MAT2500-03/04 13S 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 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). You are encouraged to use technology to check all of your hand results.
LSUnUExPVDNERzYsLSUnQ1VSVkVTRzYnNyQ3JSQiIiEhIiIkIiIhISIiJCIiISEiIjclJCIjPyEiIiQiIzUhIiIkIitLTExMTCEjNTclNyUkIiopRzQmZSIhIikkIistV1lEeiEjNSQiK2ExcSV5JCEjNTclJCIjPyEiIiQiIzUhIiIkIitLTExMTCEjNTclJCIqNzJcaCIhIikkIiljYHUhKSEiKSQiK3lFals6ISM1LSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIzUhIiItJSpUSElDS05FU1NHNiMiIiMtJSZTVFlMRUc2IyUsUEFUQ0hOT0dSSURHLSUnQ1VSVkVTRzYlN1Q3JSQiK0IpKWVkdCEjNSQiK0tHTmA4ISM1JCIrWSpvJmY7ISM2NyUkIislZTZUbSghIzUkIitmXllvOSEjNSQiKzFvdXY9ISM2NyUkIis7VlMkKXohIzUkIis3J29MZiIhIzUkIis0VjM/QCEjNjclJCIrRyEqKmZKKSEjNSQiKycqZiopRzwhIzUkIipkXGlSIyEjNTclJCIrKSkpXENtKSEjNSQiK180JmYoPSEjNSQiKz0mKVEzRiEjNjclJCIrYlRMQiEqISM1JCIrIylSXk4/ISM1JCIqOSg9aEkhIzU3JSQiK09KRCpSKiEjNSQiKykpKlsnM0EhIzUkIitAPSUqZk0hIzY3JSQiKyo9TDN6KiEjNSQiK2ouXidSIyEjNSQiK24pUTEiUiEjNjclJCIreEUoKT41ISIqJCIrInBdLmcjISM1JCIrbFYvP1chIzY3JSQiKk5oQjEiISIpJCIrJTRIOiNHISM1JCIraWQhZSpcISM2NyUkIitNLGkxNiEiKiQiKywuX2hJISM1JCIrdm1jWWMhIzY3JSQiK3NGc182ISIqJCIrdlcjPkskISM1JCIqbCc0I1EnISM1NyUkIisjM1kyPyIhIiokIiskKXlaL08hIzUkIitGeVY4cyEjNjclJCIreit4XTchIiokIit5VzE2UiEjNSQiKzcyMmAiKSEjNjclJCIrOiJ5R0kiISIqJCIrYCVHUEMlISM1JCIrVTo1OiMqISM2NyUkIiosZHJOIiEiKSQiK3d5by9ZISM1JCIrMHVhVDUhIzU3JSQiKzFzcDg5ISIqJCIrdyVcaipcISM1JCIrIykzQXg2ISM1NyUkIioiSGZzOSEiKSQiKyVwQzhVJiEjNSQiKyVRbjBMIiEjNTclJCIrYEElUmAiISIqJCIrKjNaQyllISM1JCIrQyEqKVFdIiEjNTclJCIrZHUleWYiISIqJCIrUjd6I1EnISM1JCIrI3ooeSpwIiEjNTclJCIoOldtIiEiJyQiK01LcERwISM1JCIrPVk/QD4hIzU3JSQiKyl6YlB0IiEiKiQiK0BIeDl2ISM1JCIrQE5ZckAhIzU3JSQiKyhRJilmIT0hIiokIitXIWVSOikhIzUkIis3OkthQyEjNTclJCIrOlRBIik9ISIqJCIqVjV2JSkpISIqJCIrKCpcLXVGISM1NyUkIitZdGZmPiEiKiQiKVcwKycqISIpJCIrQE5QTkohIzU3JSQiK21jQlQ/ISIqJCIraDJtVDUhIiokIitIPXpWTiEjNTclJCIpXkZFQCEiKCQiK145RUk2ISIqJCIrKkg2YSslISM1NyUkIitVdCZbQCMhIiokIitpIylSRTchIiokIisiKj47RlghIzU3JSQiKykpKkhySSMhIiokIismPjcyTCIhIiokIisnb3dvNiYhIzU3JSQiKyR5WUtTIyEiKiQiK3UoKSpRVyIhIiokIit0JTNNeSYhIzU3JSQiK3d5Ty5EISIqJCIrIm83bmMiISIqJCIrZk93T2whIzU3JSQiKjRnd2cjISIpJCIrKHlzKipwIiEiKiQiKyYpPkQpUSghIzU3JSQiKyk+KEg7RiEiKiQiKzx3Y1c9ISIqJCIrPGtsXSQpISM1NyUkIis2LVlIRyEiKiQiK3I3WSw/ISIqJCIrW11VUSUqISM1NyUkIipvUHQlSCEiKSQiK14pKnByQCEiKiQiKSkpeW01ISIoNyUkIis7ZzdxSSEiKiQiK1ElPWtOIyEiKiQiK2UtdjA3ISIqNyUkIisxKUghKT4kISIqJCIqbVtvYiMhIikkIitRSSJHTyIhIio3JSQiK25ARUpMISIqJCIrMHBLdUYhIiokIitWXUxTOiEiKjclJCIrKTNYK1okISIqJCIrSEtJNUkhIiokIitBOCk0dSIhIio3JSQiKzgpNFloJCEiKiQiK0Q1Tm1LISIqJCIreVN3bj4hIio3JSQiKzlzPmxQISIqJCIrOXY8V04hIiokIislKXkzQ0EhIio3JSQiKywjZT8jUiEiKiQiK0tealhRISIqJCIrLDMhUV4jISIqNyUkIitPVFgmMyUhIiokIit1UXRzVCEiKiQiK1dARFRHISIqNyUkIipGZGNEJSEiKSQiKVpsRlghIigkIit1eE42SyEiKjclJCIrKD5eSFYlISIqJCIrelN3N1whIiokIilSbkhPISIoNyUkIitDOGo8WSEiKiQiK2goSDFMJiEiKiQiKyRcIVstVCEiKjclJCIrbWArNVshIiokIiswei4leSYhIiokIipjdm9qJSEiKTclJCIrZFFSNV0hIiokIipsNmdGJyEiKSQiKy48KTNDJiEiKjclJCIrJm9JIj5fISIqJCIreTckKTRvISIqJCIra2xjQmYhIio3JSQiK1xPY09hISIqJCIreWcwKlEoISIqJCIrWSp5XnAnISIqLSUmQ09MT1JHNiYlJFJHQkckIiM1ISIiJCIiISEiIiQiIiEhIiItJSpUSElDS05FU1NHNiMiIiMtJiUmX0FYSVNHNiMiIiI2Ji0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUqTElORVNUWUxFRzYjIiIhLSUqVEhJQ0tORVNTRzYjIiIhLSUtVFJBTlNQQVJFTkNZRzYjJCIiISEiIi0mJSZfQVhJU0c2IyIiIzYmLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpMSU5FU1RZTEVHNiMiIiEtJSpUSElDS05FU1NHNiMiIiEtJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSYlJl9BWElTRzYjIiIkNiYtJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIS0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJStBWEVTTEFCRUxTRzYlLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEieDYiLyUnZmFtaWx5R1EhNiIvJSVzaXplR1EjMTA2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EldHJ1ZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0YWxpYzYiLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEieTYiLyUnZmFtaWx5R1EhNiIvJSVzaXplR1EjMTA2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EldHJ1ZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0YWxpYzYiLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEiejYiLyUnZmFtaWx5R1EhNiIvJSVzaXplR1EjMTA2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EldHJ1ZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0YWxpYzYiLSUrTElHSFRNT0RFTEc2I1EoTElHSFRfMzYiLSUrUFJPSkVDVElPTkc2LCQiKUsmPUckISIpJCIpeFEvQCEiKSQiKWJ2MyMqISIpJCEoYlUhKSkhIikkIigwV3gqISIoJCEpXiopPj4hIikkISgxXVMqISIoJCEpYSJwIT0hIiokIih5SVIkISIoJCIjNSEiIi0lKkFYRVNTVFlMRUc2IyUkQk9YRy0lJVJPT1RHNictJSlCT1VORFNfWEc2IyQiJCsiISIiLSUpQk9VTkRTX1lHNiMkIiQrIiEiIi0lLUJPVU5EU19XSURUSEc2IyQiJWdPISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJWdPISIiLSUpQ0hJTERSRU5HNiI=JSFHThe parametrized curve segment 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
is shown in the figure together with 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.
