2004-9-22 10:10-12:00brief introduction to calculus1 mechanics math prerequisites(i) calculus (...

2004-9-22 10:10-12:00

Brief Introduction to Calculus 1

MechanicsMechanicsMath Prerequisites(I)

Calculus(微积分 )

2004-9-22 10:10-12:00


Brief introduction to CalculusBrief introduction to Calculus• Calculus:

• Independently invented by Newton and Leibnitz;• One of the important math tools used in physics st

udy;

• 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 );

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1.1. Variable, constant and functionVariable, constant and function(( 变量、常数和函数变量、常数和函数 ))

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1. Variable, constant and function1. Variable, constant and functionA 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 Bf

x y

Variable:Variable:

Constant quantity:Constant quantity:

Function:Function:

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1. Variable, constant and function1. Variable, constant and function

x: independent variabley: dependent variable

Set A: domain of the functionSet 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)

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1. Variable, constant and function1. 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(Composite Function( 复合函数复合函数 ))::

Example: x = Acos t, x is a composite function of t , and t is the intermediate variable

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2. Derivative(2. Derivative( 导数导数 ))

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2. Derivative2. DerivativeDefinition: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

xy

xxfxxfxf

xx

00lim)()(lim)(

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.

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2. Derivative2. DerivativeNotation:Notation:

dxdf

dxdyxf )(

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.

•

y

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2. Derivative2. Derivative

xx

x

xx

x

xxxxx

x

exxe

xyxf

xexey

xex

eeeexfxxfy

exf

0lim)(

)11(

11

)1()()(

)(

For example:

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x

y=f(x)

P

Q

x x+x

y

Q

x

The geometric meaning of the derivativeThe 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/xWe 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

)(lim0

xfxy

x

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)()( 2

2

xfdxdy

dxydxf

Second Derivative(Second Derivative( 二阶导数二阶导数 ):):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 ;

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1

22

22

1

)ln().(lg7sin

1csc).(6

cos1sec).(5

sin).(cos4cos).(sin3

).(2

0).(1

axxx

xctgx

xxtgx

xxxx

nxx

c

a

nn

)(,)1().(14

)(,)1().(13

)11(,1).(arccos12

)11(,1).(arcsin11

).(10

ln).(9

).(ln8

12

12

12

12

1

xxxarcctg

xxxarctg

xxx

xxx

ee

aaa

xx

xx

xx

Derivatives of some simple functions:Derivatives of some simple functions:

c is constantn is a real number

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)0(,.4

)(.3;)(.2

)(.1

2

vvuvvu

vu

uccuuvvuuvvuvu

Rules for computing derivatives:Rules for computing derivatives:Assume u and v are functions of x

c is constant

The Quotient Rule

The Product Rule

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0)()(

1)(

yy

xf

5. x = (y) is the inverse function of y = f(x)

dxdu

dudy

dxdy

6. y = f(u), u = (x), y is the composite function of x, y = f[(x)]

The Chain rule(链式法则 )

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dxxfdyxfdxdy )()(

Extremum of a function(Extremum of a function( 函数的极值函数的极值 ):):The necessary condition for f(x) to have a minimum (maximum) at x0 is f´(x0) = 0

1. If f´´(x0) > 0, then f has a minimum at x0;2. If f´´(x0) < 0, then f has a maximum at x0

Differentiation of a function(Differentiation of a function( 函数的微分函数的微分 ):):

dy: the differentiation of function y=f(x) at point x;dx: the differentiation of variable x;dy is proportional to dx

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3. Indefinite Integral (不定积分 )

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3. Indefinite Integral (不定积分 )If the derivative of a function is given, how to determine this function?

Primitive(Primitive( 原函数原函数 ):):A continuous function F(x) is called a primitive for a function f(x) on a segment X,if for each xX

F'(x) = f(x)Example: The function F(x) = x3 is a primitive for the

function f(x) = 3x2 on the interval ( -, + ) , because

F´(x) = (x3)´ = 3x2 = f (x) For all x ( -, + )

Indefinite integral

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3. Indefinite Integral (不定积分 )It is easy to check, that the function x3 + 13 has the same derivative 3x2, so it is also a primitive for the function 3x2 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

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3. Indefinite Integral (不定积分 )Definition of indefinite integral: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

CxFdxxf )()(

where C – any constant, called a constant of integration.

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3. Indefinite Integral (不定积分 )

Cedxe

aaCaadxa

Cxdxx

nCnxdxx

Caxadx

Cdx

xx

xx

nn

.6

)1,0(,ln

.5

ln1.4

)1(,1

.3

.2

0.1

1

Caxarctg

adx

xa

Caxdx

xa

Cctgxxdxdxx

Ctgxxdxdxx

Cxxdx

Cxxdx

11.12

arcsin1.11

cscsin

1.10

seccos

1.9

sincos.8

cossin.7

22

22

22

22

Indefinite integrals of some elementary functionsIndefinite integrals of some elementary functionsAssume C, a and n are all constant

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3. Indefinite Integral (不定积分 )Rules for calculating indefinite integrals:Rules for calculating indefinite integrals:

1. If a function f (x) has a primitive on a interval X , and k – a number, then

dxxfkdxxkf )()(

2. If functions f (x) and g(x) have primitives on a interval X , then

dxxgdxxfdxxgxf )()()()(

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3. Indefinite Integral (不定积分 )3. Integration by substitution ( exchange ):

dzzfdxxgxgfdxxF )()()]([)(

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

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4. Definite integral(定积分 )

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4. Definite integral(4. Definite integral( 定积分定积分 ))

nttt 12

Concept: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 t1 to t2

• If v(t)=constant: s = v•(t2-t1)• If v(t) changes with t

1. Divide the time interval [t1, t2] into n sub-intervals of an equal length

o tt1 t2

v(t)

t

v(i)

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4. At n, sn s

n

iinnntvss

1

)(limlim

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

n

iii

n

in tvsss

11

)(

Definite integration

2. In each of the sub-intervals, we approximate v as constant

tvs ii )(

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4. Definite integral(4. Definite integral( 定积分定积分 ))Definition of definite integration: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 = x1<x2<…xi<xi+1<…xn+1 = b

Let xi= (b–a)/n = xi -xi-1 and i [xi , xi-1], where i=1,2,…, n. At n , the limit of the sum

n

iiin xfI

1

)(

is called an integral of a function f(x) from a to b or a definite integral

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n

iii

b

a nxfdxxf

1

)(lim)(

Limits of integration

an integrand

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b

c

c

a

b

a

b

a

b

a

b

a

b

a

a

b

a

b

b

a

dxxfdxxfdxxf

bccaba

dxxgdxxfdxxgxf

kdxxfkdxxfk

dxxfdxxf

)()()(

],[],,[],.[4

)()()]()([.3

)0(,)()(.2

)()(.1

Geometric meaningGeometric 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 integrationProperties of definite integration::

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b

a

b

axFaFbFdxxf )()()()(

Newton – Leibniz formula(Newton – Leibniz formula( 牛顿牛顿 -- 莱布尼茨公式莱布尼茨公式 ))::if F (x) is primitive for the function f (x) on a interval [a, b ] , then

2004-9-22 10:10-12:00brief introduction to calculus1 mechanics math prerequisites(i) calculus (...

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