1. 2004-9-22 10:10-12:00Brief Introduction to Calculus1 Mechanics Math Prerequisites(I) Calculus ( 微积分 )

2. 2004-9-22 10:10-12:00Brief Introduction to Calculus2 Brief introduction to Calculus Calculus: Independently invented by Newton and Leibnitz; One of the important math tools used in physics study; In mechanics, the motion of a body can be conveniently described by using calculus; This lecture will give a very brief introduction to calculus: The derivative and differentiation( 导数和微分 ); Indefinite integral( 不定积分 ); Definite integral( 定积分 );

3. 2004-9-22 10:10-12:00Brief Introduction to Calculus3 Mechanics Math Prerequisites(I) Calculus ( 微积分 ) 1.Variable, constant and function ( 变量、常数和函数 )

4. 2004-9-22 10:10-12:00Brief Introduction to Calculus4 1. Variable, constant and function A quantity that can assume any of a set of values Examples: time, position of moving body, … A quantity that does not vary A function is something that associates each element of a set A with an element of another set B A B f xy  Variable:  Constant quantity:  Function:

5. 2004-9-22 10:10-12:00Brief Introduction to Calculus5 1. Variable, constant and function x : independent variable y : dependent variable Set A: domain of the function Set B: codomain of the function For example: the distance s traveled by a body is a function of time t : s = s(t) x : a variable whose values are elements of set A; y : a variable whose values are elements of set B;  y is called a function of x, denoted by y = f(x)

6. 2004-9-22 10:10-12:00Brief Introduction to Calculus6 1. Variable, constant and function y is a function of z, y = f(z), and z is a function of x : z = g(x), y is called a composite function of x : y =  (x)=f[g(x)] z = g(x) : the intermediate variable  Composite Function( 复合函数 ): Example: x = Acos  t, x is a composite function of t, and  t is the intermediate variable

7. 2004-9-22 10:10-12:00Brief Introduction to Calculus7 Mechanics Math Prerequisites(I) Calculus ( 微积分 ) 2. Derivative( 导数 )

8. 2004-9-22 10:10-12:00Brief Introduction to Calculus8 2. Derivative  Definition: Let y = f(x) be a function. The derivative of f with respect to x is the function whose value at x is the limit provided this limit exists. If this limit exists for each x in an open interval A, then we say that f is differentiable on A.

9. 2004-9-22 10:10-12:00Brief Introduction to Calculus9 2. Derivative  Notation: We have used the notation f ' to denote the derivative of the function f. There are also many other ways to denote the derivative If we consider y = f(x), then y' or denotes the derivative of the function f.

10. 2004-9-22 10:10-12:00Brief Introduction to Calculus10 2. Derivative For example:

11. 2004-9-22 10:10-12:00Brief Introduction to Calculus11 2. Derivative x y=f(x) P Q x x+  x yy Q xx  The geometric meaning of the derivative ： f ´(x) is the slope of the line tangent to y = f(x) at x. Let's look for this slope at P: The secant line through P and Q' has slope  y/  x We can approximate the tangent line through P by moving Q' towards P, decreasing  x. In the limit as  x  0, we get the tangent line through P with slope

12. 2004-9-22 10:10-12:00Brief Introduction to Calculus12 2. Derivative  Second Derivative( 二阶导数 ): If the derivative of f(x) is differentiable, then the derivative of f(x) with respect to x is called the second derivative of f(x), denoted by For example: Acceleration is the derivative (with respect to time) of an object’s velocity, and is the second derivative of the object’s position Velocity is the derivative (with respect to time) of an object’s position ;

13. 2004-9-22 10:10-12:00Brief Introduction to Calculus13 2. Derivative  Derivatives of some simple functions: c is constant n is a real number

14. 2004-9-22 10:10-12:00Brief Introduction to Calculus14 2. Derivative  Rules for computing derivatives: Assume u and v are functions of x c is constant  The Quotient Rule  The Product Rule

15. 2004-9-22 10:10-12:00Brief Introduction to Calculus15 2. Derivative 5. x =  (y) is the inverse function of y = f(x) 6. y = f(u), u =  (x), y is the composite function of x, y = f[  (x)]  The Chain rule( 链式法则 )

16. 2004-9-22 10:10-12:00Brief Introduction to Calculus16 2. Derivative  Extremum of a function( 函数的极值 ): The necessary condition for f(x) to have a minimum (maximum) at x 0 is f´(x 0 ) = 0 1.If f´´(x 0 ) > 0, then f has a minimum at x 0 ; 2.If f´´(x 0 ) < 0, then f has a maximum at x 0  Differentiation of a function( 函数的微分 ): dy : the differentiation of function y=f(x) at point x ; dx : the differentiation of variable x ; dy is proportional to dx

