*continuous on a closed interval*. This is to account for limits such as

*Intermediate Value Theorem*.

*bounded*. If the function is unbounded then it is not continuous. We showed how the Intermediate Value Theorem can

*estimate the location of roots*of functions. Finally we showed with an example the restriction (in the theorem) to closed intervals is necessary.

*differentiation*: why do we do it? We derived the formula for the derivative of a function, defined a

*differentiable*function (at a point) and showed that, as expected, the derivative of a line is just . This also implies that a constant function has derivative zero. We did one more example (a quadratic). Finally we said that in some sense a differentiable function must be somewhat ‘nicer than’ or ‘as nice’ as a continuous function – as all differentiable functions are continuous. We will prove this on 1/11/10.

**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. 9 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise2.pdf

*Past Exam Papers*

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

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

Q. 3(a), 5(a) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/ms2001s2004.pdf

Q. 2(b) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/ms2001aut.pdf

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

Q. 4(b), 6(a) from http://booleweb.ucc.ie/ExamPapers/exams/Mathematical_Studies/MS2001.pdf

## Leave a comment

Comments feed for this article