Preface
This is part of the collection of lecture notes that I wrote for MATH1201 Calculus I (A) in
the academic year 19-20, which corresponds to 8 weeks (32 hours) of lectures. I would like
to acknowledge my TAs Yi-An Wu, Ting-Tsen Lin and Yu-Ming Chen for their important
contributions to the completion of these notes and many of my students for pointing out
the inaccuracies of the notes and for their constructive suggestions.
The theme of this part of the notes is on differentiation of functions in one real variable.
To be specific, we will define the derivative of a function, derive basic rules and techniques
of differentiation, analyse extrema of a function, discuss the statement and applications of
the Mean Value Theorem(s) and sketch the graph of a function.
Rather than getting into the subject from a highly theoretical perspective, the notes are
written in a concrete, example-based style as the author believes that maths is more concrete than one thinks and immersing oneself into a pool of explicit examples helps develop
a sense into the subject. For those students who seek a more theoretical approach to the
subject should consult other texts or literature.
Kwok-Wing Tsoi
1
Functions and their Limits
WEEK 1
Kwok-Wing Tsoi
1.1
Vocabulary of functions
In this course, we will study functions of one real variable.
Example. f (x) = sin(x)
Example. f (x) =
2
(
x
if x < 0,
x +1
2
if x ≥ 0
K.-W. Tsoi
Calculus 1 Class 07 (Year 113)
1.2
Exponential functions and their properties
In this section, we study functions of the form
f (x) = ax for some a > 0.
For example, the graph of y = 2x is shown in Figure 1.
Figure 1. The graph of y = 2x
Example. Sketch the graph for y =
x
1
.
2
Example. You are given the sketch of four exponential functions in the following
graph. Compare the size of a, b, c and d.
3
K.-W. Tsoi
Calculus 1 Class 07 (Year 113)
1.3
What is e - episode I
Definition 1.3.1. ‘e’ is the real number such that the slope of the tangent to y = ex
at (0, 1) equals to 1.
Remark. Some of you may have seen the following, perhaps, more ‘popular definition’
of the number e. We will discuss this in Week 3.
Definition 1.3.2 (Alternative definition).
The number ‘e’ is the limit of y =
1+
In other words,
1
x
x
when x tends to infinity.
e = lim
x→∞
1+
1
x
x
.
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Calculus 1 Class 07 (Year 113)
1.3.1
Natural logarithm ln(x)
Definition 1.3.3. The natural logarithm ln(x) is the ‘logarithm with base e’. i.e.
ln(x) = loge (x).
Example. Solve the equation ln(x) − ln(4x + 2) = 1.
K.-W. Tsoi
5
K.-W. Tsoi
Calculus 1 Class 07 (Year 113)
1.4
Limits - the basics
In this chapter, we develop the language of ‘limits’ that allows us to describe the asymptotic
behaviours of a function, which lies at the subject matter of Calculus.
1.4.1
Intuitive definition
Definition 1.4.1 (‘Definition’). We write lim f (x) = L if the values of f (x) get (arbix→a
trarily) close to L when x gets sufficiently close to a (from both sides) but not equal
to a.
x2 − 4
.
x→2 x − 2
Example. Compute the limit lim
(x + 2)(x − 2)
x2 − 4
= lim
= lim (x + 2) = 2 + 2 = 4
x→2
x→2
x→2 x − 2
x−2
Solution. lim
Example. Compute the limit limπ
x→ 2
1 + cos(2x)
.
cos(x)
Example. Compute the limit lim x tan cos−1 (x) .
x→0
6
K.-W. Tsoi
Calculus 1 Class 07 (Year 113)
1.4.2
One-sided limits
Definition 1.4.2 (‘Definitions’). We write
(a) lim+ f (x) = L if the values of f (x) get (arbitrarily) close to L when x approaches
x→a
to a from the right, but not equal to a.
(b) lim− f (x) = L if the values of f (x) get (arbitrarily) close to L when x apx→a
proaches to a from the left, but not equal to a.
Theorem 1.4.1. If lim+ f (x) = lim− f (x) and its value equals to L, then lim f (x)
x→a
exists and is equal to L as well.
