Calculus 1, Chapter 1 Fall 2021
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Table des matières
Chapter 1: Functions of one real variable: differentiability and related topics ...................................... 1
1. Inverse functions ............................................................................................................................. 1
1.1 Bijections (bijective functions, invertible functions) ............................................................. 1
1.2 Continuous bijection ................................................................................................................. 3
1.3 Examples of inverse functions ................................................................................................ 3
2. Differentiability ................................................................................................................................ 5
2.1. Differentiability at a point ........................................................................................................ 5
2.2 Digression: Variables vs functions ............................................................................................. 7
2.3. Applying derivatives to computation of limits ....................................................................... 10
Chapter 1: Functions of one real variable: differentiability and related
topics
1. Inverse functions
1.1 Bijections (bijective functions, invertible functions)
Definition: a bijection is a function between the elements of
two sets, where each element of one set is paired with exactly
one element of the other set, and each element of the other set
is paired with exactly one element of the first set. There are no
unpaired elements.
Definition 1.1.1 (formal):
a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a
set Y:
X such that
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An injective
non surjective mapping
An injective surjective
mapping (bijection)
A non-injective
surjective mapping
A non-injective non-
surjective mapping
Detecting injectivity on a graph:
This graph does not pass the horizontal line test
Theorem 1. 1. 1 and definition 1. 1. 2: Let f be a bijection from X to Y. Then there exists a unique
function denoted f -1, called the inverse of f, defined on X with values in Y, such that
.
(We can rewrite that statement in terms of functions: , , where the
identity mapping is defined by )
Non-injective, because there are two pre-images for any strictly positive
number:
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1.2 Continuous bijection
Theorem 1. 2. 1: Any function continuous on an interval I,
with values in an interval J and strictly increasing
(respectively decreasing) on I is a bijection from I to J.
Furthermore, its inverse is a continuous function, too, and it
is strictly increasing (respectively decreasing) on J
Theorem 1. 2. 2 : In a Cartesian coordinate system, the
curves of a bijection and its inverse are symmetric to each
other with respect to the diagonal line y = x
Here,
(blue line),
(red line)
1.3 Examples of inverse functions
I’ll give you a couple of classical examples of inverse functions. The second example leads to a
«new» function, non-reduceable to ordinary algebraic or trigonometric functions or xponentials.
We’ll study the properties of some of such «new» functions below.
The demonstration of their existence (a consequence of theorem 1. 2. 1) is left as an exercise for
the students.
Example 1: n-th root
Definition 1. 3. 1 : Let n be a natural nonzero number. The
function defined on by is a bijection from to
, which results in its being invertible. Its inverse is called
the n-th root, denoted
.
Remark: we also use the notation
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Example 2: arctan
Definition 1. 3. 2 : The function tan is is a bijection from
to , so that it is invertible. Its inverse function is
called arctangent, denoted arctan (in French: also arctg; in
English: also atan and ).
Properties of arctan
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2. Differentiability
2.1. Differentiability at a point
Definition 2. 1. 1: Let f be a function defined in a neighborhood of a real number a. f is differentiable
at a if there exists a real number called the derivative of f at a, denoted (so that )
such that:
This can also be written as
Theorem 2. 1. 1 and definition 2. 1. 2: Let f be a function defined in a neighborhood of a. Then the
following two statements are equivalent:
1. f is differentiable at a and its derivative at a is ,
2. in a neighborhood of a, the following equality holds:
That equality is called a polynomial approximation of order 1 (a linear approximation) of at a.
Proof (obtaining an explicit expression for ):
is differentiable at and its derivative at equals ℓ:
)=0
Consider a function defined at a neighborhood of a by ()=
. Then
()=()+(−)
+(−)() and
()=0. QCD
Remark: One can define in a similar way the notion of differentiability on the right and on the left of
a, using
and
Theorem 2. 1. 2: Any function differentiable at a point is continuous at that point.
Remark: The converse of that theorem is false:
Theorem 2. 1. 3: Let f be a function differentiable at a, with the derivative at a equal to . Let us
denote Cf the curve of f in an X-Y Cartesian coordinate system. Then the slope of the tangent line to
Cf drawn at the point whose X-coordinate is a, is .
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