a) Is the curve moving up or down in the figure as t increases? Why? Evaluate and simplify 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, (perfect square! show it simplifies to 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 ) , 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, 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| and their values (including 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) at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=.b) Write the parametrized equations of the tangent line through 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. c) Evaluate and simplify 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 and 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 . Show that 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 simplifies to 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 because of the perfect square. Evaluate 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 .
d) Write the equation of the 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjJGMg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnRjJGMkYyRjI= .f) Evaluate the unit vector 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 in the direction of 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbW92ZXJHRiQ2JS1GIzYoLUkjbWlHRiQ2J1EiQkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNi8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRjE2I1EhRidGNEY3RjlGPC1JI21vR0YkNi5RLiZSaWdodFZlY3RvcjtGJy8lLHBsYWNlaG9sZGVyR0Y2L0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGTC8lKXN0cmV0Y2h5R0Y2LyUqc3ltbWV0cmljR0ZMLyUobGFyZ2VvcEdGTC8lLm1vdmFibGVsaW1pdHNHRkwvJSdhY2NlbnRHRkwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zlbi9GWEY2LUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtbkdGJDYkUSIwRidGSEZIRkhGSA==.g) Evaluate the unit normal 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\303\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.h) Evaluate the scalar tangential projection LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRIlRGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRGQUY9Rj1GPS1GLzYjUSFGJ0Y9 along 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 of the acceleration 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 and its scalar normal projection 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 exactly. Does the sum of their squares equal the square of the magnitude of the acceleration (from part a)) as it should?
i) The center of the osculating circle has position vector: 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. Show that 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.j) Write down an integral formula for the length of the curve 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 from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GV0ZTRjk=to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk=. Evaluate it exactly (multiply out!) and then numerically to 4 decimal places. [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTEY59.8 to 1 decimal place!]solutiona) 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vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the simplest normal to the plane of the plane curve which retains the direction of 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. c) 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= for the parametrized curve and solve forLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YrRi8=PkkjdHpHNiItSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkLy1JIitHNiRGKC9JK21vZHVsZW5hbWVHRiRJL1ZlY3RvckNhbGN1bHVzRzYlRigvRjBJKFN0dWRlbnRHNiRGKUYoRig2JCMiIiIiIiRJInRHRiRGOEY6then evaluate the other two