Created by T. Madas Created by T. Madas Question 1 Evaluate the following. 0 A�+�I�S�pF(��u,Q�@�5h��8�P�k�R���5(��Q�[�?��_ �����xƫN�t�R����X3�R������\$�~**�+�YYC^����1��@}� x��Y�n�F}7��#YT��% |M44�S�Z�-�6�ڒR��3�rI��S�x�9;{���4=y��ݔ�m����d�-��"�>=]o�뇿�W����ky{W�ۻu5���g�������ypus|Ă�� K✧���>�������?> That is, yex if and only if xy ln. Differentiation refers to how a business separates itself into key components such as departments or product offerings. On completion of this tutorial you should be able to do the following. %%EOF 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. 2 • We have seen two applications: – signal smoothing – root ﬁnding • Today we look – differentation – integration endstream endobj startxref Calculus is usually divided up into two parts, integration and differentiation. Apply Newton’s rules of differentiation to basic functions. The domain of I = Z b a f(x)dx … Z b a fn(x)dx where fn(x) = a0 +a1x+a2x2 +:::+anxn. On completion of this tutorial you should be able to do the following. Numerical differentiation and interpolation Abstract Numerical integration and differentiation are some of the m ost frequently needed methods in computational physics. Each is the reverse process of the other. Integration: With a flow rate of 2x, the tank volume increases by x 2. 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 Differentiation and Integration, both operations involve limits for their determination. Explain differential coefficients. Differentiation and integration provide two possible methods for businesses to organize their operations and projects. Integration refers to how those components cooperate. Differentiation and Integration 1. 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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 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 483 0 obj <>stream d���uU�L�l*���2�±�U�غ�x%���&�VN���T|�' ���Pj��lŨ�؇��u\$���K�XJ�bKPd�gө&���K�3�\7=/���8���� �~������K}Gj7qW�uG�#O�Хsgb 3�o�ڃl�]��z�]�"Q�we�.7����ٺ�hG%��&��i�w��i �q(�k�E�u���-)��]�d�̟��R�=���gGF{� Both differentiation and integration, as discussed are inverse processes of each other. h��Vmo�6�+���� (8ɼh; ɖ? Example: with the flow in liters per minute, and the tank starting at 0 . or an integral! 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. 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. Differentiation formulas . Math Methods Numerical Integration & Differentiation Project Rev 070105 A-2 Fig. Integration is covered in tutorial 1. Each is the reverse process of the other. Integration is covered in tutorial 1. Interchange of Diﬀerentiation and Integration The theme of this course is about various limiting processes. 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 … integration and integration to differentiation. %PDF-1.5 %���� It is therefore important to have good methods to compute and manipulate derivatives and integrals. Quite often we are confronted with the need of evaluat-ing either the derivative f! Properties of the Natural Exponential Function: 1. ��t�"|"��.��d�� @�s�ZG< ���=���΂`�3����y�x��I�v��8I�,�� ާ�h"ëh"�ߢ,���Ƴ@�Y��߂h���c�0�#�(�\D��g�z����{L`dꞿo�O9�1�c�����qy|t�χdE���� +���c�P��T�,�,߇!H��Lj�s;[����,���=����m��0"�a��(�>��jD3�1�~����Q�=�U��\$�Ah��i;5i6��w|ڻxDJ\4{�dq޲�ŕ).J�v�g�&� ��OЋ�f�8�ْ�}8�5C���*���eBb ,g������*>�櫂�\rj��,H�X�WH��f��r���p��-֬u�\$�"D�^��>�-'�n�O�4��iX�^� �.{�����H{�U��p៝��. S�=��%���� �d�iF�y ��a�����w8�pj���[�;���b @,��[z�������̖�}�n��|�8eclÆ�'p��7� �E6�\>]�b��U�}Z߀����Z�v��;8�)W� �7��~�:�� g�h�b�h�����>�c�����������Z/��9��ɡ���g{�G[vĝ��Gȃ�-oPB �"Yb�������)��\$�` 9}��7�n`YOrz�!�j����͞խ�2F��c��Ny3��ʂ����S���dP�)t��<1�T The concept of understanding integrating a differential function gives the original function is very hard for a high school student.

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