Math 489/889Stochastic Processes andAdvanced Mathematical FinanceHomework 6Steve DunbarDue Monday, October 13, 2010
<Text-field style="Heading 1" layout="Heading 1">Problem 1</Text-field>PkkpRHVyYXRpb25HNiIsJiomSSNUMEdGJCIiIiwmSSJxR0YkRihJInBHRiQhIiJGLEYoKipJImFHRiRGKEYpRiwsJkYoRigpKiZGKkYoRitGLEYnRixGKCwmRihGKClGMUYuRixGLEYsprint(); # input placeholderJSFH
<Text-field style="Heading 2" layout="Heading 2">Part a</Text-field>For $ T_0 = 10 $ and $ a = 20 $ , draw a graph of theduration as a function of the probability $ q $ . LUklcGxvdEc2IjYkLUklc3Vic0clKnByb3RlY3RlZEc2JDwlL0kjVDBHRiQiIzUvSSJhR0YkIiM/L0kicEdGJCwmIiIiRjRJInFHRiQhIiJJKUR1cmF0aW9uR0YkL0Y1OyIiIUY0%;JSFHJSFH
<Text-field style="Heading 2" layout="Heading 2">Part b</Text-field>For $ a = 20 $ and $ q = 0.55 $ draw a graph of the expected duration as a function of $ T_0 $ . LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUklc3Vic0dGJTYkPCUvSSJhR0YnIiM/L0kicEdGJyQiI1ghIiMvSSJxR0YnJCIjYkY0SSlEdXJhdGlvbkdGJy9JI1QwR0YnOyIiIUYv%;JSFHJSFH
<Text-field style="Heading 2" layout="Heading 2">Part c</Text-field>For $ a = 20 $ and $ q = 0.45 $ draw a graph of the expected duration as a function of $ T_0 $ . LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUklc3Vic0dGJTYkPCUvSSJhR0YnIiM/L0kicEdGJyQiI2IhIiMvSSJxR0YnJCIjWEY0SSlEdXJhdGlvbkdGJy9JI1QwR0YnOyIiIUYv%;JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Problem 2</Text-field>The boundary condition at state 26 says that the duration of the "game" from 26 down to 18 is the sum of hte duration of the subsequent games from 26 down to 25, and then from 25 to 18. We can compute the duration of the first sub-game using Corollary 1. Then the set of 9 first-step equations in 9 unknowns is:QzQ+SSZlcW4yNkc2Ii9JJEQyNkdGJSwmKiQtSSIrRyUqcHJvdGVjdGVkRzYkIyIiIyIiJCMhIiJGMEYyIiIiSSREMjVHRiVGM0YzPkkmZXFuMjVHRiUvRjQsKEYnI0YzRjBJJEQyNEdGJUYuRjNGM0YzPkkmZXFuMjRHRiUvRjosKEY0RjlJJEQyM0dGJUYuRjNGM0YzPkkmZXFuMjNHRiUvRj8sKEY6RjlJJEQyMkdGJUYuRjNGM0YzPkkmZXFuMjJHRiUvRkQsKEY/RjlJJEQyMUdGJUYuRjNGM0YzPkkmZXFuMjFHRiUvRkksKEZEIyIiKiIjNUkkRDIwR0YlI0YzRlBGM0YzRjM+SSZlcW4yMEdGJS9GUSwoRkkjRjAiIiVJJEQxOUdGJSNGM0ZYRjNGM0YzPkkmZXFuMTlHRiUvRlksKEZRRldJJEQxOEdGJUZaRjNGM0YzPkkmZXFuMThHRiUvRmluIiIhRjM=print(); # input placeholderQyQ+SSpkdXJhdGlvbnNHNiItSSZzb2x2ZUdGJTYkPCtJJmVxbjE4R0YlSSZlcW4xOUdGJUkmZXFuMjBHRiVJJmVxbjIxR0YlSSZlcW4yMkdGJUkmZXFuMjNHRiVJJmVxbjI0R0YlSSZlcW4yNUdGJUkmZXFuMjZHRiU8K0kkRDE4R0YlSSREMTlHRiVJJEQyMEdGJUkkRDIxR0YlSSREMjJHRiVJJEQyM0dGJUkkRDI0R0YlSSREMjVHRiVJJEQyNkdGJSIiIg==print(); # input placeholderQyQtSSVzdWJzRyUqcHJvdGVjdGVkRzYkSSIlRzYiSSREMjVHRigiIiI=print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Problem 3</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Insert the trial 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into the difference equation. For s < kLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEiLEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYtUSJ+RidGL0YyL0Y2RjRGOEY6RjxGPkZARkIvRkZGREYvthe substitution yields 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so the equation is satisfied by the trial particular solution.For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlMtRiw2JVEia0YnRi9GMkY5 the difference equation becomes 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 and again the equation is satisfied identically.For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1GLDYlUSJrRidGL0YyRjk=, the difference equation becomes 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 Therefore, the particular solution is 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= 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.
