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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