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# Kalkulus II (23 - 24)

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### Kalkulus II (23 - 24)

1. 1. Kalkulus II<br />Teguh Budi P, M.Si <br />Sesion#23-24<br />JurusanFisika<br />FakultasMatematikadanIlmuPengetahuanAlam<br />
2. 2. Constantmultiple rule <br />Higher Order Derivatives<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />2<br />Outline<br />1/9/2011<br />
3. 3. Multivariable Functions and Their Derivatives(part 1)<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />3<br />1/9/2011<br />
4. 4. The derivative of a constant is zero.<br />If the derivative of a function is its slope, then for a constant function, the derivative must be zero.<br />example:<br />1/9/2011<br />4<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
5. 5. (Pascal’s Triangle)<br />If we find derivatives with the difference quotient:<br />We observe a pattern:<br />…<br />1/9/2011<br />5<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
6. 6. We observe a pattern:<br />…<br />power rule<br />examples:<br />1/9/2011<br />6<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
7. 7. constant multiple rule:<br />examples:<br />When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.<br />1/9/2011<br />7<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
8. 8. constant multiple rule:<br />sum and difference rules:<br />(Each term is treated separately)<br />1/9/2011<br />8<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
9. 9. Horizontal tangents occur when slope = zero.<br />Example:<br />Find the horizontal tangents of: <br />Plugging the x values into the original equation, we get:<br />(The function is even, so we only get two horizontal tangents.)<br />1/9/2011<br />9<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
10. 10. 1/9/2011<br />10<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
11. 11. 1/9/2011<br />11<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
12. 12. 1/9/2011<br />12<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
13. 13. 1/9/2011<br />13<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
14. 14. 1/9/2011<br />14<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
15. 15. First derivative (slope) is zero at:<br />1/9/2011<br />15<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
16. 16. product rule:<br />Notice that this is not just the product of two derivatives.<br />This is sometimes memorized as:<br />1/9/2011<br />16<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
17. 17. quotient rule:<br />or<br />1/9/2011<br />17<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
18. 18. is the first derivative of y with respect to x.<br />is the second derivative.<br />is the third derivative.<br />is the fourth derivative.<br />Higher Order Derivatives:<br />(y double prime)<br />We will learn later what these higher order derivatives are used for.<br />1/9/2011<br />18<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
19. 19. Thank You<br />1/9/2011<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />19<br />