pa�/���Y\L��{3�|c���1�|��X�!�e�:�i#��.S���8�H�>n-� �Im�^*. f()xydfdyd(f()x)Dfx() dxdxdx ¢¢===== If y= fx( )all of the following are equivalent notations for derivative evaluated at xa= . �)N< |1�xL����= ����|��.י�{�k+����t���w"�� 3 Fundamental Theorem of Calculus Part I : If fx( ) is continuous on [ab,] then () x() a gx= ò ftdt is also continuous on [ab,] and () () x a d gxftdtfx dx ¢ ==ò. Integration by Parts The standard formulas for integration … *��JD�yw� mGl?��`�V��ۏRVI�&���<�ӞD�`离��\$�\$� Ya���C�2��-�cp���G��0��"2��Go�=�J���_g� ����ʦ�ŀȖ�G4P�pV�(J\������Їr����40�4�U�?|��f7��5c���� ^����,7ѷ�F�Mq��fcsX_��yF����+�֨��[/��Y2�̝g-()����6��``+2)�c��V�2Eem};[a�nft����pf��/��n�����H�)?e>���ʨ\$�-u#���%;�VБm�W�4�O{�ƽf[�D��� ����8-��˅�]Q*&�;|��XgI��ψO�r,J ��L}�r,��4|������`���ZKJ�>�`��M+�! Linearity in diﬀerentiation 7 3. Anti-Derivative : An anti-derivative of f x( ) is a function, Fx( ), such that F x f x′( )= ( ). u Substitution Given (())() b a ò fgxg¢ xdx then the substitution u= gx( ) will convert this into the integral, (())() () bgb( ) aga òòfgxg¢ xdx= fudu . Integrals Definitions Definite Integral: Suppose f x( ) is continuous on [ab,]. +��.B����4�)����=l0�7M�D�_�(��vA'T�;����,�|���G�H���t�7U The product rule for diﬀerentiation 10 5. Then ( ) (*) 1 lim i b n a n i f x dx f x x →∞ = ∫ =∑ ∆. Standard Integration Techniques Note that all but the first one of these tend to be taught in a Calculus II class. Integrals Definitions Definite Integral: Suppose fx( ) is continuous on [ab,]. x��}K��Ǒ����Z�k��|?f���R�3�`_Sa��EɆ���s"��2)���0���}�DF>"2"2��K��K���������~��秿1]ۥ�.��~-��]��� /����x�����M,��S��\�~�||�)��x��5�~�k�����_�r��e �Z�%Ϥ/z��Z����?c��=�R�5�I����9�i\����i=��ˇ���.懧�А��X��{���r���_���A#��ݿ�k()�?�m��@p�B��q��A�r�%�kfc��� ��rI��+@{o�b^���6#Z�K�TIoP;��\��s`Pf��fj�:I�AK��v�I�ڡ�o�~���x�\�l��? The nth Derivative is denoted as n n n df fx dx and is defined as fx f x nn 1 , i.e. ]Fc��+�i�n's��9悖�ܛys��0b�-HAa�(X3)�y� ��p�A�����[iTm�۹m�i�I�-N\%�Ӿ,�br�tO��J�?W 3 Fundamental Theorem of Calculus Part I : If fx( ) is continuous on [ab,] then () x() a gx= ò ftdt is also continuous on [ab,] and () () x a d gxftdtfx dx ¢ ==ò. Anti-Derivative : An anti-derivative of fx( ) is a function, Fx( ), such that F¢(x) = fx( ). ! 322 Fundamental Theorem of Calculus Part I : If f ()x is continuous on [ab,] then () ()x a g x =∫ f tdt is also continuous on [ab,] and () () x a d g xftdtfx dx ′ ==∫. The chain rule for diﬀerentiation 13 7. �y�ٙH��꤈ ����ä ����%N���n@�ψZ���{�U�;H�=. %�쏢 Diﬀerentiation of functions deﬁned parametrically 16 9. Integral Calculus Formula Sheet Derivative Rules: 0 d c dx nn 1 d xnx dx sin cos d x x dx sec sec tan d x xx dx tan sec2 