ACS 201 — Differential Calculus for Actuarial Sciences: Lecture Note

About this lecture note

This is the complete lecture note for ACS 201 Differential Calculus for Actuarial Sciences. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.

Preview: opening of the note

1&3*9ࣆaࣆ431.3*1*&73.3,;.)*451&9+472+4754898*(43)&7> -.,-*7.389.9:9.4389:)*398

1&3*9ࣆaࣆ431.3*1*&73.3,;.)*451&9+472+4754898*(43)&7> -.,-*7.389.9:9.4389:)*398$&6'LIIHUHQWLDO&DOFXOXVIRU$FWXDULDO6FLHQFH 8QLWV&/+ &RXUVH&RGH$&6&RXUVH7LWOH'LIIHUHQWLDO&DOFXOXVIRU$FWXDULDO6FLHQFH&RXUVH8QLW8QLWV6HULHV5HODWLRQV)XQFWLRQVDQG/LPLWV6HULHV'LIIHUHQWLDWLRQ7KHRUHPVDQG7HFKQLTXHV6HULHV$SSOLFDWLRQVRI'LIIHUHQWLDWLRQ6HULHV3RO\QRPLDO$SSUR[LPDWLRQVDQG6HULHV([SDQVLRQV6HULHV&RQWLQXLW\6HTXHQFHVDQG/ߤ+RVSLWDOߤV5XOH6HULHV7JQFYNTSX+ZSHYNTSXFSI1NRNYXx3DUW5HODWLRQVDQG)XQFWLRQVR'HILQLWLRQRIUHODWLRQVR'RPDLQUDQJHDQGFRGRPDLQR,QYHUVHUHODWLRQVDQGIXQFWLRQVx3DUW/LPLWVRI)XQFWLRQVR&RQFHSWRIOLPLWVR7HFKQLTXHVIRUHYDOXDWLQJOLPLWVR([SDQVLRQPHWKRGIRUOLPLWVx3DUW&RQWLQXLW\RI)XQFWLRQVR'HILQLWLRQRIFRQWLQXLW\R7\SHVRIGLVFRQWLQXLWLHVR5HODWLRQVKLSEHWZHHQFRQWLQXLW\DQGGLIIHUHQWLDELOLW\

1&3*9ࣆaࣆ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x6/2GHILQHUHODWLRQVDQGH[SODLQWKHFRQFHSWVRIGRPDLQUDQJHDQGFRGRPDLQx6/2GHVFULEHWKHSURSHUWLHVRIIXQFWLRQVDQGGHPRQVWUDWHKRZWRGHWHUPLQHLQYHUVHIXQFWLRQVx6/2H[SODLQWKHFRQFHSWRIOLPLWVDQGDSSO\WHFKQLTXHVIRUHYDOXDWLQJOLPLWVRIIXQFWLRQVx6/2GHPRQVWUDWHKRZWRXVHWKHH[SDQVLRQPHWKRGWRHYDOXDWHOLPLWVx6/2GHILQHFRQWLQXLW\RIIXQFWLRQVDQGLGHQWLI\GLIIHUHQWW\SHVRIGLVFRQWLQXLWLHVDQGx6/2DQDO\VHWKHUHODWLRQVKLSEHWZHHQFRQWLQXLW\DQGGLIIHUHQWLDELOLW\7JQFYNTSXFSI+ZSHYNTSX)JKNSNYNTSTKWJQFYNTSX$UHODWLRQLQPDWKHPDWLFVLVDUXOHWKDWFRQQHFWVHOHPHQWVRIRQHVHWZLWKHOHPHQWVRIDQRWKHUVHW,IZHKDYHWZRVHWVVD\$DQG%DUHODWLRQLVDQ\VXEVHWRIWKH&DUWHVLDQSURGXFW$™%7KLVPHDQVWKDWDUHODWLRQSDLUVHOHPHQWVIURPVHW$ZLWKHOHPHQWVIURPVHW%5HODWLRQVDUHLPSRUWDQWEHFDXVHWKH\IRUPWKHIRXQGDWLRQIRUGHILQLQJIXQFWLRQVZKLFKDUHVSHFLDOW\SHVRIUHODWLRQV

