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v=x

t

v(t) =dr

dt

f : R R f(x) f df/dx x

f

(x0) =df(x)

dx |x0 = limh0

f(x0+h) f(x0)

(x0+h) x0

f f f(x) = x2

2 f(2) = 22 f : R R

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x0 f x0

f(x0+h) x0 f(x0)

x0

h

h x0 xo +h x0

f(x) = x2

f(x0) = limh0

f(x0+h) f(x0)(x0+h) x0

= limh0

(x0+h)2 x20(x0+h) x0

=

= limh0

(x20+ 2x0h+h2 x20

(x0+h) x0= lim

h0

2x0h+h2

h = lim

h0(2x0+h) = 2x0

2x0+h h

2x0 x2

2x

f, g: R R

(f g) =f g

(fg) =f g +fg

(f

g) =

fg fg

g2

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c = 0

(cx) =c

(xn) =nxn1

n N

n R

(sin(x)) =cos(x)

cos(x) = sin(x)

tg(x)

= 1 +tg(x)

(f(g(x))) =f(g(x)) g(x)

f(x) = sin(x3) x3 f(x) = Cos(x3) (x3) = 3x2Cos(x3)

f(x) =x3 7x 2 3x2 7 3x2 7 = 0 x=

7/3

f

f(x) = x2 f(x) = 2x 2x x2

f(x)

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F(x) = f(x)dxF(x) = f(x)

ba

f(x)dx

a b

Ox a b

ba

f(x)dx= F(b) F(a)

F f

f(x) = 2x x2 F(x) =x2 f(x)

31

2xdx= F(3) F(1) = 32 12 = 8

f(x)

f(x)

n [xi, xi+1]

xi = a+i x

x= (b a)/n

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xi xi+1 f(xi) x

A(f,a,b) =

ni=1

A(f, xi, xi+1) =

ni=1

x f(xi)

x 0

A(f,a,b) =

ba

f(x)dx= F(b) F(a)

v(t) = 2 (0, tmax) d= v tmax

v(t) =dx(t)

dt

dx(t) = v(t)dt

x

d=

tmax

0

dx(t) =

tmax

0

v(t)dt=

tmax

0

2dt= 2

tmax

0

dt= 2(tmax 0)

dx

t tmax

v(t) = t2

d=

tmax

0

dx(t) =

tmax

0

v(t)dt=

tmax

0

t2dt= t3

3|tmax0 =

tmax3

3

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t2 t3/3

=m

V

l

(x) =m

l

0 Lmax (x) = 2

(x) =dm(x)

dx

dm(x) = (x)dx

dm(x) Lmax

m=

Lmax

0

dm(x) =

Lmax

0

(x)dx=

Lmax

0

2dx= 2(Lmax 0)

(x) = Sin[x]

m=

Lmax

0

dm(x) =

Lmax

0

(x)dx=

Lmax

0

Sin(x)dx= Cos(x)|Lmax0 =Cos[Lmax]Cos[0] =Cos[Lmax

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M =J

M J

(r) = 8

J=

Lmax0

(r)r2dr =

Lmax0

r2dr= r3

3|Lmax0 =Lmax

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Lmax/2, Lmax/2

J=

Lmax/2

Lmax/2

(r)r2dr=

Lmax/2

Lmax/2

r2dr= r3

3|Lmax/2Lmax/2 =

=/3(Lmax3

8

(Lmax)3

8 ) =

Lmax3

12

(x,y,z) =dm(x,y,z)

dV

dV

dV = dxdydz

m=

xmaxxmin

ymaxymin

zmaxzmin

dm(x,y,z) =

xmaxxmin

ymaxymin

zmaxzmin

(x,y,z)dV

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(x,y,z) =constant= 0

Ox

J=

xmax

xmin

ymax

ymin

zmax

zmin

(x,y,z)x2dV =0

xmax

xmin

ymax

ymin

zmax

zmin

x2dxdydz

0 x2 y z

J=... = 0

xmax

xmin

x2

dx(

ymax

ymin

zmax

zmin

dydz)

xy

ymax

ymin

zmax

zmin

dydz = z|zmaxzmin y|ymaxymin = (zmax zmin)(ymax ymin) = S

S

J=0S

xmaxxmin

x2dx= 0Sx3

3 |xmaxxmin =0S(xmax xmin)(

xmax3 xmin3

3(xmax xmin))

=mxmax3 xmin3

3(xmax xmin)

S(xmaxxmin)

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