tag:blogger.com,1999:blog-55524032045883942972024-02-19T02:12:25.290-08:00core pure 3 notesdon stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.comBlogger41125tag:blogger.com,1999:blog-5552403204588394297.post-82599628865199555962012-01-01T13:07:00.000-08:002012-01-01T13:07:01.919-08:00integration by parts: examplescheck that you can understand how the solutions are obtained:<br />
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<a href="http://www.youtube.com/watch?v=dqaDSlYdRcs">integration by parts (patrickJMT)</a> parts 1 and 2 (not part3)<br />
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<a href="http://www.youtube.com/watch?v=zGGI4PkHzhI">integration by parts definite integral </a>(patrickJMT)<br />
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many (good) examples in <a href="http://integralcalc.com/integration-by-parts/#%21prettyPhoto">integralcalc.com</a>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-81449014688861097212011-12-31T13:54:00.000-08:002011-12-31T13:54:38.238-08:00integrating trigonometric functions<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4shthlf78mEwYFmkYOSlmWKUZVtuoLljmJUFXez9VPdcSPP-OLGw1MkTUfbbEu8hDcB6ZaYJNnNg-Zy5cJfr9aSLJgssj2KWDHCp0gEOKAO37F00bdLMCNftzTetgB-bm4TLuj4h3YdQ/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="252" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4shthlf78mEwYFmkYOSlmWKUZVtuoLljmJUFXez9VPdcSPP-OLGw1MkTUfbbEu8hDcB6ZaYJNnNg-Zy5cJfr9aSLJgssj2KWDHCp0gEOKAO37F00bdLMCNftzTetgB-bm4TLuj4h3YdQ/s400/Picture1.png" width="400" /></a></div><br />
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<li>some integrations can be done normally, even though they look like a complex case, requiring u-substitution</li>
</ul><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhu8yDQarXrZxyuXcrSzbS5z5L7a2p8wvjic7Cb7yGtrVeBWwYy9HhbFhK_CYm9-ekacx8-vQ4VYJkgTTng0aiIDTgEXt2sOLBMDKVtue80ebBAEyiXot2DGkM6aiZ2evACheGszGd6CtA/s1600/Picture2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="266" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhu8yDQarXrZxyuXcrSzbS5z5L7a2p8wvjic7Cb7yGtrVeBWwYy9HhbFhK_CYm9-ekacx8-vQ4VYJkgTTng0aiIDTgEXt2sOLBMDKVtue80ebBAEyiXot2DGkM6aiZ2evACheGszGd6CtA/s400/Picture2.png" width="400" /></a></div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivhaHODzE5kIxWQ0RB_dI88aKYI1nSb207VYRFisuzEa3A0fhadHk0B9-iVAqmTh3239vLru8-xvzmDgHvAegpdDbfMvxlY-G56E6kHl8MNy-KIlkF2JJbdL205ZuDSJqxuBYiHbfubQ0/s1600/Picture3.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="268" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivhaHODzE5kIxWQ0RB_dI88aKYI1nSb207VYRFisuzEa3A0fhadHk0B9-iVAqmTh3239vLru8-xvzmDgHvAegpdDbfMvxlY-G56E6kHl8MNy-KIlkF2JJbdL205ZuDSJqxuBYiHbfubQ0/s400/Picture3.png" width="400" /></a></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-43565834405389847872011-12-28T13:53:00.000-08:002011-12-30T12:11:48.951-08:00u-substitution: function with derivativewatch the youtube clip if you need help with this: <a href="http://www.youtube.com/watch?v=qclrs-1rpKI">integration by u-substitution</a><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsWj1VbFXdfkVNKE_dtXbkBrXcr2iO6jnLHLtbOx3Md8NqeQLc44eiQ5ETXjVOkm0Vg9OPYQ3Mu96HjKIQ5ZVPNUvVLPoNXK2s30GXYsaNokrdgj0z5J9fZctBYOEDKF235ev_SV37uuw/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsWj1VbFXdfkVNKE_dtXbkBrXcr2iO6jnLHLtbOx3Md8NqeQLc44eiQ5ETXjVOkm0Vg9OPYQ3Mu96HjKIQ5ZVPNUvVLPoNXK2s30GXYsaNokrdgj0z5J9fZctBYOEDKF235ev_SV37uuw/s400/Picture1.png" width="400" /></a></div><br />
