJ2ZXBsdEc2Ii1JK3NwYWNlY3VydmVHRiU2JS1JInJHRiU2I0kidEdGJS9GLDsiIiEkIiM8ISIiL0kqdGhpY2tuZXNzR0YlIiIjRjI=We scale down the normal so it does not distort the picture: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By rotating the picture so that the cyan vector is directly out of the screen, we look perpendicular to the plane of the first and second derivative vectors, i.e., the acceleration plane, where we see the black vector is the orthogonal projection of the acceleration perpendicular to the first derivative. I set the orientation to do this automatically. Rotate it around to see how the vectors are related to each other and the curve.JSFH