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Chapter 2

Chapter 2. The differential calculus and its applications (for single variable). The idea of derivation is first brought forward by French. mathematician Fermat. The founders of calculus:. Englishman : Newton. German :Leibniz. derivative.

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Chapter 2

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  1. Chapter 2 The differential calculus and its applications (for single variable)

  2. 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

  3. 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

  4. So the instantaneous velocity at is 1.Introductionexamples Instantaneous velocity Suppose that variation law of a moving object is Then the average velocity in the interval ( ) is

  5. (when ) 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:

  6. Instantaneous velocity The common character: Slope of the tangent The limit of the quotient of the increments Similar question: acceleration linear density electric current

  7. 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.

  8. note: If the limit above does not exist , Especially, we say the derivation of at is infinite.

  9. derived function of Denoted by : note:

  10. 3.One-sided derivatives If the limit exists, then the limit is called the right (left) derivative of denoted by i.e. For instance, for

  11. exists It is easy to know

  12. For instance，

  13. (5) for

  14. i.e.

  15. P109 T2(2)Suppose that exists, find the limit

  16. Example , find the value of a ,such that exists in and find Solution: and So when

  17. 5.Geometric interpretation of derivative

  18. At the point The tangent line: The normal line:

  19. 6.Relationship between and continuity and derivability Th2.1.1. Note：the converse is not necessarily true. is derivable on [a , b]

  20. Derivative the rate of change

  21. 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 :

  22. ？ Have a think and difference: is a function , is a value . relation: attention:

  23. exists, then 2. If 3. We have then

  24. Spare questions find 1. Suppose exists and Solu: so

  25. P112 T1.Suppose exists， prove: is continuous at and is derivable at

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