MA1521 Calculus for Computing

Lecture 1

Wong Yan Loi

National University of Singapore

January 11, 2022

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Assessment

1 Final exam (open book via Zoom proctoring) - 50%

2 Midterm (open book via Zoom proctoring) - 35% (schedule on the
first lecture of week 8 (i.e. 8 March 2022, 8-10am). Material to be
tested includes chapter 1 to 5, and tutorial 1 to 5.)
3 Three assignments - each 5% (due on week 4,9,12 Wed noon)

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1 All learning material will be uploaded at Luminus.
2 Learning schedule is at Luminus.
3 Textbook: Thomas’ Calculus, Weir and Hass, Fourteenth edition.
2018 Pearson, ISBN 9780134438986

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Table of Contents

1 Real Numbers and Functions

2 Absolute Value

3 Functions

4 Polynomials

5 Rational Functions

6 Trigonometric Functions

7 Exponential and Logarithmic Functions

8 The Range of a Function

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Chapter 0: Real numbers and functions
Read Thomas’ Calculus, Chapter 1.

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 Real Numbers and Functions

The collection of all real numbers is denoted by R. Thus R includes the

integers
. . . , −2, −1, 0, 1, 2, 3 . . . ,

√ where p and q are integers (q 6= 0), and the

the rational numbers, p/q,
irrational numbers, like 2, π, e, etc.

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 Real Numbers and Functions

a ∈ R means a is a member of the set R. In other words, a is a real

number.

Given two real numbers a and b with a < b, the closed interval [a, b]
consists of all x such that a ≤ x ≤ b, and the open interval (a, b)
consists of all x such that a < x < b.

Similarly, we may form the half-open intervals [a, b) and (a, b].

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 Absolute Value

The absolute value of a number a ∈ R is written as |a| and is defined

as 
a if a ≥ 0
|a| =
−a if a < 0.
For example, |2| = 2, | − 2| = 2.

Some properties of |x| are summarized as follows:

1 | − x| = |x|, for all x ∈ R.
2 |xy| = |x||y |, for all x, y ∈ R.
3 −|x| ≤ x ≤ |x|, for all x ∈ R.
4 For a fixed r > 0, |x| < r if and only if x ∈ (−r , r ).
√
5 x 2 = |x|, x ∈ R.
6 (Triangle Inequality) |x + y | ≤ |x| + |y| for all x, y ∈ R.

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 Absolute Value

Example
2x − 1
Solve the inequality < 1.
2x + 1

Solution.
2x − 1
<1
2x + 1
2x − 1
⇔0 < 1 −
2x + 1
2x + 1 − 2x + 1
⇔0 <
2x + 1
2
⇔0 <
2x + 1
⇔ 0 < 2x + 1
1
⇔ − < x.
2


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 Absolute Value

Example
Solve the inequality |x + 1| ≤ |2x − 1|.

Solution.

|x + 1| ≤ |2x − 1|
⇔ |x + 1|2 ≤ |2x − 1|2
⇔ x 2 + 2x + 1 ≤ 4x 2 − 4x + 1
⇔ 0 ≤ 3x 2 − 6x
⇔ 0 ≤ 3x(x − 2)
⇔ x ≤ 0 or x ≥ 2
⇔ x ∈ (−∞, 0] ∪ [2, ∞).

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 Absolute Value

Exercise
Let r > 0. Prove that |x − a| < r if and only if x ∈ (−r + a, a + r ).

Exercise
Prove the triangle inequality |x + y| ≤ |x| + |y|.

Exercise
Prove that for any x, y ∈ R, ||x| − |y|| ≤ |x − y|.

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 Absolute Value

Example
Prove the triangle inequality |x + y| ≤ |x| + |y|.

Solution.

|x + y| ≤ |x| + |y|
⇔ |x + y|2 ≤ (|x| + |y|)2
⇔ (x + y)2 ≤ (|x| + |y |)2
⇔ x 2 + y 2 + 2xy ≤ |x|2 + |y |2 + 2|x||y|
⇔ x 2 + y 2 + 2xy ≤ x 2 + y 2 + 2|x||y|
⇔ xy ≤ |x||y |
⇔ xy ≤ |xy|.

Since the last inequality xy ≤ |xy| is true, we have proved the triangle
inequality.


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 Functions

A function f : A −→ B is a rule that assigns to each a ∈ A one specific

member f (a) of B. Symbolically we may denote the function by
a 7→ f (a). We can specify a function f by giving the rule for f (x).

Example
f (x) = x 2 /(1 − x) assigns the number x 2 /(1 − x) to each x 6= 1 in R.

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 Functions

The set A is called the domain of f and B is the codomain of f .

The range of f is the subset of B consisting of all the values of f . That
is, the range of f = {f (x) ∈ B | x ∈ A}.
Given f : A −→ R, it means that f assigns a value f (x) in R to each
x ∈ A.
Such a function is called a real-valued function.

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 Functions

For a real-valued function f : A −→ R defined on a subset A of R, the

graph of f consists of all the points (x, f (x)) in the xy-plane.

