The differential calculus and its applications

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Chapter 2
The differential calculus and its applications
(for single variable)
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The idea of derivation is first brought forward
by French
mathematician Fermat.
The founders of calculus
Englishman Newton
German Leibniz
derivative
Describe the speed of change of the function
differential
differential
Describe the degree of the change of the function
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2.1 Concept of derivatives
1.Introduction examples
2.Definition of derivatives
3.One-sided derivatives
4.The derivatives of some elementary
functions 5.Geometric interpretation of derivative
6.Relationship between derivability and
continuity
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1.Introduction examples
Instantaneous velocity
Suppose that variation law of a moving object
is Then the average velocity in the interval (
) is
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The slope of a tangent line to a plane curve
the tangent line MT to
the limiting position of the secant line M N
The slope of MT
The slope of MN
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The common character
Instantaneous velocity
Slope of the tangent
The limit of the quotient of the increments
Similar question
acceleration
linear density
electric current
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2.Definition of derivatives
Definition2.1.1 .Suppose that
is defined in
if
is said to be derivable at ,and the limit is
then
exists,
called the derivative of
,denoted by
i.e.
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note
If the limit above does not exist ,
Especially,
we say the derivation of at
is infinite.
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Denoted by
note
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3.One-sided derivatives
If the limit
then the limit is called the right (left)
derivative of
exists,
denoted by
i.e.
For instance, for
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It is easy to know
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For instance,
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(5) for
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i.e.
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P109 T2(2) Suppose that
exists, find the limit
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Example
, find the value of a ,such that
exists in
and find
Solution
and
So when
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5.Geometric interpretation of derivative
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At the point
The tangent line
The normal line
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6.Relationship between and continuity and
derivability
Th2.1.1.
Notethe converse is not necessarily true.
is derivable on a , b
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• Derivative the rate of change

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Summarize
1. The definition of derivative
2.
3. Geometric meaning
slope of the tangent line
4. Derivable continuous
whether continuous
5. How to judge the derivability
by the definition
one-sided derivatives
6. Important derivatives
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Have a think
and
difference
is a function ,
is a value .
relation
attention
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2. If
exists, then
3. We have
then
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Spare questions
find
1. Suppose
exists and
Solu
so
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P112 T1. Suppose
exists,
prove
is continuous at
and
is derivable at