Differentiation and integration are basic mathematical operations with a wide range of applications in many areas of science. Two integrals of the same function may differ by a constant. I = Z b a f(x)dx … Z b a fn(x)dx where fn(x) = a0 +a1x+a2x2 +:::+anxn. Chapter 5: Numerical Integration and Differentiation PART I: Numerical Integration Newton-Cotes Integration Formulas The idea of Newton-Cotes formulas is to replace a complicated function or tabu-lated data with an approximating function that is easy to integrate. 9}��7�n`YOrz�!�j����͞խ�2F��c��Ny3��ʂ����S���dP�)t��<1�T Derivative: If the tank volume increases by x 2, then the flow rate must be 2x. Calculus is usually divided up into two parts, integration and differentiation. Integration refers to how those components cooperate. {wp�ȴ���i�5z=�g!��<6{���@�K�M��0���Z�i� � +�4��՜��R�@�%~)�^"V����2��5�ƅ�萉��Ҡ�=ļ5��f��P�z�� Each is the reverse process of the other. On completion of this tutorial you should be able to do the following. Differentiation & Integration Formulas DIFFERENTIATION FORMULAS dx d (sin u) = cos u dx du (csc u) = −csc u cot u (cos u) = −sin u (sec u) = sec u tan u (tan u) = sec² u (cot u) = −csc² u (ln u) = 1⁄ u (e u) = eu (log a u) = 1⁄ u log a e INTEGRATION FORMULAS Note: a, b and c are constants; k is the integration … Math Methods Numerical Integration & Differentiation Project Rev 070105 A-2 Fig. The domain of That is, yex if and only if xy ln. integration and integration to differentiation. DIFFERENTIATION . Differentiation refers to how a business separates itself into key components such as departments or product offerings. Integration: With a flow rate of 2x, the tank volume increases by x 2. Explain differential coefficients. endstream endobj startxref Explain differential coefficients. 483 0 obj <>stream 2 • We have seen two applications: – signal smoothing – root finding • Today we look – differentation – integration Differentiation and integration provide two possible methods for businesses to organize their operations and projects. A�+�I�S�pF(��u,Q�@�5h��8�P�k�R���5(��Q�[�?��_ �����xƫN�t�R����X3�R������$�~**�+�YYC^����1��@}� 452 0 obj <> endobj The derivative of any function is unique but on the other hand, the integral of every function is not unique. 461 0 obj <>/Filter/FlateDecode/ID[<2E4C56239E86E34FBA67A222F313302C>]/Index[452 32]/Info 451 0 R/Length 60/Prev 411030/Root 453 0 R/Size 484/Type/XRef/W[1 2 1]>>stream endstream endobj 456 0 obj <>stream �fʐ��� �8YF����C0�j��� 7+J����P�L���8/+�f���=N�5�q7�w�p�ׁ�`c8 -��|pٷ;�m�)�̺��� �=�����:s:;��ř��nh�v����Q(Wt����2�Z9��{=U��LƽEH�*�c���@}fl$H��i�Z�3��J�&Hs�V��V|%�B���nW��{|o��xF�,��/������Y� Differentiation and Integration 1. Result of calculations. 0 h�bbd``b`� $k@��`��ׂ$�AbK���I �H02Y��&�3�0 � When performed over a full cycle and plot, the result is a sine wave like that shown in Fig. Chapter 5: Numerical Integration and Differentiation PART I: Numerical Integration Newton-Cotes Integration Formulas The idea of Newton-Cotes formulas is to replace a complicated function or tabu-lated data with an approximating function that is easy to integrate. I = Z b a f(x)dx … Z b a fn(x)dx where fn(x) = a0 +a1x+a2x2 +:::+anxn. �7��~�:�� g�h�b�h�����>Ž�c�����������Z/��9��ɡ���g{�G[vĝ��Gȃ�-oPB �"Yb�������)��$�` Quite often we are confronted with the need of evaluat-ing either the derivative f! h�b```� fN/a�13�0p,��(ʠ8���%GZ�~a�= ���&CÌL׀J�=������cg��;��a���}�6�`�h`�`�h`�h�`�h��h�� ��:l��4;s������|��'��� �e`S��Y��7f`s� �� p�5T You proba-bly learnt the basic rules of differentiation and integration … Interchange of Differentiation and Integration The theme of this course is about various limiting processes. Numerical differentiation and interpolation Abstract Numerical integration and differentiation are some of the m ost frequently needed methods in computational physics. 5.4 Exponential Functions: Differentiation and Integration Definition of the Natural Exponential Function – The inverse function of the natural logarithmic function f x xln is called the natural exponential function and is denoted by f x e 1 x. Both differentiation and integration, as discussed are inverse processes of each other. Numerical Integration *** 3/1/13 EC What’s Ahead • A Case Study on Numerical Differentiation: Velocity Gradient for Blood Flow • Finite Difference Formulas and Errors • Interpolation-Based Formulas and Errors • Richardson Extrapolation Technique • Finite Difference and Interpolation-based Formulas for Second Derivatives The concept of understanding integrating a differential function gives the original function is very hard for a high school student. endstream endobj 453 0 obj <>/Metadata 49 0 R/Pages 450 0 R/StructTreeRoot 68 0 R/Type/Catalog>> endobj 454 0 obj <>/MediaBox[0 0 595.4 842]/Parent 450 0 R/Resources<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI]>>/Rotate 0/StructParents 0/Tabs/S/Type/Page>> endobj 455 0 obj <>stream After 3 minutes (x=3): the flow rate has reached 2x = 2×3 = 6 liters/min, (���%�R����z��x����G��x$��x�ˆ2�n�U�X�&Nrh �LK,�Ryᚣʃ �e�[+@��{f@D06�2 ��� k4R�k@�锵D�4ʁ�����cDx�A�Dr�\�w��~�ˆ8�����]�l�b�^f��y�zl��^�ٙ����٪&R�yU6՗p�a=��8�1)?e������A�O�_�6Y��w�{�]4ٺx���uN��gˢ\%���K���M�o~��v�͓�kWl�jG�…]8?�%���2��h�/�c[[�!�)/ڲ�G��