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<div class="separator" style="clear: both; text-align: center;">watch this explained <a href="http://www.youtube.com/watch?v=W1WVFJ8iMqQ&feature=related">here</a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2rQiVJeNV-lPd3syWC2IsuK3cfT-zoRXWbvj95J6GYmREgKd9QVKrGWtq-QQZ6tobVJQlAvqxwEjE0UAVNu5dqfOfocL6r2Zm3Ri0cB6n_MS6OVF5_HJj_rPkoApEusHyI-ERyq1wT98/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="303" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2rQiVJeNV-lPd3syWC2IsuK3cfT-zoRXWbvj95J6GYmREgKd9QVKrGWtq-QQZ6tobVJQlAvqxwEjE0UAVNu5dqfOfocL6r2Zm3Ri0cB6n_MS6OVF5_HJj_rPkoApEusHyI-ERyq1wT98/s400/Picture1.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmAVYGRZvDWGdUucS1lrys4gv5_q_ZD7lSXvurXhDe8BOQiuUfq005vbYwbWOam0cGhT6-sg7V_tSh535Shtx5XXgxQZ5yN3FnJ4GismtoPPE2Oi1quGcAQodrp06C0qQH5c-bDr-Xka0/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="295" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmAVYGRZvDWGdUucS1lrys4gv5_q_ZD7lSXvurXhDe8BOQiuUfq005vbYwbWOam0cGhT6-sg7V_tSh535Shtx5XXgxQZ5yN3FnJ4GismtoPPE2Oi1quGcAQodrp06C0qQH5c-bDr-Xka0/s400/Picture1.png" width="400" /></a></div><br />
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<div class="separator" style="clear: both; text-align: center;"></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-12761508384221024552011-12-28T13:45:00.000-08:002011-12-29T12:34:51.337-08:00simple u-substitution<div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdJ2weD8eLO7NlXlmYwtS_heurZzAYO7ON6Sn0huz_X0j71j-SUPli_0zmh9AJ4tIjzNseo5M-ma-mMQbBbEY6NETp66Or5k7e3ZmG4VNcO59ZFHKizi3p-oRxpQ2q6qdHVaQM3gjh3lA/s1600/Picture2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="283" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdJ2weD8eLO7NlXlmYwtS_heurZzAYO7ON6Sn0huz_X0j71j-SUPli_0zmh9AJ4tIjzNseo5M-ma-mMQbBbEY6NETp66Or5k7e3ZmG4VNcO59ZFHKizi3p-oRxpQ2q6qdHVaQM3gjh3lA/s400/Picture2.png" width="400" /></a></div><br />
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</a>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-37120205881712031292011-12-28T13:41:00.001-08:002011-12-28T13:41:52.687-08:00integration overview<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLDarXduL9KQmoxxTQxuR15cErw2LFJbJHMmJwv8ucy2B1h6NZfLogQquiTIVmqcJ6eO0GXAXqX3lY3rJDX1OScO-yEGbCKFIhEtMy8iUH1O6sufp5WNHvTaIo0efgsR_rYW3kylhVAs0/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="303" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLDarXduL9KQmoxxTQxuR15cErw2LFJbJHMmJwv8ucy2B1h6NZfLogQquiTIVmqcJ6eO0GXAXqX3lY3rJDX1OScO-yEGbCKFIhEtMy8iUH1O6sufp5WNHvTaIo0efgsR_rYW3kylhVAs0/s400/Picture1.png" width="400" /></a></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-20707578847297407352011-11-23T11:52:00.000-08:002011-11-23T11:53:38.877-08:00Casio calculator<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivxl-lDnqj7A2Zh_CcKODBj6bTELmTOjZztWRUSUHYD4fF1sfKD1g9ZMtxiKtRihE0mXy69A2STm1kBrlwP_w4lJ3OAcH_Hrs_WBgQjtoqnJ_vauuJ-i7ajWvAD9ZtDtKupCW5eSz3O2c/s1600/Casio-fx115ES-5564.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivxl-lDnqj7A2Zh_CcKODBj6bTELmTOjZztWRUSUHYD4fF1sfKD1g9ZMtxiKtRihE0mXy69A2STm1kBrlwP_w4lJ3OAcH_Hrs_WBgQjtoqnJ_vauuJ-i7ajWvAD9ZtDtKupCW5eSz3O2c/s320/Casio-fx115ES-5564.jpg" width="238" /></a></div>the Casio fx-115ES<br />
