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.