MATE 3021 Practice problems
You are responsible for the problems posted in the syllabus . You do not have to spend the
same amount of time on every problem, but you must identify groups of
problems corresponding to each topic, and you must spend enough time
on as many from each group, as you think is necessary for you to grasp
that concept. On quizzes and tests, you are graded for carefully
justifying the steps, not simply for getting the right
answer. Clusters of problems posted here will be identified by the
posting date; problems posted on, say, Wednesday, are fair game for an
unannounced quiz on the following Monday.
(18/08) (Dates are coded by dd/mm): section 1.2, problems 3--67 of
syllabus. Topics: (i) When are functions equal (ii) Symmetries (iii)
Domain and range. Additional exercise: using the graphical method
shown in class, find the range of the function f(x) = x squared when
the domain is:
a) The interval [-3,0]
b) The interval [-3,4].
(21/08) Section 1.2, all problems. Section 2.1: exponential growth and
decay, recursions. Problems from syllabus.
(23/08) Section 2.2: sequences and their limits.
(28/08) Sequences defined recursively. Finding limit of a sequence by
finding its fixed points.
(11/09) Definition of limit: problems 1, 3, 5, 6, 7, 8 p. 129. For
each problem, illustrate using the graphical method shown in class,
using three colours: one for the y-interval, one for the arc on the
graph, one for the x-interval. Limit laws.
(22/09) Continuity. Limits at infinity. Sandwich theorem and limits of
trigonometric functions.
(9/10) Property of continuous functions. Definition of derivative,
differentiability and continuity.
(14/10) Elementary differentiation rules. Power, product and quotient
rule.
(20/10) Chain rule. Implicit differentiation.
(24/10) Higher derivatives.
(27/10) Derivatives of trigonometric functions.
(31/10) Derivatives of exponential functions. Derivatives of inverse
functions.
(11/11) Derivatives of logarithmic functions, logarithmic
differentiation.
We showed how to obtain, graphically, the direct image or the inverse
image of a set of values under a function. We denoted the inverse
image of B under f, f^(-1)(B), but since many of you are still
confusing this notation with that of inverse function, we will simply
refer to that set as "inverse image of B under f". Solving these
problems (graphically) is a skill which I consider important enough
that you can expect problems of this kind on the remaining partial
tests, and on the final exam. Here is a set of problems for further
practice: find the direct image, and the inverse image, of the
interval [0,1] under each of the following functions:
i) f(x) = x squared
ii) g(x) = | x | - 2
iii) h(x) = x(x+1)(x-1). In this case, show how to obtain the solution
graphically, without indicating the exact values of interval
endpoints.
iv) j(x) = x/(2x+2).
For further reinforcement, repeat the same steps, this time replacing
the interval [0,1] with the union of the intervals [0,1] and
[-2,-1].
(17/11) Properties of functions: monotonicity and curvature,
characterization through the first and second
derivative. Graphing. Asymptotes.
(27/11) Practice final exam ,
due 2/12 (in class for section 050, in review session, M-118, at 18:00
for section 100). The practice exam will count as two quizzes.
Antiderivative.
(1/12) Update on the previous announcement: for section 100, we can
start the review session during the regular class period (16:30-17:45)
and decide whether to continue in the evening.
Last modified: Sun Dec 1 16:15:50 AST 2013