cos(2x) cos(x) cos(x) sin(x) cos(3x) sin(x)3
222
f(x) = 5x7+ 3x34x2+ 9x+ 5 g(x) = xex
h(x) = 6x+ 5
x2+ 1 i(x) = cos x
j(x) = sin(x) log(x)k(x) = qp1 + x2
l(x) = 6 sin x+ 5
(sin x)2+ 1 m(x) = 1/xx
f(x) = xx+x1x > 0f(0) = 0 fR+
E
1
f(x) = E(x)2
g(x) = (x1)E(x)
h(x) = (x1)2E(x)
f(x) = (xE(x)) sin πx
2xR
f
n f n
f
f:RR0f(0) = 0 f0(0) = 2
f(sin x)
xx0
nlimx1f(xn1)f(x1)
x1
limx0f(2x)2f(3x2)
x2
xR
+\ {1}log x < x 1
u(x) = xev(x) = exx > 0
u v R
+log y=x/e
u v
Py=x2
y=ax +bP4b+a2= 0
D1D2Pa16=a2
a1a2D1D2
Py=1/4
f:RRf(x) = 2x3+ 4x210x+ 9
f(2) f(3) f0
6x2+ 8x10 = 0 ] 2,3[
g:x7→ exsin(x)R+
x g(x) = 0 g0
sin(x) = 2xcos(x) [0,4]
P P (x) = xn+ax +b n N\ {0,1}a b R
P0(x) = 0 n a
f:RRk k 2
f0k1
P n n
a f : [a, +[R[a, +[ ]a, +[
limx→∞ f(x) = f(a)c]a, +[f0(c) = 0
g: [0, ea]Rg(0) = f(a)g(x) = f(log x)x]0, ea]
g[0, ea] ]0, ea[
g0g(0) g(ea)
c]a, +[f0(c) = 0
f: ]0,1[R
limx0f(x) = limx1f(x) = +
a < b f : [a, b]RGff
xRPx(x, 0)
a= 0 b= 1 f(x) = x(x1) f(a) = f(b) = 0 Gf
GfP1
2P1
2P3
f f(a) = f(b) = 0 c /[a, b]
GfPc
f[a, b]x0[a, b]
Gf(x0, f(x0)) Pcf0(x0)(x0c)f(x0) = 0
f(a) = f(b) = 0
f0(x)(xc)f(x)a b
c /[a, b]c=a
f: [0,2π]Rf(x) = ecos(2x)sin(x)
cRGfPc
Hn=Pn
k=1
1
kRn=Pn
k=1
1
2knN
x > 01
x+ 1 <log(x+ 1) log x < 1
x
log nHnlog n+ 1
x1x+ 1 x < 1
2x<xx1
Rn2
a b
c a3b3= 3c2(ab)
a2+ab +b2a=b= 0
f:IRKR|f(x)| ≤ K x I
f:RRf0
K0|f(x)f(0)| ≤ K|x|xR
K≥ |f(0)|
x7→ f(x)
1 + |x|
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