does integrations between limitsdon stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-71816637488694191752011-11-23T11:44:00.000-08:002011-11-23T11:44:32.312-08:00appleif you have or have access to an iphone or ipad you might want to get these applets:<br />
<a href="http://itunes.apple.com/gb/app/quick-graph/id292412367?mt=8">Quick Graph</a> is free<br />
<a href="http://itunes.apple.com/us/app/symbolic-calculator/id333252515?mt=8">Symbolic calculator</a> is cheap<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjS6efPo0D5tItOWTP7a07o8rkMRKeQB6RNnxor8VnFi_e95xwl3OwyYtkCuXD0M38bGvIRsTjJ1giO3pei0pyXA7amj_fu2l5rHGhrRqzZBfd8kaIQN70h8cFpdNCw9fdbDdywW-QkV90/s1600/quick-graph.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjS6efPo0D5tItOWTP7a07o8rkMRKeQB6RNnxor8VnFi_e95xwl3OwyYtkCuXD0M38bGvIRsTjJ1giO3pei0pyXA7amj_fu2l5rHGhrRqzZBfd8kaIQN70h8cFpdNCw9fdbDdywW-QkV90/s1600/quick-graph.png" /></a></div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgd4QxKSpD_VJceO10mNRWiSxbIEvo8JJTlyZNuCVXKaS-oYbVvJnecs1pOXT5JNDGVVk6kjDS6bxd7VBo2dOm2qXZUJ0JbYE59nc35wA1DIojQNfU8D2Dv5Zq6tpJcU1K1CheHt-O37EA/s1600/1977-1-symbolic-calculator.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgd4QxKSpD_VJceO10mNRWiSxbIEvo8JJTlyZNuCVXKaS-oYbVvJnecs1pOXT5JNDGVVk6kjDS6bxd7VBo2dOm2qXZUJ0JbYE59nc35wA1DIojQNfU8D2Dv5Zq6tpJcU1K1CheHt-O37EA/s320/1977-1-symbolic-calculator.jpg" width="213" /></a></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-3925559668085439782011-11-19T13:09:00.000-08:002011-11-19T13:09:29.898-08:00implicit differentiation<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUHMHvAZlpMWj7WutnWRfrcmrsbhhpVMrwQo8vdBM8g7AKstOz1PG48Z9KZFq9pN2HNEJqBoh0J8JheD0urt9-hHIzipf_X9J86Da1sHhVh7A1eqmwlJE8p6iTxwhiiCVpAOKq-U0E0pM/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="265" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUHMHvAZlpMWj7WutnWRfrcmrsbhhpVMrwQo8vdBM8g7AKstOz1PG48Z9KZFq9pN2HNEJqBoh0J8JheD0urt9-hHIzipf_X9J86Da1sHhVh7A1eqmwlJE8p6iTxwhiiCVpAOKq-U0E0pM/s400/Picture1.png" width="400" /></a></div><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3WH0dq2yPwRaaQ_uZvQy2Cp271_Ny4650t_d6Md6QNqSImGWe2WoU4c40_TZI_vhgNAPZvlNdr7DBQNV5UkG1LpHTLJP34yWYQmD3QrehtPZJpCslqgEFLJyg0O6PcudztMgqG5i6I_U/s1600/Picture3.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="273" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3WH0dq2yPwRaaQ_uZvQy2Cp271_Ny4650t_d6Md6QNqSImGWe2WoU4c40_TZI_vhgNAPZvlNdr7DBQNV5UkG1LpHTLJP34yWYQmD3QrehtPZJpCslqgEFLJyg0O6PcudztMgqG5i6I_U/s400/Picture3.png" width="400" /></a></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-69948423331192481152011-11-19T13:03:00.000-08:002011-11-19T13:03:12.946-08:00derivative plotterthis <a href="http://mathdl.maa.org/images/upload_library/4/vol4/kaskosz/derapp.html">applet </a>enables you to see how the graph of a trig function (put the x in brackets e.g. sin(x)) relates to the differential of this function<br />
