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 $f(x)=|x|$. 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 $x^n$ and $x^{-n}$ for $n\geq 1$. As a corollary we showed that polynomials are differentiable everywhere. Finally we wrote down the Chain Rule.
In the tutorial nobody asked any questions, so I threw the exercises up on the projector and allowed ye work away. Ye put up your hand if ye wanted assistance. In the end I did questions 1 & 2 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise3.pdf.
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 $\sin x$, $\cos x$, $\tan x$, $e^x$ and $\log x$ (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

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

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