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CORNELL MATH 1110 - Review (Chapter 1)

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Review (Chapter 1)Week 2 DiscussionMATH 1110WorksheetMathematical definitions are often technical and use specialized notation; a well-written definition does notallow for ambiguities. Once we understand a well-written definition, we can determine with certainty if aparticular object (or objects) satisfy the given definition. You will practice this today, while reviewing somepre-requisite material for Calculus.Definition: Suppose that functions f and g have the same domain D. We say the function f is equalto the function g, if for all x in D, f(x) = g(x).Definition: A function p is a polynomial ifp(x) = anxn+ an−1xn−1+ ··· + a1x + a0,where n is a nonnegative integer, and the numbers a0, a1, a2, . . . anare real constants, called the coef-ficients of the polynomial. If the leading coefficient an6= 0, then n is the degree of the polynomial.Definition: A rational function is a quotient f(x) =p(x)q(x), where p and q are polynomials.1. Let f be defined by f(x) =x2− 4x − 2, g by g(x) = x + 2, and h by h(u) = u + 2.(a) True or False? The function f is equal to the function g.(b) True or False? The function g is equal to the function h.Created by the Cornell Active Learning in Math team.c 1/5Review (Chapter 1)Week 2 DiscussionMATH 1110Worksheet2. For each of the following below, answer the following questions. Be prepared to rigorously justify youranswer using the formal definitions of even, odd, polynomial, and rational functions.• Is the function even?• Is the function odd?• Is the function a polynomial? If so, what is its degree?• Is the function a rational function?(a) f(x) =√x(b) f(x) =x32+ 1(c) f(x) =x2x5+ x(d) f(x) =8x2+ 1(e) f(x) = 7(f) f (x) = x−5+ 3x3+ πx2/5Review (Chapter 1)Week 2 DiscussionMATH 1110Worksheet3. A piecewise-defined function is a function that is defined differently over several intervals. Thefunction f(x) = |x| is a piecewise-defined function described as follows:|x| =(−x x ≤ 0x x > 0Rewrite each of the following functions as a piecewise-defined function.(a) g(x) = |x − 3|(b) h(x) = |x2− 4|(c) k(x) = |2x −1|+ |2x + 1|4. Let f(x) = cos x and g(x) = sec x =1cos x. Are f and g inverses of one another? Use the definition ofinverse function to justify your answer.3/5Review (Chapter 1)Week 2 DiscussionMATH 1110Worksheet5. Consider the following diagram.1↵<latexit sha1_base64="+wSBPeL8nxBdvzPXA2qswhGhfpg=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oUy2m3btZhN2N0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqKGvSWMSqE6BmgkvWNNwI1kkUwygQrB2Mb2d++4kpzWP5YCYJ8yMcSh5yisZKrR6KZIT9csWtunOQVeLlpAI5Gv3yV28Q0zRi0lCBWnc9NzF+hspwKti01Es1S5COcci6lkqMmPaz+bVTcmaVAQljZUsaMld/T2QYaT2JAtsZoRnpZW8m/ud1UxNe+xmXSWqYpItFYSqIicnsdTLgilEjJpYgVdzeSugIFVJjAyrZELzll1dJq1b1Lqq1+8tK/SaPowgncArn4MEV1OEOGtAECo/wDK/w5sTOi/PufCxaC04+cwx/4Hz+AIzPjxw=</latexit><latexit sha1_base64="EpwadKl79nuFFVBzWqBryCC0A38=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGCaQttKJvtpl262YTdiVBCf4MXD4p49Qd589+4bXPQ1gcDj/dmmJkXplIYdN1vZ219Y3Nru7RT3t3bPzisHB23TJJpxn2WyER3Qmq4FIr7KFDyTqo5jUPJ2+H4bua3n7g2IlGPOEl5ENOhEpFgFK3k90KOtF+pujV3DrJKvIJUoUCzX/nqDRKWxVwhk9SYruemGORUo2CST8u9zPCUsjEd8q6lisbcBPn82Ck5t8qARIm2pZDM1d8TOY2NmcSh7YwpjsyyNxP/87oZRjdBLlSaIVdssSjKJMGEzD4nA6E5QzmxhDIt7K2EjaimDG0+ZRuCt/zyKmnVa95lrf5wVW3cFnGU4BTO4AI8uIYG3EMTfGAg4Ble4c1Rzovz7nwsWtecYuYE/sD5/AHFJI6o</latexit>Ccos <latexit sha1_base64="lXpXIxzJ2EXckop2R19As3Ir36c=">AAAB8nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/YAklM122y7dZMPuRCihP8OLB0W8+mu8+W/ctjlo64OBx3szzMyLUikMuu63s7a+sbm1Xdop7+7tHxxWjo7bRmWa8RZTUuluRA2XIuEtFCh5N9WcxpHknWh8N/M7T1wboZJHnKQ8jOkwEQPBKFrJD5gyeRBxpNNeperW3DnIKvEKUoUCzV7lK+grlsU8QSapMb7nphjmVKNgkk/LQWZ4StmYDrlvaUJjbsJ8fvKUnFulTwZK20qQzNXfEzmNjZnEke2MKY7MsjcT//P8DAc3YS6SNEOesMWiQSYJKjL7n/SF5gzlxBLKtLC3EjaimjK0KZVtCN7yy6ukXa95l7X6w1W1cVvEUYJTOIML8OAaGnAPTWgBAwXP8ApvDjovzrvzsWhdc4qZE/gD5/MHpYORfQ==</latexit>sin <latexit sha1_base64="LpjJuya/XJof2++wfiIkpOtt+F0=">AAAB8nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/YAklM122y7dbMLuRCihP8OLB0W8+mu8+W/ctjlo64OBx3szzMyLUikMuu63s7a+sbm1Xdop7+7tHxxWjo7bJsk04y2WyER3I2q4FIq3UKDk3VRzGkeSd6Lx3czvPHFtRKIecZLyMKZDJQaCUbSSHxih8iDiSKe9StWtuXOQVeIVpAoFmr3KV9BPWBZzhUxSY3zPTTHMqUbBJJ+Wg8zwlLIxHXLfUkVjbsJ8fvKUnFulTwaJtqWQzNXfEzmNjZnEke2MKY7MsjcT//P8DAc3YS5UmiFXbLFokEmCCZn9T/pCc4ZyYgllWthbCRtRTRnalMo2BG/55VXSrte8y1r94arauC3iKMEpnMEFeHANDbiHJrSAQQLP8ApvDjovzrvzsWhdc4qZE/gD5/MHrVSRgg==</latexit>`1<latexit sha1_base64="dG6bzKC2dg3qnAR/DemvCj8B9X4=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2k3btZhN2N0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHssHM0nQj+hQ8pAzaqzU6qEQfa9frrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42v3ZKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjJ7nQy4QmbExBLKFLe3EjaiijJjAyrZELzll1dJq1b1Lqq1+8tK/SaPowgncArn4MEV1OEOGtAEBo/wDK/w5sTOi/PufCxaC04+cwx/4Hz+ADZqjuM=</latexit>`2<latexit sha1_base64="K5liZQWh8pqCmK5MS6PBkZdgT44=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2k3btZhN2N0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHssHM0nQj+hQ8pAzaqzU6qEQ/Vq/XHGr7hxklXg5qUCORr/81RvELI1QGiao1l3PTYyfUWU4Ezgt9VKNCWVjOsSupZJGqP1sfu2UnFllQMJY2ZKGzNXfExmNtJ5Ege2MqBnpZW8m/ud1UxNe+xmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBlSyIXjLL6+SVq3qXVRr95eV+k0eRxFO4BTOwYMrqMMdNKAJDB7hGV7hzYmdF+fd+Vi0Fpx85hj+wPn8ATfujuQ=</latexit>`3<latexit sha1_base64="dDNK0Q5KKnZkmcJNUxOw0l3JsM0=">AAAB7XicbVBNS8NAEJ34WetX1aOXxSJ4Kkkr6LHoxWMF+wFtKJvtpF27yYbdjVBC/4MXD4p49f9489+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpuaZkqhk0mhVSdgGoUPMam4UZgJ1FIo0BgOxjfzvz2EyrNZfxgJgn6ER3GPOSMGiu1eihEv9Yvld2KOwdZJV5OypCj0S999QaSpRHGhgmqdddzE+NnVBnOBE6LvVRjQtmYDrFraUwj1H42v3ZKzq0yIKFUtmJD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzHSWowZotFYSqIkWT2OhlwhcyIiSWUKW5vJWxEFWXGBlS0IXjLL6+SVrXi1SrV+8ty/SaPowCncAYX4MEV1OEOGtAEBo/wDK/w5kjnxXl3Phata04+cwJ/4Hz+ADlyjuU=</latexit>`4<latexit sha1_base64="cOUqkwr9YQLb4qNGJ7TiYi0iPjE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lqQY9FLx4r2FpoQ9lsJ+3azSbsboQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTjm5n/8IRK81jem0mCfkSHkoecUWOldg+F6Nf75Ypbdecgq8TLSQVyNPvlr94gZmmE0jBBte56bmL8jCrDmcBpqZdqTCgb0yF2LZU0Qu1n82un5MwqAxLGypY0ZK7+nshopPUkCmxnRM1IL3sz8T+vm5rwys+4TFKDki0WhakgJiaz18mAK2RGTCyhTHF7K2EjqigzNqCSDcFbfnmVtGtV76Jau6tXGtd5HEU4gVM4Bw8uoQG30IQWMHiEZ3iFNyd2Xpx352PRWnDymWP4A+fzBzr2juY=</latexit>`5<latexit


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