Example. Let f (x) =
(
x
if x < 0
x +1
2
x→a
x→a
if x ≥ 0
. Compute lim+ f (x) and lim− f (x).
x→0
x→0
Solution.
lim f (x) = lim+ (x2 + 1)
x→0+
x→0
=1
lim f (x) = lim− x
x→0−
x→0
=0
Example (Absolute values). Compute the limits lim+
x→0
x
x
and lim−
.
|x|
x→0 |x|
7
K.-W. Tsoi
Calculus 1 Class 07 (Year 113)
x2 − 4x + 3
.
Example (Trap). Evaluate the limit lim− p
x→3
(x − 3)2
Example (Greatest integer function). Compute the limits lim+ JxK and lim− J−xK.
x→2
x→2
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K.-W. Tsoi
Calculus 1 Class 07 (Year 113)
1.5
9
Infinite limits and vertical asymptotes
Definition 1.5.1 (‘Definitions’). We write
(a) lim f (x) = ∞ if the values of f (x) get arbitrarily large positively when x apx→a
proaches to a, but not equal to a.
(b) lim f (x) = −∞ if the values of f (x) get arbitrarily large negatively when x
x→a
approaches to a, but not equal to a.
Analogously, we can define lim+ f (x) = ±∞ and lim− f (x) = ±∞. Indeed,
x→a
x→a
Definition 1.5.2. We say that x = a is a vertical asymptote for the graph y = f (x) if
at least one of the following is valid :
lim f (x) = ∞
x→a+
lim f (x) = −∞
x→a+
lim f (x) = ∞
x→a−
lim f (x) = −∞
x→a−
lim f (x) = ∞
x→a
lim f (x) = −∞.
x→a
Example (I. Logarithm). Consider the function f (x) = ln(x).
Example (II. Denominator explosion). Consider the function f (x) =
1
.
x2
Remarks.
The ‘precise’ definitions of infinite limits are discussed in
Worksheet 2.
Calculus 1 Class 07 (Year 113)
Example (III. Tangent). Consider the function f (x) = tan(x).
Using these, we can read off the vertical asymptotes of an arbitrary function easily.
1
Example. Let f (x) = ln 1 −
.
(x − 1)
Write down all the vertical asymptotes of y = f (x).
K.-W. Tsoi
10
K.-W. Tsoi
Calculus 1 Class 07 (Year 113)
1.6
Techniques of computing limits (I)
1.6.1
An important limit : lim
x→0
The following is a sketch of y =
sin x
x
sin(x)
.
x
sin(x)
=
x→0
x
Theorem 1.6.1 (Sine trick). lim
Proof. We will postpone the proof of this limit to a few pages later.
1.6.2
Skills 1 : Algebras of limits
Theorem 1.6.2 (Algebras of limits). Suppose both the limits lim f (x) and lim g(x)
x→a
x→a
exist. Then the following are valid.
(1) lim (f (x) ± g(x)) = lim f (x) ± lim g(x),
x→a
x→a
x→a
(2) lim (f (x) · g(x)) = lim f (x) · lim g(x),
x→a
x→a
x→a
lim f (x)
f (x)
= x→a
whenever lim g(x) ̸= 0.
x→a g(x)
x→a
lim g(x)
(3) lim
x→a
Example. Compute lim
x − sin(x)
x→0 x + tan(x)
.
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Calculus 1 Class 07 (Year 113)
1.6.3
Skills 2 : Rationalization
To compute limits that involve sums or differences of two functions that grow at a similar
rate, it is often useful to do so-called ‘rationalisation’ - the key identity here is
(A + B)(A − B) = A2 − B 2 .
√
Example. Compute lim
x→0
Solution.
√
lim
x→0
x2 + 16 − 4
.
x2
x2 + 16 − 4
= lim
x→0
x2
√
x2 + 16 − 4
x2
The technique of ‘rationalization’ also sheds light on computing the following limit.
Example. Compute lim
x→0
1 − cos x
.
x2
Solution.
lim
x→0
1 − cos x
1 − cos x
= lim
x→0
x2
x2
K.-W. Tsoi
12
K.-W. Tsoi
Calculus 1 Class 07 (Year 113)
√
3
Example (Cubic rationalization). Let b > 0. Compute the limit lim
x→0
1 + bx − 1
x
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