University of Toronto Mississauga
MAT157Y5Y: Analysis 1, 2018-19
Pre-exam sessions in DH 3000, on:
Monday Apri l8th, 2-3 pm (Belal);
Thursday April 11th, 11 am - 1 pm (Yael); and
Friday April 12, 5:30-7:30 pm (Belal).
Last term test marks distribution - linked below.
Guidelines for the final exam.
Pre-exam sessions, in DH 3000:
Monday April 8th, 2pm-3pm (Belal);
Thursday April 11th, 11am-1pm (Yael);
Friday April 12th, 5:30pm-7:30pm (Belal).
Also do these before the final exam:
(In Assignment 7, please assume—without proof—the
formulas for the derivatives of the sine and cosine functions.)
(In Assignment 8 (1), restrict to the subset of [0,5]
where the function is defined.
If the function has a min or max, find them, and if not, say so.)
"Taylor series - a matter of life or death".
Supplementary notes, 1st term:
Postulates for the real numbers and some consequences.
The least upper bound property.
Properties of continuous functions
that follow from the least upper bound property of the real numbers.
Applications of derivatives;
More applications of derivatives.
Supplementary notes, 2nd term:
Length of a curve.
The Riemann integral.
The length of a
continuously differentiable curve.
Definitions of sine and cosine.
The logarithm and exponential functions.
Lim-sup and lim-inf.
Criteria for convergence of series.
Notes on uniform convergence.
11 April 2019: fixed an incorrect formula in Example (2) on Page 1.
1st term test,
2nd term test with comments;
3rd term test with comments;
4th term test;
First term test:
Second term test:
Third term test:
(with practice problems from earlier material).
The third term test was converted to homework (Assignment 13)
because of a snow storm.
Last term test:
test (with the clarification
to Question 5 that was given during the test);
Research opportunity in McMaster,
Summer school in Germany.
LaTeX is not required for this course, but I strongly recommend
that you eventually learn it.
Here is a three-page sample file and the resulting pdf:
I myself prepare my LaTeX source file on an ubuntu terminal window
with the "vi" editor;
from a second terminal window
I run "pdflatex" to create the pdf file and "evince" to view it.