1&3*9ࣆaࣆ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x'RPDLQ6HWRIDOOSRVVLEOHLQSXWVx&RGRPDLQ6HWRIDOOSRVVLEOHRXWSXWVx5DQJH6HWRIDFWXDORXWSXWVSURGXFHG$,SURPSWVXJJHVWLRQߧ*HQHUDWHDWH[WER[OLVWLQJGRPDLQFRGRPDLQDQGUDQJHZLWKGHILQLWLRQVߨ6HDUFKVWULQJߧGRPDLQUDQJHFRGRPDLQIXQFWLRQV2(5ߨ

1&3*9ࣆaࣆ431.3*1*&73.3,;.)*451&9+472+4754898*(43)&7> -.,-*7.389.9:9.4389:)*398.S[JWXJWJQFYNTSXFSIKZSHYNTSX$QLQYHUVHUHODWLRQUHYHUVHVWKHGLUHFWLRQRIDJLYHQUHODWLRQ,IDUHODWLRQPDSVD!$WRE!%WKHQWKHLQYHUVHUHODWLRQPDSVEEDFNWRD)RUIXQFWLRQVWKHLQYHUVHIXQFWLRQH[LVWVRQO\LIWKHRULJLQDOIXQFWLRQLVRQHWRRQH ELMHFWLYH ,QWKDWFDVHWKHLQYHUVHIXQFWLRQXQGRHVWKHDFWLRQRIWKHRULJLQDOIXQFWLRQ)RUH[DPSOHLII [ [WKHQLWVLQYHUVHLVIࡁ \ \ࡁ)LOOLQWKHEODQN$QLQYHUVHIXQFWLRQH[LVWVRQO\LIWKHRULJLQDOIXQFWLRQLV???B)LJXUH,QYHUVHIXQFWLRQPDSSLQJ5NQTYVZJXYNTSX'HILQHDUHODWLRQLQPDWKHPDWLFV)LOOLQWKHEODQN$UHODWLRQLVDQ\VXEVHWRIWKH???BSURGXFWRIWZRVHWV6WDWHWKHGLIIHUHQFHEHWZHHQFRGRPDLQDQGUDQJH)LOOLQWKHEODQN7KHVHWRIDOODFWXDORXWSXWVRIDIXQFWLRQLVFDOOHGWKH???B([SODLQZKHQDQLQYHUVHIXQFWLRQH[LVWV)LOOLQWKHEODQN$QLQYHUVHIXQFWLRQH[LVWVRQO\LIWKHRULJLQDOIXQFWLRQLV???B

1&3*9ࣆaࣆ431.3*1*&73.3,;.)*451&9+472+4754898*(43)&7> -.,-*7.389.9:9.4389:)*3981NRNYXTK+ZSHYNTSX(TSHJUYTKQNRNYX$OLPLWGHVFULEHVWKHEHKDYLRXURIDIXQFWLRQDVLWVLQSXWDSSURDFKHVDSDUWLFXODUYDOXH,WKHOSVXVXQGHUVWDQGKRZIXQFWLRQVEHKDYHQHDUSRLQWVRILQWHUHVWHYHQZKHQWKHIXQFWLRQLVQRWH[SOLFLWO\GHILQHGDWWKRVHSRLQWV)RUH[DPSOHWKHOLPLWRII [ [ࡁ[ࡁ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

