calculus cheat sheet derivatives and integrals pdf

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! 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"�� If a function f is differentiable in an interval I, i.e., its derivative f ′exists at each point of I, then a natural question arises that given f ′at each point of I, can we determine the function? Indefinite Integral :ò f (x )d =+Fxc where Fx ( )is ant -der vative of fx. +��.B����4�)����=l0�7M�D�_�(��vA'T�;����,�|���G�H���t�7U *���q ���9 u Substitution Given (())() b a ò fgxg¢ xdx then the substitution u= gx( ) will convert this into the integral, (())() () bgb( ) aga òòfgxg¢ xdx= fudu . ��E*��� ���� Integrals Definitions Definite Integral: Suppose fx( ) is continuous on [ab,]. Then ( ) (*) 1 lim i b n a n i f x dx f x x →∞ = ∫ =∑ ∆. The quotient rule for diﬀerentiation 11 6. ^�S�w�4Q�����F��b T`�\$��V.jɘv:ج�(; �%�m���ۡ��j8ӥi�a�x�� �B��y�M���T��S��NT�*� Calculus Cheat Sheet Integrals Definitions Definite Integral: Suppose f x is continuous on [], a b. Divide [], a b into n subintervals of width x D and choose * i x from each interval. pa�/���Y\L��{3�|c���1�|��X�!�e�:�i#��.S���8�H�>n-� �Im�^*. The nth Derivative is denoted as n n n df fx dx and is defined as fx f x nn 1 , i.e. ��5�)}(��| �%���w;��.�V^7�q�5G#����z����'��h�"2�w7�Y>�Я_�p�Ǐ�)��֍n>?�[�w?��g*dU�C����\$�e�������.b�f�J�P%F�^�{���Q�����y��Q�b3��� ��)���C? Common Derivatives and Integrals Anti-Derivative : An anti-derivative of f x( ) is a function, Fx( ), such that F x f x′( )= ( ). %PDF-1.3 vB�Tգ�����`�?%���B�P'#��?�7� 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 ′ ==∫. 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. <> Diﬀerentiation of functions deﬁned parametrically 16 9. Then () * 1 lim i b a n i f x dx f x x fi¥ = ¥ = D ° ±. Derivatives Definition and Notation If y= fx( ) then the derivative is defined to be ( ) ( ) 0 lim h fxhfx fx ﬁ h +-¢= . 5 0 obj 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: Indefinite Integral :ò f (x )d =+Fxc where Fx ( )is ant -der vative of fx. Diﬀerentiation of functions deﬁned implicitly 15 8. The product rule for diﬀerentiation 10 5. 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. [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. Tables of derivatives and integrals 4 1. Indefinite Integral :∫f (xdx F x c) =+( ) where F ()x is an anti-derivative of f (x). *��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+�! 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. ߵ:���HPa ����1d�:�bԔ,�uޢ��/t'[g��p5���A����\$����� If y f x= ( ) then all of the following are equivalent notations for the derivative. Anti-Derivative : An anti-derivative of fx( ) is a function, Fx( ), such that F¢(x) = fx( ). Integral Calculus is motivated by the problem of defining and calculating the area of the region bounded by the graph of the functions. %� ��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�� �y�ٙH��꤈ ����ä ����%N���n@�ψZ���{�U�;H�=. Integration Formulas Z dx = x+C (1) Z xn dx = xn+1 n+1 +C (2) Z dx x = ln|x|+C (3) Z ex dx = ex +C (4) Z ax dx = 1 lna ax +C (5) Z lnxdx = xlnx−x+C (6) Z sinxdx = −cosx+C (7) Z cosxdx = sinx+C (8) Z tanxdx = −ln|cosx|+C (9) Z cotxdx = ln|sinx|+C (10) Z secxdx = ln|secx+tanx|+C (11) Z cscxdx = −ln |x+cot +C (12) Z sec2 xdx = tanx+C (13) Z csc2 xdx = −cotx+C (14) Z secxtanxdx = secx+C Anti-Derivative : An anti-derivative of fx( ) is a function, Fx( ), such that F¢(x) = fx( ). Divide [ab,] into n subintervals of width D x and choose * xi from each interval. 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. ]Fc��+�i�n's��9悖�ܛys��0b�-HAa�(X3)�y� ��p�A�����[iTm�۹m�i�I�-N\%�Ӿ,�br�tO��J�?W Derivatives of basic functions 5 2. Standard Integration Techniques Note that all but the first one of these tend to be taught in a Calculus II class. 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��? stream integral, (())() () bgb( ) aga òòfgxg¢ xdx= fudu . ~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"��� Then () (*) 1 lim i b a n i fxdxfxx ﬁ¥ = ¥ ò =Då. Anti-Derivative : An anti-derivative of fx( ) is a function, Fx( ), such that F¢(x) = fx( ). The chain rule for diﬀerentiation 13 7. Integration by Parts The standard formulas for integration … If y= fx( ) then all of the following are equivalent notations for the derivative. f x y( ) df dy d (fx fxD( )) ( ) dx dx dx ′′ = = = = = Ify f x= ( )all of the following are equivalent notations for derivative evaluated at xa= . the derivative of Derivatives Definition and Notation If y fx= ( ) then the derivative is defined to be( ) ( ) ( ) 0 lim h fx h fx fx → h +− ′ = . Linearity in diﬀerentiation 7 3. The nth Derivative is denoted as n n n df fx dx and is defined as fx f x nn 1 , i.e. the derivative of the first derivative, fx . Integrals Definitions Definite Integral: Suppose f x( ) is continuous on [ab,]. ��)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