d x x dx cos sin d x x dx csc csc cot d x xx dx cot csc2 d x x dx d aaaxxln dx d eex x dx dd cf x c f x dx dx ddd f x gx f x gx dx dx dx fg f g f g 2 f fg fg gg d fgx f gx g x dx Properties of Integrals: ��)7�"��\$AɁvpqE�sv��"Q�rZJz3O˞Ni||��P�:��VC�EWZ����nn���(1� @K��G�n>?YMh��6�6������UA,���,� hU��>�\�R�[�V9�{������g̝g-(9�8�-RW=�T���3e�3Vء&�j�NLZ O�t� (Ԡ� integral, (())() () bgb( ) aga òòfgxg¢ xdx= fudu . the derivative of %� ��A��i������+�r�!3If�-��R>� J,���o����Nc[�왶��d쯓4k�G3�^QD.�0덖��G0���\T���{�OG̈e_89���yw�Z~Y/�������G-GR,�^��f1�#r��9 (O�q�;�c1�- �qDfۂ�4���f�h���y�bç����Z�r�� Integration by Parts The standard formulas for integration by parts are, bbb aaa òudv=uv-vduòòudv=-uvvdu Choose u and dv and then compute du by differentiating u and compute v by using the fact that v= òdv. The quotient rule for diﬀerentiation 11 6. Divide [ab,] into n subintervals of width ∆x and choose * x i from each interval. the derivative of the first derivative, fx . stream Derivatives of basic functions 5 2. *�T;��R��e�Qx��ASR���o��,��s���&���\$������1CQgb;#N�р�C��?M]�L��:;��B�I�"�}�Ao5�hB ��;d��q�~�-V�;�4߇�64���&\$�-� �����V��?��[�R�nqy��_X\$��u`F|�F�}�u���&;R�;DX4Ʊ�VL?��e����\$�.�iHdۗosv�@S�S��'�_�?',�����%в! [u�y��>���A��B�k��8����s���#ɦd(�M�9�{���+r��dP��0��50N*b ��S���0J�EϢ� ��2Gzj�L��YSF�݄�����th z���)A�c P=�h���_��|q�d|��#H.�D���`�X��� 0S��ǜ}H�;��7f+��!�����]ujds�P��%@sp,/^��f��XW��Xx��L�&pa�j}�_ As���Y���V�����m��9����A����ċ��K��o�TOup������\Ho��4Cךy��|�`h��%��a�C+`��s�t잇c��7����r�T}҇2*�2�s ��+ɿ�ۂִ. Anti-Derivative : An anti-derivative of fx( ) is a function, Fx( ), such that F¢(x) = fx( ). ��E*��� ���� Tables of derivatives and integrals 4 1. Divide [ab,] into n subintervals of width D x and choose * xi from each interval. Diﬀerentiation of functions deﬁned implicitly 15 8. ~i�|=�f����|�lT���K��.�ot����|5� �#M�з-��`R��g��6�]`�Q;5���6-�Vy���M�8 G>��Wru]��:_=��04V�:W���:KJ�����K5xzp�rh�E�A�Q�k���_�uX;:O�܉��^~���ij3Z+>d�Җ��"��a�U`�#1"��� Indefinite Integral :∫f (xdx F x c) =+( ) where F ()x is an anti-derivative of f (x). Then () * 1 lim i b a n i f x dx f x x fi¥ = ¥ = D ° ±. Indefinite Integral :ò f (x )d =+Fxc where Fx ( )is ant -der vative of fx. Higher Order Derivatives The Second Derivative is denoted as 2 2 2 df fx f x dx and is defined as fx fx , i.e. �#o�B�\$�+���1m�-�X����(�Y8��ա�Y� L�BA�������P �q���KjWe�T��f�Ũ�:ͽjt&-Gy�v�i�u�)j9Je��%�d.��Ld�st���ٲ�v�Z\$������o�V�ra�ϩ(Wś�G,�ZZ���X�qC�;�:�/�-5���F�5���(Z�rݬ/Y�a��ʘZt��Fǌ%�_�1��Q� �b@`jh���K�4��7G��2U�����/ee=>e{� �w�� ��˶�t��\�r��!�KٗO�uj�1㠧��R\$2_k��Say��"j-_�A�>�x0�l6u���Bi:kQ�V괞���!fK�y��Y���g����9h=�����Ǖ3v 9P�4S��#`�

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