<Text-field style="Heading 1" layout="Heading 1">Problem 4</Text-field>PkkvRXhwZWN0ZWRWaXNpdHNHNiItSSIqRyUqcHJvdGVjdGVkRzYkIiIjNyMsJiooSSJzR0YkIiIiSSJTR0YkISIiLUkkc3VtR0YkNiQqJkkia0dGJEYuLCZGL0YuRjVGMEYuL0Y1O0YuLCZGL0YuRjBGLkYuRi4tRjI2JComRjVGLiwmRi1GLkY1RjBGLi9GNTtGLiwmRi1GLkYwRi5GMA==print(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyc2ltcGxpZnl+c3ltYm9saWNGJ0YvRjJGQw==LUkkbWFwRyUqcHJvdGVjdGVkRzYlSSlzaW1wbGlmeUc2JEYkSShfc3lzbGliRzYiNyMsKCooSSJzR0YpIiIiSSJTR0YpISIiLCYqJEYvIiIkI0YuIiInRi8jRjBGNUYuIiIjKiRGLUYzI0YwRjNGLSNGLkYzLkkpc3ltYm9saWNHRik=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=PkksU3VtSW50ZWdlcnNHNiItSSRzdW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JEkia0dGJC9GKzsiIiJJIk5HRiQ=print(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyc2ltcGxpZnl+c3ltYm9saWNGJ0YvRjJGQw==LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwoKiQsJkkiTkdGJyIiIkYtRi0iIiMjRi1GLkYsIyEiIkYuRjBGLS5JKXN5bWJvbGljR0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=PkkrU3VtU3F1YXJlc0c2Ii1JJHN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkKiRJImtHRiQiIiMvRiw7IiIiSSJOR0Ykprint(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyc2ltcGxpZnl+c3ltYm9saWNGJ0YvRjJGQw==LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwqKiQsJkkiTkdGJyIiIkYtRi0iIiQjRi1GLiokRisiIiMjISIiRjFGLCNGLSIiJ0Y0Ri0uSSlzeW1ib2xpY0dGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHPkktU3VtUXVhZHJhdGljRzYiLUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQqJkkia0dGJCIiIiwmSSJNR0YkRi1GLCEiIkYtL0YsO0YtSSJOR0Ykprint(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyc2ltcGxpZnl+c3ltYm9saWNGJ0YvRjJGQw==LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwuKiZJIk1HRiciIiIsJkkiTkdGJ0YsRixGLCIiIyNGLEYvKiZGK0YsRi1GLCMhIiJGLyokRi0iIiQjRjNGNSokRi1GL0YwRi4jRjMiIidGOEYsLkkpc3ltYm9saWNHRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=PkkyQ29tbmluYXRpb25Qb3dlcnNHNiIsJiomSSJNR0YkIiIiSSxTdW1JbnRlZ2Vyc0dGJEYoRihJK1N1bVNxdWFyZXNHRiQhIiI=print(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyc2ltcGxpZnl+c3ltYm9saWNGJ0YvRjJGQw==LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwsKiZJIk1HRiciIiIsKCokLCZJIk5HRidGLEYsRiwiIiMjRixGMUYwIyEiIkYxRjNGLEYsRiwqJEYvIiIkI0Y0RjZGLkYyRjAjRjQiIidGOEYsLkkpc3ltYm9saWNHRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Problem 5</Text-field>PkkiQ0c2IioqLCZJIktHRiQiIiIqKkkickdGJEYoSSJTR0YkIiIkSSJ4R0YkRigsJkYoRigqJEYtIiIjISIiRigjRihGLEYoRishIiNGLUYxLCZGKEYoRi1GMUYxprint(); # input placeholderQyQtJkkmcGxvdHNHNiI2I0ksaW50ZXJhY3RpdmVHRiZGJiIiIg==JSFH
<Text-field style="Heading 2" layout="Heading 2">Part a</Text-field>QyQ+SSVkQ2R4RzYiLUklZGlmZkclKnByb3RlY3RlZEc2JEkiQ0dGJUkieEdGJSIiIg==print(); # input placeholderJSFHJSFH
<Text-field style="Heading 2" layout="Heading 2">Part b</Text-field>QyQ+SSVkQ2RTRzYiLUklZGlmZkclKnByb3RlY3RlZEc2JEkiQ0dGJUkiU0dGJSIiIg==print(); # input placeholderJSFH