17. 2004-9-22 10:10-12:00Brief Introduction to Calculus17 Mechanics Math Prerequisites(I) Calculus ( 微积分 ) 3. Indefinite Integral ( 不定积分 )

18. 2004-9-22 10:10-12:00Brief Introduction to Calculus18 3. Indefinite Integral ( 不定积分 ) If the derivative of a function is given, how to determine this function?  Primitive( 原函数 ): A continuous function F(x) is called a primitive for a function f(x) on a segment X, if for each x  X F'(x) = f(x) Example: The function F(x) = x 3 is a primitive for the function f(x) = 3x 2 on the interval ( - , +  ), because F´(x) = (x 3 )´ = 3x 2 = f (x) For all x  ( - , +  )  Indefinite integral

19. 2004-9-22 10:10-12:00Brief Introduction to Calculus19 3. Indefinite Integral ( 不定积分 ) It is easy to check, that the function x 3 + 13 has the same derivative 3x 2, so it is also a primitive for the function 3x 2 for all x  ( - , +  ) It is clear, that instead of 13 we can use any constant. Thus, the problem of finding a primitive has an infinite set of solutions. This fact is reflected in the definition of an indefinite integral

20. 2004-9-22 10:10-12:00Brief Introduction to Calculus20 3. Indefinite Integral ( 不定积分 )  Definition of indefinite integral: Indefinite integral of a function f(x) on a segment X is a set of all its primitives. This is written as where C – any constant, called a constant of integration.

21. 2004-9-22 10:10-12:00Brief Introduction to Calculus21 3. Indefinite Integral ( 不定积分 )  Indefinite integrals of some elementary functions Assume C, a and n are all constant

22. 2004-9-22 10:10-12:00Brief Introduction to Calculus22 3. Indefinite Integral ( 不定积分 )  Rules for calculating indefinite integrals: 1.If a function f (x) has a primitive on a interval X, and k – a number, then 2.If functions f (x) and g(x) have primitives on a interval X, then

23. 2004-9-22 10:10-12:00Brief Introduction to Calculus23 3. Indefinite Integral ( 不定积分 ) 3.Integration by substitution ( exchange ): Then the function F( x ) = f [ g (x)] g' (x) has a primitive in Х and f (z) has a primitive at z  Z Function z = g(x) has a continuous derivative at x  X, and g(X)  Z

24. 2004-9-22 10:10-12:00Brief Introduction to Calculus24 Mechanics Math Prerequisites(I) Calculus ( 微积分 ) 4. Definite integral( 定积分 )

25. 2004-9-22 10:10-12:00Brief Introduction to Calculus25 4. Definite integral( 定积分 )  Concept: Suppose a particle moves along a straight line with velocity v(t), calculate the displacement s of the particle in the time interval from t 1 to t 2 If v(t) =constant: s = v(t 2 -t 1 ) If v(t) changes with t 1.Divide the time interval [ t 1, t 2 ] into n sub-intervals of an equal length o t t1t1 t2t2 v(t)v(t) tt v(i)v(i)

26. 2004-9-22 10:10-12:00Brief Introduction to Calculus26 4. Definite integral( 定积分 ) 4.At n  , s n  s Where  i is a point in the sub-interval 3.The displacement s approximately equals to the sum of the displacements in the n sub-intervals  Definite integration 2.In each of the sub-intervals, we approximate v as constant

27. 2004-9-22 10:10-12:00Brief Introduction to Calculus27 4. Definite integral( 定积分 )  Definition of definite integration: Consider a continuous function y = f (x), given on a interval [ a, b ]. Divide the interval [ a, b ] into n sub- intervals of an equal length by points: a = x 1 <x 2 <…x i <x i+1 <…x n+1 = b Let  x i = (b–a)/n = x i -x i-1 and  i  [x i, x i-1 ], where i=1,2,…, n. At n  , the limit of the sum is called an integral of a function f(x) from a to b or a definite integral

28. 2004-9-22 10:10-12:00Brief Introduction to Calculus28 4. Definite integral( 定积分 ) Limits of integration an integrand

29. 2004-9-22 10:10-12:00Brief Introduction to Calculus29 4. Definite integral( 定积分 )  Geometric meaning: Gives the area of a curvilinear trapezoid bounded by a graph of function f(x), a segment [ a, b ] and straight lines x = a and x = b  Properties of definite integration:

30. 2004-9-22 10:10-12:00Brief Introduction to Calculus30 4. Definite integral( 定积分 )  Newton – Leibniz formula( 牛顿 - 莱布尼茨公式 ): if F (x) is primitive for the function f (x) on a interval [ a, b ], then