calculus cheat sheet derivatives and integrals pdf

calculus cheat sheet derivatives and integrals pdf

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 ′ ==∫. ߵ:���HPa ����1d�:�bԔ,�uޢ��/t'[g��p5���A����$����� pa�/���Y\L��{3�|c���1�|��X�!�e�:�i#��.S���8�H�>n-� �Im�^*. Integral Calculus is motivated by the problem of defining and calculating the area of the region bounded by the graph of the functions. Indefinite Integral :ò f (x )d =+Fxc where Fx ( )is ant -der vative of fx. Standard Integration Techniques Note that all but the first one of these tend to be taught in a Calculus II class. Integrals Definitions Definite Integral: Suppose fx( ) is continuous on [ab,]. Higher Order Derivatives The Second Derivative is denoted as 2 2 2 df fx f x dx and is defined as fx fx , i.e. %PDF-1.3 Anti-Derivative : An anti-derivative of f x is a function, F x, such that () F x f x ¢ =. Integrals Definitions Definite Integral: Suppose f x( ) is continuous on [ab,]. Higher derivatives 9 4. ��5�)}(��| �%���w;��.�V^7�q�5G#����z����'��h�"2�w7�Y>�Я_�p�Ǐ�)��֍n>?�[�w?��g*dU�C����$�e�������.b�f�J�P%F�^�{���Q�����y��Q�b3��� ��)���C? Derivatives of basic functions 5 2. 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"��� �#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$2_k��Say��"j-_�A�>�x0�l6u���Bi:kQ�V괞���!fK�y��Y���g����9h=�����Ǖ3v 9P�4S��#`� �y�ٙH��꤈ ����ä ����%N���n@�ψZ���{�U�;H�=. Derivatives Definition and Notation If y= fx( ) then the derivative is defined to be ( ) ( ) 0 lim h fxhfx fx fi h +-¢= . f()xydfdyd(f()x)Dfx() dxdxdx ¢¢===== If y= fx( )all of the following are equivalent notations for derivative evaluated at xa= . Then () (*) 1 lim i b a n i fxdxfxx fi¥ = ¥ ò =Då. %�쏢 Divide [ab,] into n subintervals of width ∆x and choose * x i from each interval. 3 Fundamental Theorem of Calculus Part I : If fx( ) is continuous on [ab,] then () x() a gx= ò ftdt is also continuous on [ab,] and () () x a d gxftdtfx dx ¢ ==ò. ! ^�S�w�4Q�����F��b T`�$��V.jɘv:ج�(; �%�m���ۡ��j8ӥi�a�x�� vB�Tգ�����`�?%���B�P'#��?�7� Anti-Derivative : An anti-derivative of fx( ) is a function, Fx( ), such that F¢(x) = fx( ). *��JD�yw� the derivative of Then () * 1 lim i b a n i f x dx f x x fi¥ = ¥ = D ° ±. Tables of derivatives and integrals 4 1. *�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! 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+�! 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 ]Fc��+�i�n's��9悖�ܛys��0b�-HAa�(X3)�y� ��p�A�����[iTm�۹m�i�I�-N\%�Ӿ,�br�tO��J�?W Linearity in differentiation 7 3. If y= fx( ) then all of the following are equivalent notations for the derivative. Common Derivatives and Integrals Indefinite Integral :∫f (xdx F x c) =+( ) where F ()x is an anti-derivative of f (x). The quotient rule for differentiation 11 6. Divide [ab,] into n subintervals of width D x and choose * xi from each interval. 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. 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.

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