coordinates:LUklc3Vic0clKnByb3RlY3RlZEc2JC9JInRHNiItSSIqRzYkRiQvSSttb2R1bGVuYW1lR0YoSS9WZWN0b3JDYWxjdWx1c0c2JUYkL0YtSShTdHVkZW50RzYkSShfc3lzbGliR0YoRiRGJDYkIiIjIyIiIiIiJC1JJDwsPkdGMzYlLUkiK0dGKzYkRjUtRio2JEY1RictRj02JEY/RjctRj02JEY2Ric=by differentiationQystSSVldmFsRyUqcHJvdGVjdGVkRzYkLUklZGlmZkdGJTYkLUkiVEc2IjYjSSJ4R0YsRi4vRi5JInRHRiwiIiItSSVzcXJ0RzYkRiVJKF9zeXNsaWJHRiw2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYlL0krbW9kdWxlbmFtZUdGLEksVHlwZXNldHRpbmdHRjQ2JEYjRiNGMS1JKXNpbXBsaWZ5R0Y0NiNJIyUlR0YsRjEtRj82Iy1GODYkSSIlR0YsRkZGMS1JIipHNiRGJS9GO0kvVmVjdG9yQ2FsY3VsdXNHNiVGJS9GO0koU3R1ZGVudEc2JEY1RiVGJTYkRkEqJEZGISIiLUkiKkc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJL1ZlY3RvckNhbGN1bHVzRzYlRiUvRidJKFN0dWRlbnRHNiRJKF9zeXNsaWJHRihGJUYlNiQtSSVldmFsR0YlNiQtSSVkaWZmR0YlNiQtSSJUR0YoNiNJInhHRihGOS9GOUkidEdGKCokLUkpc2ltcGxpZnlHNiRGJUYuNiMtSSVzcXJ0R0Y/NiMtSTBkZWxheURvdFByb2R1Y3RHNiRGJS9GJ0ksVHlwZXNldHRpbmdHRj82JEYwRjAhIiI=Unit normal by cross productQyUtSSJCRzYiNiNJInRHRiUiIiItSSJUR0YlRiY=Qyk+SSJORzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkyZGVsYXlDcm9zc1Byb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSSxUeXBlc2V0dGluZ0c2JEYvSShfc3lzbGliR0YlNiQtSSJCR0YlNiM5JC1JIlRHRiVGOEYlRiVGJSEiIi1GJEYnIiIiLUkpc2ltcGxpZnlHRjM2I0kiJUdGJUY+LUYkNiMiIiE=g) Length of curve is integral of speed.QyctSSZzcGVlZEc2IjYjSSJ0R0YlIiIiLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiRGIy9GJzsiIiFGKEYoL0kiTEdGJS1JJmV2YWxmR0YsNiNJIiVHRiU=g)LUklZXZhbEclKnByb3RlY3RlZEc2JC1JJWRpZmZHRiQ2JS1JInJHNiI2I0kieEdGK0YtRi0vRi0iIiE=QyY+SSVhbWFnRzYiLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLUkwZGVsYXlEb3RQcm9kdWN0RzYkRikvSSttb2R1bGVuYW1lR0YlSSxUeXBlc2V0dGluZ0dGKDYkLUklZXZhbEdGKTYkLUklZGlmZkdGKTYlLUkickdGJTYjSSJ4R0YlRjxGPC9GPEkkdGF1R0YlRjMiIiItSSZldmFsZkdGKTYjSSIlR0YlRj8=Note that the tangential acceleration equals the speed for this curve.QyUtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUkwZGVsYXlEb3RQcm9kdWN0RzYkRiYvSSttb2R1bGVuYW1lR0YoSSxUeXBlc2V0dGluZ0dGJTYkLUklZXZhbEdGJjYkLUklZGlmZkdGJjYlLUkickdGKDYjSSJ4R0YoRjpGOi9GOkkidEdGKC1JIlRHRig2I0Y8IiIiLUkmc3BlZWRHRihGPw==QyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUkwZGVsYXlEb3RQcm9kdWN0RzYkRiYvSSttb2R1bGVuYW1lR0YoSSxUeXBlc2V0dGluZ0dGJTYkLUklZXZhbEdGJjYkLUklZGlmZkdGJjYlLUkickdGKDYjSSJ4R0YoRjpGOi9GOkkidEdGKC1JIk5HRig2I0Y8IiIiQyotSSJURzYiNiNJJHRhdUdGJSIiIj5JI2FUR0YlLUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUksVHlwZXNldHRpbmdHNiRGLkkoX3N5c2xpYkdGJTYkLUklZXZhbEdGLjYkLUklZGlmZkdGLjYlLUkickdGJTYjSSJ4R0YlRj5GPi9GPkYnRiNGKC1JMmRlbGF5Q3Jvc3NQcm9kdWN0R0YtNiRGI0Y1Rig+SSNhTkdGJS1GLDYkRjUtSSJOR0YlRiZGKA==JSFHa check: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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.making the diagram and 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 we include the part of the tangent line.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 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: TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzQyODg5NDYyWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJC1JJGV4cEc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IyUidEciIiMtRis2IywkRjBGMSwkLUYrNiMsJEYwIiIkIyIiIkY5RiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzQyODg5ODIyWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJC1JJGV4cEc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IyUidEciIiMsJC1GKzYjLCRGMEYxRjEtRis2IywkRjAiIiRGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzQyODkwMTgyWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJC1JJGV4cEc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IyUidEciIiMsJC1GKzYjLCRGMEYxIiIlLCQtRis2IywkRjAiIiRGO0YlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Nzc1ODU0OTY2WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQiIiMiIiIjRioiIiRGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Nzc1ODQzMDU0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQiIiNGKSIiIkYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Nzc1ODQzNDE0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQiIiMiIiUiIiRGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Nzc1ODEwMTY2WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJCokLCYtSSRleHBHNiQlKnByb3RlY3RlZEclKF9zeXNsaWJHNiMsJCUidEciIiMiIiJGNEY1ISIiRjQsJComRitGNi1GLTYjRjNGNUY0KiZGK0Y2RixGNUYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Nzc1MzgyMTI2WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJComLUkkZXhwRzYkJSpwcm90ZWN0ZWRHJShfc3lzbGliRzYjLCQlInRHIiIjIiIiLUYsNiMsJEYyIiIkRjRGMywkKiYtRiw2I0YyRjRGNUY0ISIlLCQqJkY7RjRGK0Y0IiIlRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0NzYxMDIwNzEwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJC1JJGV4cEc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IywkJSJ0RyIiJiIiIywkLUYrNiMsJEYxIiIlISIlLCQtRis2IywkRjEiIiRGOEYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0NzYxMDIxOTEwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQqJiwmLUkkZXhwRzYkJSpwcm90ZWN0ZWRHJShfc3lzbGliRzYjLCQlInRHIiIjIiIiRjNGNCEiIkYrRjQsJComRipGNS1GLDYjRjJGNCEiIywkKiRGKkY1RjNGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Nzc1ODAzNDE0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJC1JJGV4cEc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IywkJSJ0RyIiJiIiIywkLUYrNiMsJEYxIiIlISIlLCQtRis2IywkRjEiIiRGOEYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2Nzg4ODcwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQhIzciIiciIzdGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2NzgzNDU0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQjISIjIiIkIyIiIkYrIyIiI0YrRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2Nzc2NTgyWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJComLCYtSSRleHBHNiQlKnByb3RlY3RlZEclKF9zeXNsaWJHNiMsJCUidEciIiMiIiJGNEY1ISIiLUYtNiNGM0Y1ISIjLCQqJiwmRixGNUY5RjVGNUYrRjZGNiwkRipGNEYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2Nzc2ODIyWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQjISIjIiIkIyIiIkYrIyIiI0YrRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Nzc1MzU0MDU0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJComLCYtSSRleHBHNiQlKnByb3RlY3RlZEclKF9zeXNsaWJHNiMsJCUidEciIiMiIiJGNEY1ISIjRixGNSEiJSwmKihGK0Y2LUYtNiNGM0Y1RixGNUY3KiZGKyEiIkY6RjVGNCwmKiZGK0Y2RixGNEY2KiZGK0Y9RixGNUY0RiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Nzc1MzU0MTc0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJComLCYtSSRleHBHNiQlKnByb3RlY3RlZEclKF9zeXNsaWJHNiMsJCUidEciIiMiIiJGNEY1ISIjRixGNSEiJSwkKigtRi02I0YzRjUsJkYsRjVGNkY1RjVGK0Y2RjYsJEYqIiIlRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Nzc2NzA0MTc0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJComLCYtSSRleHBHNiQlKnByb3RlY3RlZEclKF9zeXNsaWJHNiMsJCUidEciIiMiIiJGNEY1ISIiLUYtNiNGM0Y1ISIjLCQqJiwmRixGNUY5RjVGNUYrRjZGNiwkRipGNEYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Nzc2NzgzOTE4WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQjISIjIiIkIyIiIkYrIyIiI0YrRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2OTg3MDU0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJComLUkkZXhwRzYkJSpwcm90ZWN0ZWRHJShfc3lzbGliRzYjJSJ0RyIiIiwmLUYsNiMsJEYxIiIjRjJGN0YyRjIhIiIsJi1GLDYjLCRGMSIiJSNGOEY3RjdGMkYqRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2OTg4NDk0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJi1JJGV4cEc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IyUidEciIiMqJkYqIiIiLCYtRis2IywkRjBGMUYzRjFGM0YzISIiLChGNUYzLUYrNiMsJEYwIiIlI0Y4RjFGMUYzLCYtRis2IywkRjAiIiQjRjNGQ0YyRjNGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2OTg4NjE0