On Monday we proved that if a function is differentiable then it is continuous (today I stated that a rough word explaining differentiable is

*smooth*). We showed that a continuous function need not be differentiable by showing the counterexample

. We presented and proved the sum, scalar, product and quotient rules of differentiation. The proof of the quotient rule is on this page. We did the derivative of

and

for

. As a corollary we showed that polynomials are differentiable everywhere. Finally we wrote down the Chain Rule.

On Wednesday we wrote down the Chain rule again, stated the proof was up here and gave a very dodgy explanation of why we must multiply by the derivative of the ‘inside’ function. We stated and proved the derivatives of

,

,

,

and

(the last two proved non-rigorously). Finally we wrote down Rolle’s Theorem.

**Problems**

*You need to do exercises – all of the following you should be able to attempt. Do as many as you can/ want in the following order of most beneficial:*

*Wills’ Exercise Sheets*

Q. 1-10 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise3.pdf

*More exercise sheets*

Section 3 from Problems

*Past Exam Papers*

Q. 1(c), 3(b), 4(b) from http://booleweb.ucc.ie/ExamPapers/exams2010/MathsStds/MS2001Sum2010.pdf

Q . 1(c), 3(a), 4 from http://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/MS2001s09.pdf

Q. 1(c), 3(a), 4 from http://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/Autumn/MS2001A09.pdf

Q. 1(c), 3(b), 4(a) from http://booleweb.ucc.ie/ExamPapers/exams2008/Maths_Stds/MS2001Sum08.pdf

Q. 1(c) from http://booleweb.ucc.ie/ExamPapers/Exams2008/MathsStds/MS2001a08.pdf

Q. 1(c), 3 from http://booleweb.ucc.ie/ExamPapers/exams2007/Maths_Stds/MS2001Sum2007.pdf

Q. 3(b), 4(b), 5(a) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/MS2001Sum06.pdf

Q. 3(b), 4(b) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/Autumn/ms2001Aut.pdf

Q. 3(b), 4(a) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001.pdf

Q. 4, 5(a) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001Aut05.pdf

Q. 4 from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/ms2001s2004.pdf

Q. 4(b) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/MS2001aut.pdf

Q. 4(b) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/MS2001.pdf

Q. 4 from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/ms2001aut.pdf

Q. 4, 5(a), 6(a) from http://booleweb.ucc.ie/ExamPapers/exams2002/Maths_Stds/ms2001.pdf

Q. 1(b), 4(b), 5(b) from http://booleweb.ucc.ie/ExamPapers/exams2001/Maths_studies/MS2001Summer01.pdf

Q. 1(b), 4(b) from http://booleweb.ucc.ie/ExamPapers/exams/Mathematical_Studies/MS2001.pdf

*From the Class*

1. *Prove Proposition 4.1.4 (ii)*

*2. Prove Proposition 4.1.9 (ii)*

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