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Chapter 4 Transcendental Functions Outline 1. Exponential functions 2. Logarithmic functions 3. Hyperbolic functions (Read)
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1. Exponential functions
Consider the function .
The above numbers comprise the set of all rational numbers denoted by .
All other real numbers are irrational numbers.
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On the left, the graph has gaps at irrational numbers. By filling the gaps, i.e.
we get a continuous function
It is called the exponential function and is called the base.
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Exponential Function Let be a constant.
If
we define
If is an irrational number we define
The continuous functions
is called the exponential function with base . Note that
Dom and Rng .
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Remark
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Algebraic Properties
Proof From elementary math, all the above identities are true when are rational numbers.
They are true when are irrational numbers by limit laws.
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Euler Constant
There is a constant between 2 and 3 such that the tangent line to the graph of
at has slope 1. It is called the Euler constant. Note that .
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EX Find the following limits
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By the definition of the Euler constant , we have
This is the same as
Derivative formula of
So if is a function of then
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Proof
where we have used the fact that
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EX Differentiate the functions
and
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EX Find if
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Integration formula of
Proof Since
, it follows by the
definition of indefinite integral that
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EX Evaluate the integral
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EX (Substitution rule) Evaluate the indefinite integrals
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2. Logarithmic Functions
For each constant with or , the function
is one-to-one so it has the inverse.
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Def The inverse of is the continuous function denoted
It is called the log(arithmic) function with base .
In the special case , it is denoted
and is called the natural log function.
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Remark has
Dom and Rng observe that
If then
If then
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Properties
Remark The identity
gives rise to the so-called taking log method for solving equation.
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EX (Some prelim. algebra) Evaluate the limit
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EX (Taking log) If
then
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Derivative formula of
Proof We only have to prove the first formula. The rest follows from the chain rule and that .
Let . Then
so the desired formula is true.
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EX Find the derivative of
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EX (Logarithmic differentiation) Evaluate the derivative if
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Derivative formula of
Proof We employ log differentiation:
So
The other formula follows from the chain rule.
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EX Evaluate the derivative of
Remark Do not use the diff. formula for power functions:
or the formula for exp functions:
to differentiate the function
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Integration formula of
Proof We have to show that
If then
If then
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EX (Substitution rule) Evaluate the integrals
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EX (Prelim. algebra & Substitution rule) Evaluate the integrals
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Integration formula of
Proof We have to show that
The left hand side is equal to
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EX Evaluate the integrals