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scroll to user function<br />
change the scales of the graph by entering values into the boxes then pressing 'graph'<br />
use the slider to see the gradient functiondon stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-87827947648729535822011-11-19T12:48:00.001-08:002011-12-28T14:40:07.041-08:00differentiating trig functions<div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcF4KCVgbdhQXBDSGdvNundv0wOsDj7SbCocC0xvskZqrBMGQ1JwBma_coXnjthoKE9syg6qz5cTkYAWu1n15VL6tGqG3sUbU3Ravz3TWHWuh8oSwlF67IXGe8ZAzNaLJwEhZm2YP0aks/s1600/PreviewScreenSnapz019.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcF4KCVgbdhQXBDSGdvNundv0wOsDj7SbCocC0xvskZqrBMGQ1JwBma_coXnjthoKE9syg6qz5cTkYAWu1n15VL6tGqG3sUbU3Ravz3TWHWuh8oSwlF67IXGe8ZAzNaLJwEhZm2YP0aks/s1600/PreviewScreenSnapz019.jpg" /></a></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifxEF19bLkRSpNk_GcFVGfjr-rI8QsdV9re-1YVN-Ld-WMEAXwQ-_6QxhtTL7X7srEPkKAv3BjoWJPFXukPy-g4piQ08E3QYj20l0sXqsc1fqOVxJ0ei31oUKjUqySYbRGfXrImiPgROs/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="288" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifxEF19bLkRSpNk_GcFVGfjr-rI8QsdV9re-1YVN-Ld-WMEAXwQ-_6QxhtTL7X7srEPkKAv3BjoWJPFXukPy-g4piQ08E3QYj20l0sXqsc1fqOVxJ0ei31oUKjUqySYbRGfXrImiPgROs/s400/Picture1.png" width="400" /></a><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCrcFku7wEs_Ofa7dB2yxL6_Q9lgbg95_4alZW4n3wmS6n9KtE-y0lbfIIo_1k1Rg7KhskkIH4MLzVWyWiNVC76HvTHVBa3gI1nxStwAaoBb5-PCeK3xcBceIsj-Ee8EvTMszJpcbmEW0/s1600/Picture4.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="303" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCrcFku7wEs_Ofa7dB2yxL6_Q9lgbg95_4alZW4n3wmS6n9KtE-y0lbfIIo_1k1Rg7KhskkIH4MLzVWyWiNVC76HvTHVBa3gI1nxStwAaoBb5-PCeK3xcBceIsj-Ee8EvTMszJpcbmEW0/s400/Picture4.png" width="400" /></a></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-91101987743221063462011-11-19T12:45:00.000-08:002011-12-28T14:20:52.440-08:00differentiating natural logarithms and exponentials<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhezu0Vlj-FyPhI-CNFa5anQKJiSJ-nAETxjRNuMl49iOVDT5YQoLu7aEhybWPXhW8oCBc6FsDthLfbTfal0-USGjWmZSnlOopDLo2QM6fkTjssHtC-Gk2mLCnguQOij-XC0-8ugibaM0/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="305" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhezu0Vlj-FyPhI-CNFa5anQKJiSJ-nAETxjRNuMl49iOVDT5YQoLu7aEhybWPXhW8oCBc6FsDthLfbTfal0-USGjWmZSnlOopDLo2QM6fkTjssHtC-Gk2mLCnguQOij-XC0-8ugibaM0/s400/Picture1.png" width="400" /></a></div><br />
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<div class="separator" style="clear: both; text-align: center;"></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-38778460716897156092011-11-19T10:46:00.000-08:002011-11-19T10:46:29.113-08:00differentiating w.r.t. other variables<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZx7Q4dMJom4dAclqkFrYdCvR4NEZeBC8DH7rTvR3gmg2crwQWAbVVAOj6JzIqkevr8O3hop-F4i5ppnz7o3Qh4cgQC4mV7GAJStFWGqtqinPFZo5-SFyHYf6DT_95ISrZj8cmipKA9F8/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="275" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZx7Q4dMJom4dAclqkFrYdCvR4NEZeBC8DH7rTvR3gmg2crwQWAbVVAOj6JzIqkevr8O3hop-F4i5ppnz7o3Qh4cgQC4mV7GAJStFWGqtqinPFZo5-SFyHYf6DT_95ISrZj8cmipKA9F8/s400/Picture1.png" width="400" /></a></div><br />
