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Brief Introduction to Calculus 1
MechanicsMechanicsMath Prerequisites(I)
Calculus(微积分 )
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Brief Introduction to Calculus 2
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 );
2004-9-22 10:10-12:00
Brief Introduction to Calculus 3
MechanicsMechanicsMath Prerequisites(I)
Calculus(微积分 )
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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Brief Introduction to Calculus 4
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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Brief Introduction to Calculus 5
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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Brief Introduction to Calculus 6
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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Brief Introduction to Calculus 7
MechanicsMechanicsMath Prerequisites(I)
Calculus(微积分 )
2. Derivative(2. Derivative( 导数导数 ))
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Brief Introduction to Calculus 8
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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Brief Introduction to Calculus 9
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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Brief Introduction to Calculus 10
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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Brief Introduction to Calculus 11
2. Derivative2. Derivative
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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Brief Introduction to Calculus 12
2. Derivative2. Derivative
)()( 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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Brief Introduction to Calculus 13
2. Derivative2. Derivative
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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Brief Introduction to Calculus 14
2. Derivative2. Derivative
)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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Brief Introduction to Calculus 15
2. Derivative2. Derivative
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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Brief Introduction to Calculus 16
2. Derivative2. Derivative
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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Brief Introduction to Calculus 17
MechanicsMechanicsMath Prerequisites(I)
Calculus(微积分 )
3. Indefinite Integral (不定积分 )
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Brief Introduction to Calculus 18
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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Brief Introduction to Calculus 19
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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Brief Introduction to Calculus 20
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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Brief Introduction to Calculus 21
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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Brief Introduction to Calculus 22
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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Brief Introduction to Calculus 23
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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Brief Introduction to Calculus 24
MechanicsMechanicsMath Prerequisites(I)
Calculus(微积分 )
4. Definite integral(定积分 )
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Brief Introduction to Calculus 25
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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Brief Introduction to Calculus 26
4. Definite integral(4. Definite integral( 定积分定积分 ))
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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Brief Introduction to Calculus 27
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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Brief Introduction to Calculus 28
4. Definite integral(4. Definite integral( 定积分定积分 ))
n
iii
b
a nxfdxxf
1
)(lim)(
Limits of integration
an integrand
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Brief Introduction to Calculus 29
4. Definite integral(4. Definite integral( 定积分定积分 ))
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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Brief Introduction to Calculus 30
4. Definite integral(4. Definite integral( 定积分定积分 ))
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