<Text-field style="Heading 2" layout="Heading 2">Part c</Text-field>QyQ+SSdvcHRpbWFHNiItSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkPCQvSSVkQ2RTR0YlIiIhL0klZENkeEdGJUYvPCRJIlNHRiVJInhHRiUiIiI=print(); # input placeholderQyQtSSphbGx2YWx1ZXNHNiI2I0knb3B0aW1hR0YlIiIiprint(); # input placeholderInspect carefully, one solution set with is real, the other terms have complex factors 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiUtSSNtb0dGJDYtUSomdW1pbnVzMDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiRRIjFGJ0Y3RjdGNy1GIzYlLUkmbWZyYWNHRiQ2KEZOLUYjNiUtRk82JFEiM0YnRjcvJSdpdGFsaWNHUSV0cnVlRicvRjhRJ2l0YWxpY0YnLyUubGluZXRoaWNrbmVzc0dGUS8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zfby8lKWJldmVsbGVkR0Y8RmZuRmluLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y3.We can also solve the partial derivative eqations "by hand".Set the derivative LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSVkQ2RTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo= equal to 0, then clear denominators.LyoqSSJyRzYiIiIiSSJTR0YlIiIkSSJ4R0YlRiYsJkYmRiYqJEYpIiIjISIiRiYtSSIqRyUqcHJvdGVjdGVkRzYkRiwsJkkiS0dGJUYmRiMjRiZGKA==print(); # input 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JSFHSubstitute this into the derivative with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.QyQtSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kiS0c2IiwkKipJInJHRikiIiJJIlNHRikiIiRJInhHRilGLSwmRi1GLSokRjAiIiMhIiJGLSNGLSIiJ0klZENkeEdGKUYtprint(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyc2ltcGxpZnl+c3ltYm9saWNGJ0YvRjJGQw==LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwoKiosJiooSSJyR0YnIiIiSSJTR0YnIiIkLCZGLkYuKiRJInhHRiciIiMhIiJGLiNGLkYwKihGLUYuRi9GMEYzRjQjISIjRjBGLkYvRjlGM0Y1LCZGLkYuRjNGNUY1Ri4qLEYtRi5GL0YuRjNGNUYxRi5GOkY1I0Y1RjQqKkYtRi5GL0YuRjFGLkY6RjkjRi5GNC5JKXN5bWJvbGljR0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Now this derivative is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv when 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. Note that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjBGJ0Y5Rjk= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjFGJ0Y5Rjk= are not realistic values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, that is, the values LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjBGJ0Y5Rjk= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjFGJ0Y5Rjk= are not in the domain so the denominator is never LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv in the domain. Likewise, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= are not LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv. Then substitute 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 into the expression for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=LUklc3Vic0clKnByb3RlY3RlZEc2JC9JInhHNiIjIiIiIiIkL0kiS0dGKCwkKipJInJHRihGKkkiU0dGKEYrRidGKiwmRipGKiokRiciIiMhIiJGKiNGKiIiJw==print(); # input 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LUkoaXNvbGF0ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkL0kiS0dGJywkKiZJInJHRiciIiJJIlNHRiciIiQjIiIlIiMiKUYvJSFHQyQtSSphbGx2YWx1ZXNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiJUdGKCIiIg==print(); # input placeholderJSFHJSFHJSFH