MATE 3031 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 topic. 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.
(27/01) Topics: (i) Functions and their graphs; domain, range, set
notation, symmetries. (ii) Injectivity, surjectivity, existence of
inverse. (iii) Family of exponential functions and their properties;
also, logarithmic functions. Problems: those I indicated in class.
(5/02) (iv) Limit of a function, limit laws: problems from sections
2.2, 2.3 of syllabus. (v) Definition of limit in terms of the 2-player
game: exercises 1--4 of section 2.4. Note that "delta" in these
problems is simply the radius chosen by player I. Also, exercises 15,
17, 19, 23, 24, 27 of the same section. In each case, state the
formula giving player I's strategy (in response to player II's choice
of r), and illustrate graphically, drawing horizontal strips and
vertical slashed strips. (vi) Graphical method for solving
inequalities f(x) <= 0 ( or f(x) >= 0 ). Perform on the following
examples:
* x squared - 9 <= 0.
* | x-2 | >= 4.
(10/02) (vii) Continuity: definition, rules, intermediate value
theorem. Exercises from syllabus. (viii) Limit at infinity, infinite
limit (including definition). Exercises from syllabus, including 30
and 62, to test your grasp of the definition. For #30, specify the
neighbourhood of infinity chosen by player I in response to the move y
> r of player II (thereby revealing that the limit is indeed
infinity). For #62, "epsilon" is the radius of the strip chosen by
player II.
(19/02) (ix) derivative and rate of change. (x) Derivative as a
function. (xi) Higher derivatives. (xii) Two algebraic rules of
derivation. (xiii) Derivative of power and exponential function.
Derivative of polynomials. Exercises from syllabus.
(25/02) (xiv) Product and quotient rule. (xv) Derivative of
trigonometric functions. Exercises from syllabus.
(3/03) (xvi) Chain rule. Unless indicated otherwise, exercises from
the syllabus.
(10/03) (xvii) Implicit differentiation. (xviii) Logarithmic functions
and differentiation.
(13/03) (xix) Rates of change.
(9/04) (xx) Related rates. (xxi) Linear approximations. (xxii) Extrema
and local extrema. (xxiii) Mean value theorem. (xxiv) Derivatives and
the shape of the graph.
Sample third test
(14/04) (xxv) L'Hospital's rule.
(20/04) (xxvi) Curve sketching.
(25/04) (xxvii) Optimisation problems. (xxviii) Antiderivative.
(28/04) Sample fourth test
(29/04) (xxix) Area and distance.
(30/04) (xxx) Definite integral.
(8/05) (xxxi) Fundamental theorem of calculus. (xxxii) Substitution
rule.
(14/05) (xxxiii) Area between curves.
Last modified: Wed May 15 00:12:14 AST 2013