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for this function, the turning point is identified (which can be related to the graph) and then the equations of the tangent and normal are found at the point where x = 9:<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJJ0GeqUFSxQzGM8OlpGx5NMuTcfrm2HrFjwLMqBFt0njkd6uto6I5PDx-azaYQma5pPt4_ZaQw9kQd5lsfbggD_sdfRWaYGsb-abG3yNCTEaG279oWhdTmLbLgrArcN3CABMLdZOEcGo/s1600/ee234d10a64b4cfbba39ec633e7a901a.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="195" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJJ0GeqUFSxQzGM8OlpGx5NMuTcfrm2HrFjwLMqBFt0njkd6uto6I5PDx-azaYQma5pPt4_ZaQw9kQd5lsfbggD_sdfRWaYGsb-abG3yNCTEaG279oWhdTmLbLgrArcN3CABMLdZOEcGo/s400/ee234d10a64b4cfbba39ec633e7a901a.png" width="400" /></a></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-47168763794571902022011-10-13T10:34:00.001-07:002011-11-09T15:09:22.635-08:00product rule examples<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZdlVUO4S5FdI3IoZ2ZJlFiD2ZcTKfBXYA0VlVuaK8a6-BwcFKZbClJDe5h0_ihbRwiCxBYvrNbXI2Y7p0_V0qVb58HB4FW4u-G2XH_FcmK-iVpFygVeuoYKxFqakSiRkxX5sm8i59pEU/s1600/Picture5.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><br />
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<div class="separator" style="clear: both; text-align: left;">(differentiate first) times (the second) then + (the first) times (differential of the second)</div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5-wSgBnW7VzXs8xdUFNF97CjBLUB7pVr0qI3RgKoYzCSj6Cc67_RzC77r9SzewHfH0D91Crl2vQvnblKW-AwfFPtRnFW3CpVk-TQHQpnmstTj8OthNj_3VVq4TZ9VbqNY7k7ji5gVUI8/s1600/Picture2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="255" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5-wSgBnW7VzXs8xdUFNF97CjBLUB7pVr0qI3RgKoYzCSj6Cc67_RzC77r9SzewHfH0D91Crl2vQvnblKW-AwfFPtRnFW3CpVk-TQHQpnmstTj8OthNj_3VVq4TZ9VbqNY7k7ji5gVUI8/s400/Picture2.png" width="400" /></a></div><br />
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when factorising, extract the lowest power of a common function<br />
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<div class="separator" style="clear: both; text-align: center;"> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGh4Eg8Y9vMlOasW21RBqlHMvEhtAMc2u-P8CqD-KWajqcGoCfIUA-kGJrqPmw6GwWm_wvrBaOkBurRD9Ogd_iZxqs6biDLLb0EXa8unvtYOjAVxWVA4jt7VCCEe50gedVRQ9ISAZbXEY/s1600/Picture3.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGh4Eg8Y9vMlOasW21RBqlHMvEhtAMc2u-P8CqD-KWajqcGoCfIUA-kGJrqPmw6GwWm_wvrBaOkBurRD9Ogd_iZxqs6biDLLb0EXa8unvtYOjAVxWVA4jt7VCCEe50gedVRQ9ISAZbXEY/s400/Picture3.png" width="400" /></a></div><br />
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in this question, use the product rule to find the differential and then relate the turning points to the graph<br />
