DATE 
LECTURE SUMMARY 
ASSIGNMENT
(due in tutorial of the week listed)

PROBLEM OF THE WEEK 
DOCUMENTS 





Week 1: 1/9  1/13

Tuesday's Lecture:
 Administratrivia (see syllabus for details)
 Some study strategies (click help button for details)
 Staircase paradox
 CS connection: measuring the complexity of an algorithm depends on
knowing how quickly a function grows. For example, is it
possible to get a good approximation to the size of 100! (100
factorial) without actually computing it?
 Is the function sin x continuous? Started a proof of this.
Thursday's Lecture:
 Review of functions (a function is a
set; definition of domain and range; some examples)
 Zoom in on the function f(x) = x^{2} at (0,0), and at
(1,1). Both times, if zoom in close enough, f looks like a line.
Which line? We already know a point on that line, so all that remains
is to figure out the slope. This turned out to be not so hard.


We figured out that the function
f: R^{2} → R^{2}
defined by
f(x,y) = (y,x)
rotates the plane 90 degrees
counterclockwise. What function rotates by 30 degrees instead?
Or in general, by x degrees?
Don't look this up online  try to
do it on your own!

Course Syllabus

Week 2: 1/16  1/20

Tuesday's Lecture:
 First midterm on Monday, January 30th!
 Considered the general problem of zooming in on the point
(a,a^{2}) in the function f(x) = x^{2}. If zoom in
enough on this point, f looks like a line. We figured out the equation
of this line. The hardest part of the process is finding the slope, so
we gave the slope a special name: the derivative of f at
a.
 Considered the same problem for a more general function f.
We gave a mathematical definition of the derivative of f at a, which
we denoted by f'(a).
 A few examples of derivatives, including g(x) = 5x  2 (at 2);
h(x) = sin x with x in degrees (at 0); and F(x) = sin x with x in
radians (at any point).
 Slick proof that sin x is continuous (including a derivation of
the formula for sin x  sin y).
Thursday's Lecture:
 Differentiability  when does a function not look
like a line, no matter how close you zoom in?
 The derivative of f(x) = x^{3} at every point.
 The derivative of f as a function in its own right. Second, third,
and higher derivatives.


Prove (without using any facts you don't know) the formula for sin(a+b).


Week 3: 1/23  1/27

Tuesday's Lecture:
 Proof that
lim_{x→a} f(x) =
lim_{h→0} f(a + h)
 Proof that if f(x) ≤ g(x), then
lim_{x→a} f(x) ≤
lim_{x→a} g(x)
 Continued discussing differentiability. In particular, proved that
if f is differentiable at a, then it must be continuous at a.
We also saw that the converse of this statement is false.
 By using the fact that the cosine function is simply a shift
of the sine function, we figured out the derivative of cos x.
Thursday's Lecture:
 Discussed the distinction between f'(2) and d/dx (f(2)).
More generally, the interpretation of constants as numbers or
functions.
 Discussed the difference between f'(cx) and d/dx (f(cx)). We
(almost finished) proving that d/dx (f(cx)) = c f'(cx).
 In the process of the above, we proved a useful lemma: if
lim_{h→0} f(h) exists, then for any constant c,
lim_{h→0} f(ch) =
lim_{h→0} f(h).

Chapter 9 # 16, 7(a) and 7(d), 810, 14, 16, 17, 22(a)



MONDAY, JAN. 30

Midterm Exam # 1 5:007:00 pm
If your last name begins with AY, your exam is in IC 130.
If your last name begins with Z, your exam is in IC 120.

Material covered
Limits, continuity, and differentiation

Midterm solutions

Study guide

Week 4: 1/30  2/3

Tuesday's Lecture:
 Let f(x) = sin x, where the sine function interprets x in
degrees.
We evaluated f'(0), and then more generally f'(a).
(Note that f'(a) is NOT cos a!) The common
theme was that when evaluating the derivative, we obtained a limit
with quantities which tend to 0 at different rates. To get around
this, we used two tricks. First, we multiplied by a funny form of
1 (i.e. we multiplied and divided by the same factor). Second, we
employed our `scaling' lemma from before: that if G(x) tends to a
limit L as x tends to 0, and c is any constant, then G(cx) tends
to the same limit L as x tends to 0.
 We used these same tricks to determine the derivatives of f(cx)
and, somewhat more difficult, of f(x^{2}).
 We then took these tricks to the next level and proved the Chain
Rule. Although the proof is somewhat more technical, the idea is
the same as in every other proof from today's lecture.
 Finally, you applied the chain rule to find derivatives of several
complicated functions.
Thursday's Lecture:

Study for the midterm exam!



