record of the 2011-2012 lectures will be placed below.
: First example of a limit: a derivative. See Examples 8.1(a), (b) and (c)
for more worked examples of this kind of limit.
Second example of a limit: an integral. You'll find more examples like this
in Semester II of the course.
The (rough) definition of a limit, and evaluation of
limits. See Example
another related worked example.
Limits of trigonometric functions. See Example 7.31(b) for another
related worked example.
The derivative of a function. Definition and worked
examples. See Section 8.2 for
material and worked examples.
Limits at infinity, vertical and horizontal asymptotes, and
to sketch the graph of a function. See Examples 2.36 and 2.37 for more
worked examples on asymptotes and graph sketching.
Continuitinuous functions. Read Sections 7.91 and 7.92 for a
more detailed account of continuous functions.
Precise statement of the Intermediate Value Theorem. See Section 7.9.3 for real
engineering application of the Intermediate Value Theorem to
(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.
Section 8.3 in the book for more examples
L'Hopital's rule. Applications of differentiation to rates of change problems.
More applications to rates of change problems. Applications of differentiation to optimization problems.
More applications to optimization problems. Applications of differentiation to curve sketching.
[Dr J. Ward] More applications to curve sketching. See also last year's lecture
Maxima, minima, concavity and points of inflection.
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.)
Rigorous treatment of logarithms. Also: logarithmic differentiation as a speedy way to compute derivatives of certain products and quotients.
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.)
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.)
[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