### Aim:

In the first semester of MA140 we'll digest concepts such as "limit", "function", "continuity", "derivative" and illustrate their relevance to engineering problems involving rates of change.

Textbook:

"(Modern) Engineering Mathematics" by Glyn James. There are copies in the bookshop. For just over 80 euros you can buy this book together with the book for Second Year. The books also cover the algebra modules in First and Second Year Engineering.

### Lectures:

A pdf record of the 2011-2012 lectures will be placed below.

Lecture 1: First example of a limit: a derivative. See Examples 8.1(a), (b) and (c) for more worked examples of this kind of limit.

Lecture 2: Second example of a limit: an integral. You'll find more examples like this in Semester II of the course.

Lecture 3: The (rough) definition of a limit, and evaluation of limits. See Example 7.31(a)  for another related worked example.

Lecture 4: Limits of trigonometric functions. See Example 7.31(b) for another related worked example.

Lecture 5: The derivative of a function. Definition and worked examples. See Section 8.2 for more related material and worked examples.

Lecture 6: Limits at infinity, vertical and horizontal asymptotes, and how to sketch the graph of a function. See Examples 2.36 and 2.37 for more worked examples on asymptotes and graph sketching.

Lecture 7: Continuitinuous functions. Read Sections 7.91 and 7.92 for a more detailed account of continuous functions.

Lecture 8: Precise statement of the Intermediate Value Theorem. See Section 7.9.3 for real engineering application of the Intermediate Value Theorem  to locating zeros. (You might find Section 7.9.3 heavy going: in which case just skip it!)
Two examples of the Intermediate Value Theorem: (i) atmospheric pressure, (ii) old exam questions.

Lecture 9: Techniques of differentiation. See Section 8.3 in the book for more examples

Lecture 10: L'Hopital's rule. Applications of differentiation to rates of change problems.

Lecture 11: More applications to rates of change problems. Applications of differentiation to optimization problems.

Lecture 12: More applications to optimization problems. Applications of differentiation to curve sketching.

Lecture 13: [Dr J. Ward] More applications to curve sketching. See also last year's lecture by me.

Lecture 14: Maxima, minima, concavity and points of inflection.

Lecture 15: Intuitive understanding of logarithms and exponentiation. (See Sections 2.7.1 and 2.7.2 of the book for more on the intuitive understanding of logarithms.)

Lecture 16: Rigorous treatment of logarithms. Also: logarithmic differentiation as a speedy way to compute derivatives of certain products and quotients.

Lecture 17: Netwon's law for cooling objects and a differential equation. (See pages 765-770 in the book for a gentle introduction to the use of differential equations in engineering problems, and to a range of applications.)

Lecture 18: Radioactive decay as a second application of the differential equation of Lecture 17. Plus a bit of culture: the formal mathematical definition of a limit. (See Section 7.8.1 in the book for more on this formal definition.)

Lecture 19 [Prof S. Leen]: An example of the use of differential equations in mechanical engineering research.

In Lectures 20, 21, 22 I will go over past examination questions.

In lecture 23 we'll have a 50-minute class test.