Week 5: 2/6  2/10

Tuesday's Lecture:
 Discussed and proved the product rule
 Used this to establish that
d/dx (x^{a}) = a x^{a1}
for all a∈Q
 Discussed what 2^π might mean
 Proved the quotient rule
 Discussed maxima, minima, local maxima, local minima
 Proved that if g is defined on [a,b], has a max or min on (a,b) at
c, and is differentiable at c, then g'(c) = 0.
Thursday's Lecture:
 Continued discussing maxima and minima, by exploring
examples (e.g. 3x^{2}  24x + 50 and x + sin x).
 Outlined a general strategy for finding all local maxima and
minima of g(x) on [a,b]:
 Step 1: Determine all c in (a,b) for which g'(c) =
0
 Step 2: For each such c, determine whether g has a max,
a min, or neither at c. (We discussed a couple ways of
doing this: comparing values of g(c) to each other,
studying the behaviour of g' near c, ...)
 Step 3: Separately consider the behaviour of g at the
endpoints of the
interval.

Chapter 10 # 1, 2(i)(v) and (xvi), 6, 10, 11, 15, 17, 22, 24, 31



Week 6: 2/13  2/17

Tuesday's Lecture:

Suppose I'm traveling, and during the time interval between 2pm
and 3pm I travel 113 km. Then I definitely broke the speed limit
(100 km/hr) at some point. Actually, more is true: I must have
been going at 113 km/hr
at some point during the trip. This last statement is a special
case of what's known as the mean value theorem: that during any
trip, at some point during the trip you will be travelling at
precisely the same speed as your average speed during the course
of the trip.
Before discussing this in more mathematical terms, we had a short
digression on average velocity vs. instantaneous velocity. More
precisely: suppose I'm traveling from point A to point B in a
straight line, and s(t) outputs my position (i.e. my distance from
A) at time t (i.e. t hours into the trip). After some discussion,
we decided that my average velocity during the time interval from
t=a and t=b is [s(b)  s(a)] / [b  a], while my (instantaneous)
velocity at time t=c is s'(c).

Before getting to the formal version of the Mean Value Theorem,
we spent a substantial amount of time trying to prove the
following
Conjecture: If F'(x) = 0 on some nonempty interval, then F
must be constant on the interval.
We were unable to prove this conjecture. The main difficulty we
encountered was that F'(x) is written in terms of a certain limit,
and it is not at all obvious how to deduce anything precise about
the behaviour of F from the knowledge of this limit. Having
convinced ourselves that this problem is much harder than it
looks, we moved on to the Mean Value Theorem, which turns out to
be the key to proving the conjecture.

Mean Value Theorem: suppose F(x) is differentiable on the open
interval (a,b) and is continuous at both endpoints (i.e. F is
continuous on [a,b]).
Then there exists c in (a,b) such that F'(c) = [F(b)  F(a)] / [b  a].
The example above gives a physical interpretation of this
theorem, which is a good way to remember the statement. We noted
that continuity at the endpoints is a necessary hypothesis, since
otherwise the function defined by F(x) = 1 on (0,1] and F(0) = 0
would be a counterexample.

Before proving the MVT, we used it to resolve our earlier
conjecture. Indeed, suppose F'(x) = 0 for all x in some
nonempty interval I.
Given any a,b in I, the MVT guarantees the existence
of a c between a and b such that F'(c) = [F(b)  F(a)] / [b  a].
But F'(c) = 0 by hypothesis, whence F(a) = F(b). Since a and b were
arbitrary, we see that F must be constant.

As a second application of MVT, we proved that if
F'(x) > 0 for
all x in a nonempty interval I, then F is increasing on I.
(Recall: F is increasing means that
F(b) > F(a) whenever b > a.)
The proof is quite similar to the first corollary, so you should
try to reconstruct it on your own.

Philosophically, the point of the MVT is this. The definition of F'(x)
involves a limit of a certain quotient. Using the MVT, we can
remove the lim symbol... at a cost (we cannot control
where the derivative is taken).

Properly motivated, we now prove the MVT. We began with a special case
called Rolle's theorem: if h(x) is differentiable on (a,b) and
continuous on the endpoints, and if h(a) = h(b), then there exists
some c in (a,b) such that h'(c) = 0. The proof was not too hard.
By Theorem 72, h(x) has a max and a min on [a,b]. There are two
cases: either h(x) has a max or a min in (a,b), or else the
max and the min both occur at the endpoints. Both cases easily
lead to the conclusion of the theorem. In the former case, we know
the derivative vanishes at maxes and mins; in the latter case, we
saw that h(x) must equal h(a) for all x in [a,b].