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 Functions

If f : A → B and g : B → C, then the composite function of f and g is

the function g ◦ f : A → C given by g ◦ f (x) = g(f (x)).

Example
1
Let f (x) = x and g(x) = x 2 − 1. Find g ◦ f and f ◦ g.

Solution. g ◦ f (x) = g(f (x)) = g( x1 ) = ( x1 )2 − 1 = 1

x2
− 1.

1
f ◦ g(x) = f (g(x)) = f (x 2 − 1) = x 2 −1
.


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 Functions

Let f : A → B. If g : B → A is a function such that f (g(x)) = x for all

x ∈ B and g(f (x)) = x for all x ∈ A, then g is called the inverse of f .
Similarly, f is the inverse of g.
The inverse function of f is usually denoted by f −1 .
Let f : A → B.
f is called an injective function if for any x, y ∈ A, f (x) = f (y) ⇒ x = y.
f is called a surjective function if for any z ∈ B, there is an x ∈ A such
that f (x) = z.
f is called a bijective function if f is injective and surjective.

Exercise
Prove that if f −1 exists, then f is a bijective function.

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 Functions

The graph of f −1 is obtained by reflecting the graph of f about the line

y = x.

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 Polynomials

A function of the form p(x) = an x n + an−1 x n−1 + · · · + a1 x + a0 , where

a0 , . . . , an are constants, is called a polynomial of degree n.
For example, a quadratic function p(x) = ax 2 + bx + c is a polynomial
of degree 2.
A polynomial of degree n can be factored as a product of linear and
quadratic factors.
For example, x 4 − 1 = (x 2 + 1)(x + 1)(x − 1).
In general a polynomial of degree n has at most n real roots.
For example, x 4 − 1 has only two real roots −1 and 1.

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 Rational Functions

p(x)
A rational function is a function of the form , where p(x) and q(x)
q(x)
p(x)
are polynomials. The domain of consists of all real numbers
q(x)
except the roots of q(x).

x3 + 3
For example, the domain of is R \ {−1, 1}.
x4 − 1

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 Trigonometric Functions

The 6 trigonometric functions are sin x, cos x, tan x, csc x, sec x, cot x.
They are periodic functions of period 2π.

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 Trigonometric Functions

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 Trigonometric Functions

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 Trigonometric Functions

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 Trigonometric Functions

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 Trigonometric Functions

Exercise
Sketch the graph of csc(x).

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 Exponential and Logarithmic Functions

A function of the form f (x) = ax , where a > 0 is called an exponential

function. It’s inverse function, denoted by loga x is called the
logarithmic function to the base a.

Let e = 2.718281828459045235360287 be the Euler constant. Then

the inverse of the exponential function ex is the natural logarithm ln x.

We have eln x = x for x > 0 and ln ex = x for all x.

The domain of ex is R and the range is the set R+ of all positive real
numbers.

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 Exponential and Logarithmic Functions

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 Exponential and Logarithmic Functions

Example
4
Sketch the graph of y = x+2 − 3.

Solution. The domain of the function y is R \ {−2}.

When x > −2 and close to −2, the value of y is large and tends to
positive infinity.
When x < −2 and close to −2, the value of y is large and tends to
negative infinity.
4
When x is large and positive, the term x+2 is positive and small, the
value of y is bigger than −3 but close to −3.
4
When x is large and negative, the term x+2 is negative and small, the
value of y is smaller than −3 but close to −3.
Also if x2 > x1 , then x24+2 − 3 < 4
x1 +2 − 3. That is y2 < y1 , meaning y is
a decreasing function.
With this information, we can sketch he graph of y as follow.
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 Exponential and Logarithmic Functions


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 The Range of a Function

In general it is not so easy to determine the range of a function. In

some simple cases, basic algebraic techniques can be used to find the
range of a function.

Example
1
Find the maximal domain and the range of f (x) = x−1 .

Solution. The maximal domain of f is R \ {1}.

Recall that the range of f = {f (x) ∈ R | x 6= 1}.
1
To find the range of f , let y = f (x). That is y = x−1 . Solving for x, we
1
get x = 1 + y . From this we see that if y 6= 0 then we may choose
1
x =1+ y to get f (x) = y . Thus the range of f is R \ {0}.


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 The Range of a Function

Example
Find the maximal domain and range of f (x) = x 2 − x + 1.

Solution. The maximal domain of f is R.

To find the range of f , let y = f (x). That is y = x 2 − x + 1. Solving for
x, we get
x = 12 (1 ± 1 − 4(1 − y)). That is x = 21 (1 ± 4y − 3)).
p p

From this we see that if y ≥ 43 then we may choose

x = 21 (1 ± 4y − 3)) to get f (x) = y. Thus the range of f is [ 34 , ∞).
p

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 The Range of a Function

Exercise
Let f (x) = x + 5 and g(x) = x 2 − 3. Find the maximal domain and
range of g(f (x)).

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 The Range of a Function

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 The Range of a Function

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 The Range of a Function

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