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJC1JJGV4cEc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IywkJSJ0RyIiJCEiIiwoLUYrNiMsJEYxIiIjIiIiLUYrNiMsJEYxIiIlI0YzRjhGOEY5LCZGKiNGPUYyLUYrNiNGMUY4RiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2OTg5NDU0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQhIiIjIiImIiIjIyIjNSIiJEYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2OTg5NTc0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQiIiMiIiIjRioiIiRGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2Nzg1MTUwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQiIiNGKSIiIkYlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2Nzg1NTEwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQiIiMiIiUiIiRGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2NzgyODU0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQjIiIjIiIkRikjIiIiRitGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2Nzg0MDU0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQiIiMhIiUiIiVGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2Nzc3MDc4WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQiIiMhIiUiIiVGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2Nzc3MTk4WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQiIiMiIiIjRioiIiRGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2Nzc4MTU4WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJiUieEciIiIhIiNGKywmJSJ5R0YrISIiRissJiUiekdGKyNGLyIiJEYrRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2Nzc2NzAyWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQlInhHJSJ5RyUiekdGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2NzY5NzI2WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJiIiIyIiIiUidEdGKiwmRixGKkYrRissJiNGKyIiJEYrRixGK0YlTTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2NzcyNjA2WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQjIiM1IiIkIyIiKEYrIiIiRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2NzY2NTkwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJComLCYtSSRleHBHNiQlKnByb3RlY3RlZEclKF9zeXNsaWJHNiMsJCUidEciIiMiIiJGNEY1ISIjRixGNSEiJSwmKihGK0Y2LUYtNiNGM0Y1RixGNUY3KiZGKyEiIkY6RjVGNCwmKiZGK0Y2RixGNEY2KiZGK0Y9RixGNUY0RiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2NzUyNzQyWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJComLCYtSSRleHBHNiQlKnByb3RlY3RlZEclKF9zeXNsaWJHNiMsJCUidEciIiMiIiJGNEY1ISIjRixGNSEiJSwkKigtRi02I0YzRjUsJkYsRjVGNkY1RjVGK0Y2RjYsJEYqIiIlRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2NzU1MzgyWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQhIiIsJCooLUkkZXhwRzYkJSpwcm90ZWN0ZWRHJShfc3lzbGliRzYjLCQlInRHIiIjRiktRi02I0YzIiIiLCZGLEY3ISIjRjdGNyNGKUY0RjdGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2NzQ4MTUwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJCooKiYsJi1JJGV4cEc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2IywkJSJ0RyIiIyIiIkY1RjYhIiNGLUY2IyEiIkY1RixGN0YtRjZGNywkKiZGK0Y4LCYqKEYsRjctRi42I0Y0RjZGLUY2ISIlKiZGLEY5Rj5GNkY1RjYjRjZGNSwkKiZGK0Y4LCYqJkYsRjdGLUY1RjcqJkYsRjlGLUY2RjVGNkZCRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0MzU2NzQwMDk0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQqJiwmLUkkZXhwRzYkJSpwcm90ZWN0ZWRHJShfc3lzbGliRzYjLCQlInRHIiIjIiIiRjNGNCEiIkYrRjQsJComRipGNS1GLDYjRjJGNCEiIywkKiRGKkY1RjNGJQ==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TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0Nzc1ODMxNjA2WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQiIiMiIiUiIiRGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0NzczODU2OTc0WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQjIiIjIiIkRikjIiIiRitGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc0NzczODY3NTY2WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQjIiIjIiIkIyEiJUYrIyIiJUYrRiU=