Next, we proved the MVT by using a trick. We're given F(x) which is
differentiable on (a,b) and continuous at the endpoints. We
cannot apply Rolle's theorem to F because F(a) might not equal
F(b). Instead, we construct a function h(x) which mimics the
behaviour of F, but satisfies h(a) = h(b). How do we do this?
Write
h(x) = F(x) + (...) (xa), so that h(a) = F(a).
We wish to determine (...) in such a way that h(b) = h(a), i.e.
we want h(b) = F(a). A little thought shows that letting (...)
= F(a)  F(b) gives us what we want  that h(x) satisfies the
hypotheses of Rolle's theorem. It follows that there exists a
point c such that h'(c) = 0. Writing out h'(c) in terms of F and
F' yields the MVT.

Finally, we briefly discussed Cantor's Ternary Function, which is
a crazy continuous function C : [0,1] → [0,1] satisfying C(0)
= 0, C(1) = 1, and C'(x) = 0 whenever C'(x) is defined. (You
should convince yourself that this is pretty weird.)
Thursday's Lecture:
We sketched the graph of F(x) = (1+x^2)^{1} by using
derivatives. This involved the following steps:

Find all places where F'(x) = 0. (Just x=0.)

For each of these, determined whether F has a max, a min, or neither
there. (Since F'(x) > 0 for x < 0, and F'(x) < 0 for x > 0, we
must have that F is increasing to the left of 0, and decreasing to
the right of 0, i.e. F has a max at 0.)

Determined when F is positive and negative. (In this case, always
positive.)

Determined the longterm behaviour of F,
i.e. the limits as x → ± ∞
(in this case, both limits tend to 0).
 Sketch!

Next we discussed the Second Derivative Test; click
here
for a pdf.

Finally, we talked about L'Hôpital's Rule, discussed in
this pdf.

Chapter 11 exercises (starting on page 205)
# 1, 4(a) and (b), 5, 6, 7,
8, 11, 13, 15, 18, 21, 25, 29, 34



Reading week: 2/20  2/24

No classes

Catch up on the reading!



Week 7: 2/28  3/2

Tuesday's Lecture:
Thursday's Lecture:

Chapter 11 exercises (starting on page 205)
# 3, 23, 26, 28(a), 30, 36, 39, 41(a),
51, 52, 55, 64, 65, 66, 67(a). (If you have time, think about #57, but
this won't be on the quiz.)



Week 8: 3/5  3/9

Tuesday's Lecture:
Thursday's Lecture:

Study for the midterm!



FRIDAY, MAR. 9

Midterm Exam # 2 4:006:00 pm IC130

Material covered
Differentiation, maxima / minima, mean value theorem and related
results, inverse functions

Midterm solutions

Study guide

Week 9: 3/12  3/16

Tuesday's Lecture:
 Reviewed the calculation from last time: the integral from 0
to 5 of the function f(x) = x.
 Trick for quickly guessing the integral from 2 to 2 of the same
function f(x).
 Integrated the function g from 0 to 9, where g(x) = 2 everywhere
except at x=1, and g(1) = 5.
 New notation for integrals (to help tell what's the variable of
integration).
 Covered four basic properties of integrals:
 The integral on [a,b] equals the integral on [a,c] plus
the integral on [c,b];
 The integral of the sum is the sum of the integrals;
 The integral of Cf(x) equals C times the integral of
f(x); and
 If f is bounded between m and M on the interval [a,b], then
the integral of f on [a,b] is bounded between m(ba) and
M(ba).
 Discovered the fundamental theorem of calculus by looking at a
picture.
Thursday's Lecture:
 Stated and proved the "first form" of the fundamental theorem
of calculus
 Observed that a modification of the proof shows that the
function F(x), defined to be the integral from a to x of f, is
continuous even if f is not.
 Proved that if f is continuous and integrable on [a,b], and if
g' = f, then the integral from a to b of f equals g(b) 
g(a).

Read Chapters 13 and 14.



Week 10: 3/19  3/23

Tuesday's Lecture:
Thursday's Lecture:

Chapter 13 (starting on page 272) # 1, 5, 6, 7(i)(iv), 8(i)(iii) and
(vi), 11(a)(c), 1417, 20, 21, 25(a)(f). If you want a challenge,
try
7(vi).



SUNDAY, MAR. 25

Drop Deadline




Week 11: 3/26  3/30

Tuesday's Lecture:
Thursday's Lecture:

Chapter 13 # 2629, 3335.
Chapter 14 # 1(i),(iv),(v), 2(i),(ii),(iii), 3,
12, 13, 19, 25, 27



Week 12: 4/2  4/6

Tuesday's Lecture:
Thursday's Lecture:

Chapter 15 # 16, 17, 19
Chapter 18 # 1(ix) and (x), 2(a), 3, 5(vi), 6(i), 11, 17(a)(c),
20, 24
Chapter 19 # 1, 2, 3, 4(i)(iii), 5, 6, 12.



MONDAY, APR. 23

Final Exam 7:0010:00 pm
IC 130

Material covered
Calculus

Final Exam solutions

Study guide
