First thing on Monday we tried to rearrange the Monday 6 December lecture. We failed miserably but someone had the bright idea to swap with MS 2003. I have emailed the lecturer and she agrees in principle so we should have a definite plan soon. The second test is also slated for Wednesday 8 December.
Anyway, we stated and proved Rolle’s Theorem. We stated and proved the Mean Value Theorem and verified the Mean Value Theorem for a quadratic.
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 5 (iii),(iv) & 6 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise3.pdf.
On Wednesday we proved the intuitively true theorem that the sign of a derivative determines whether the function is increasing or decreasing. We used this theorem to prove that x^n is increasing on (0,\infty); a corollary of which is the existence of positive nth roots. We defined rational powers. Finally we introduced the idea of the curve; and showed a few examples on the projector.

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. 11-13 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise3.pdf

Q. 1 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise4.pdf

Past Exam Papers

Q. 4(b) from http://booleweb.ucc.ie/ExamPapers/exams2008/Maths_Stds/MS2001Sum08.pdf

Q. 4 from http://booleweb.ucc.ie/ExamPapers/Exams2008/MathsStds/MS2001a08.pdf

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

Q. 5(a) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/Autumn/ms2001Aut.pdf

Q 5(a) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/MS2001aut.pdf

Q. 5(a) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/MS2001.pdf

Q. 5(a) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/ms2001aut.pdf

Q. 5(a) from http://booleweb.ucc.ie/ExamPapers/exams2001/Maths_studies/MS2001Summer01.pdf

Q. 5 from http://booleweb.ucc.ie/ExamPapers/exams/Mathematical_Studies/MS2001.pdf

From the Class

1. Prove Proposition 4.2.1 for the case that the minimum, m differs from f(a).

2. Drawings can be deceptive! Draw a function that is continuous on a closed interval but not differentiable at any point in the interval. What does your drawing suggest? Now see http://en.wikipedia.org/wiki/Weierstrass_function

3. Prove Proposition 4.2.3 (iii)