1&3*9ࣆaࣆ431.3*1*&73.3,;.)*451&9+472+4754898*(43)&7> -.,-*7.389.9:9.4389:)*398%R[7HFKQLTXHVIRUHYDOXDWLQJOLPLWVx'LUHFWVXEVWLWXWLRQx)DFWRULVDWLRQx5DWLRQDOLVDWLRQx([SDQVLRQPHWKRGV FRYHUHGLQWKHQH[WVXE3DUW $,SURPSWVXJJHVWLRQߧ*HQHUDWHDWH[WER[OLVWLQJWHFKQLTXHVIRUHYDOXDWLQJOLPLWVߨ6HDUFKVWULQJߧWHFKQLTXHVIRUHYDOXDWLQJOLPLWV2(5ߨ*]UFSXNTSRJYMTIKTWQNRNYX7KHH[SDQVLRQPHWKRGLQYROYHVUHZULWLQJIXQFWLRQVXVLQJVHULHVH[SDQVLRQVWRVLPSOLI\WKHHYDOXDWLRQRIOLPLWV)RUH[DPSOHWKHH[SDQVLRQRIVLQ[QHDU]HURLV[ࡁ[ 8VLQJWKLVH[SDQVLRQWKHOLPLWRIVLQ[[DV[DSSURDFKHVFDQEHVKRZQWREH7KLVPHWKRGLVSDUWLFXODUO\XVHIXOZKHQGHDOLQJZLWKWULJRQRPHWULFDQGH[SRQHQWLDOIXQFWLRQVZKHUHGLUHFWVXEVWLWXWLRQRUIDFWRULVDWLRQPD\QRWZRUN)LOOLQWKHEODQN7KHH[SDQVLRQRIVLQ[QHDU]HURLV[ࡁ[ ZKLFKKHOSVVKRZWKDWOLP[࠸VLQ[[ BBB)LJXUH([SDQVLRQPHWKRGIRUHYDOXDWLQJOLPLWV

1&3*9ࣆaࣆ431.3*1*&73.3,;.)*451&9+472+4754898*(43)&7> -.,-*7.389.9:9.4389:)*3985NQTYVZJXYNTSX'HILQHWKHFRQFHSWRIDOLPLW)LOOLQWKHEODQN$OLPLWGHVFULEHVWKHEHKDYLRXURIDIXQFWLRQDVLWVLQSXW???BDSDUWLFXODUYDOXH6WDWHWZRWHFKQLTXHVIRUHYDOXDWLQJOLPLWV)LOOLQWKHEODQN:KHQGLUHFWVXEVWLWXWLRQOHDGVWRLQGHWHUPLQDWHIRUPVRWKHU???BPXVWEHDSSOLHGWRUHVROYHWKHOLPLW([SODLQKRZWKHH[SDQVLRQPHWKRGLVXVHGWRHYDOXDWHOLPLWV)LOOLQWKHEODQN7KHH[SDQVLRQRIVLQ[QHDU]HURLV[ࡁ[ ZKLFKKHOSVVKRZWKDWOLP[࠸VLQ[[ BBB(TSYNSZNY^TK+ZSHYNTSX)JKNSNYNTSTKHTSYNSZNY^$IXQFWLRQLVVDLGWREHFRQWLQXRXVDWDSRLQWLIWKHOLPLWRIWKHIXQFWLRQDVWKHLQSXWDSSURDFKHVWKDWSRLQWLVHTXDOWRWKHYDOXHRIWKHIXQFWLRQDWWKDWSRLQW,QRWKHUZRUGVWKHUHLVQRVXGGHQMXPSRUEUHDNLQWKHJUDSKRIWKHIXQFWLRQ)RUH[DPSOHWKHIXQFWLRQI [ [LVFRQWLQXRXVHYHU\ZKHUHEHFDXVHLWVJUDSKLVVPRRWKZLWKRXWLQWHUUXSWLRQV)LOOLQWKHEODQN$IXQFWLRQLVFRQWLQXRXVDWDSRLQWLIWKHOLPLWRIWKHIXQFWLRQDVWKHLQSXWDSSURDFKHVWKDWSRLQWLVHTXDOWRWKH???BRIWKHIXQFWLRQDWWKDWSRLQW)LJXUH&RQWLQXLW\RIDIXQFWLRQ

This is the opening of the note. Read the full note or download the PDF to continue.

Video lectures for this course

• Differential Calculus for Actuarial Science
• 1.2.3 Expansion Method for Limits
• Differentiation Theorems and Techniques
• Part 2.2: Trigonometric, Exponential and Logarithmic Functions
• Applications of Differentiation
• 3.2.3 Second Derivative Test
• Polynomial Approximations and Series Expansions
• Part 4.4: Interval of Convergence
• Continuity, Sequences, and L'Hospital's Rule
• 5.2.2 Convergence and Divergence of Sequences