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some other examples, involving exponential, logarithmic and trig functions<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhY0mhWG5pMiVcoDWNgltUWEuNtqxwLlDTrnQExpzu_I5pUtGZyj_VvwSHPq5dI5QSaWxgewccCyUOTuZLe1nk41VsHZEegNQbDqITpGFsefPQDh_IY_DCRFrl9W2hmWGxN9L113ZcfV4/s1600/Picture1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="276" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhY0mhWG5pMiVcoDWNgltUWEuNtqxwLlDTrnQExpzu_I5pUtGZyj_VvwSHPq5dI5QSaWxgewccCyUOTuZLe1nk41VsHZEegNQbDqITpGFsefPQDh_IY_DCRFrl9W2hmWGxN9L113ZcfV4/s400/Picture1.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-35522805991481435092011-10-13T10:33:00.002-07:002011-11-05T13:04:07.408-07:00product and quotient rule<div class="separator" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" height="276" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggZ0qZ_UQrI6hoX3jR04mvKC7gHH2SAbvhGXfh2UwzG7Bp5mvevUSyieGSMd2mLNELMiOkbaH5ZGfEmjAbqi9q2_DhNRSHPXC9z_Or7Oie4Fwr1XWmVKsFo_ezsUW2fSrCvuJ5VY9MOXg/s400/Picture3.png" width="400" /> </div><div class="separator" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;">an example of the <a href="http://vimeo.com/7078384">product rule</a> on Vimeo</div><div class="separator" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;">an example of the <a href="http://vimeo.com/21810402">quotient rule</a> on Vimeo</div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0tag:blogger.com,1999:blog-5552403204588394297.post-27412613793810672142011-10-13T10:33:00.000-07:002011-11-05T14:53:22.255-07:00chain rule<div class="separator" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" height="396" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiC5vr_QK3m4cpm3cya-hrON80zOdiwrJchfdhusesgUKz4wu0l1qwTakbVNdQ-CBOFypVHKTuDzI6CW2b3F7tFqx8qe5VvcCV5sFjxzMrs-yuToqDYmXp0aYU2TTpkTQNlcp1bjnDLrwA/s400/Picture1.png" width="400" /></div>see this explained on <a href="http://vimeo.com/30756259">Vimeo</a><br />
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</div><div style="text-align: left;">here they use the terms 'inside function' and 'outside function'<br />
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the derivative is<br />
the derivate of the outside function x<br />
derivative of inside function</div><div style="text-align: left;"><br />
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to see how the chain rule works, from some simple examples:</div><div style="text-align: left;"><br />
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a couple of simple examples:</div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe1QQoPEh0YRJB7XkjN75DDIOSnqiHhWvGs4in6OzJFOdwonrEX5p-MaRDcisqCiZe_WmmRf5u-WHMMEKpQMXIUKycaOcSuAkytwoHEKXoaiqjNMYoL2ys9Rqw6dWjnoeHBQGPT4jWyuA/s1600/Picture2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="330" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe1QQoPEh0YRJB7XkjN75DDIOSnqiHhWvGs4in6OzJFOdwonrEX5p-MaRDcisqCiZe_WmmRf5u-WHMMEKpQMXIUKycaOcSuAkytwoHEKXoaiqjNMYoL2ys9Rqw6dWjnoeHBQGPT4jWyuA/s400/Picture2.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioexpJ2SUSTgi7NPCA_ZXbXPP4eg6I2vwSLK0vqWoLgcDyjazZgUZT7koVApxdm-rNQ9IAk94ZBE7IZmm4V6pg18kW-5snM2ceMRfgblwXnAx3NLlkuA-b5rSePd4lwq2Te_qXJpTzLcE/s1600/Picture4.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="355" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioexpJ2SUSTgi7NPCA_ZXbXPP4eg6I2vwSLK0vqWoLgcDyjazZgUZT7koVApxdm-rNQ9IAk94ZBE7IZmm4V6pg18kW-5snM2ceMRfgblwXnAx3NLlkuA-b5rSePd4lwq2Te_qXJpTzLcE/s400/Picture4.png" width="400" /></a></div>don stewardhttp://www.blogger.com/profile/04868382949680830399noreply@blogger.com0