AP微积分BC公式大全

∞
∑ cn xn = c0 + c1x + c2x2 + c3x3 + ... + cnxn + ....(n = 0 to ∞)
n=0
(2)Taylor series
f (x) = f (a) + f '(a)(x − a) + f ''(a) (x − a)2 + ... + f (n) (a) (x − a)n + ...;
(2)Integral : If an = f (n) is positive, continuous and decreaing for x ≥ 1,
∞
∞
then∑ an and ∫1 f (x)dx both convergence or divergerence;
n=1
(3)Comparison : Let 0 ≤ an ≤ bn for all n,
f +′( x0
)
=
lim
x → x0+
f (x) − f (x0 ) x − x0
= lim ∆x→0+
f (x0 + ∆x) − ∆x
f (x0) ;
2. 求导公式和法则
(1)Derivative formulas of five basic elementry functions :
(xn
)
'
a
a
a
∫ f (x)dx = 0( f (x) odd ) −a
∫−a f (x)dx = 2∫0 f (x)dx( f (x)even)
5. 微积分两条基础理论
b
b
∫a f '(x)dx = f (b) − f (a) or f (b) = f (a) + ∫a f '(x)dx
dA( x) dx
1
1. 导数定义式
(1) f
'(x0 )
=
∆y lim ∆x→0 ∆x
=
lim
∆x→0
f
( x0
+ ∆x) − ∆x
f
(x0 )
(2)One − side :
f−′(x0
)
=
lim
x → x0−
f (x) − f (x0 ) x − x0
= lim ∆x→0−
f (x0 + ∆x) − ∆x
f (x0 ) ;
=
nxn−1,
(1 x
)
'
=
−
1 x2
,
(
x)'= 1 ; 2x
(ax ) ' = ax (ln a), (ex ) ' = ex;
1
1
(loga x) ' = x ln a , (ln x) ' = x ;
(sin x)′ = cos x, (cos x) ' = − sin x, (tan x) ' = sec2 x,
���� , vector(r(t)) ' =<
f '(t), g '(t) >
dx f '(θ ) cosθ − r sinθ
Chapter3. Integral
2
1. 不定积分定义式
definition∫ f '(x)dx = f (x) + C method ∫ df (x) = f (x) + C;
3. 反常积分的两种形式
(1)Integral on an Infinite Interval
+∞
b
∫ ∫ f (x)dx = lim f (x)dx
a
b→+∞ a
b
b
∫ ∫ f (x)dx = lim f (x)dx
−∞
a→−∞ a
+∞
+∞
+∞
∫ ∫ ∫ f (x)dx = f (x)dx + f (x)dx
(4)Logarithmic differentiation
(5)Inverse function dy = 1 dx dx dy
dy dy dt d 2 y d (dy dx) dt
parametric =
,
=
dx dx dt dx2
dx dt
polar dy =
f '(θ ) sinθ + rcosθ
∫ a[ f (x) ± g(x)]dx = ∫ a f (x)dx ±∫ a g(x)dx
b
b
∫ a kf (x)dx = k ∫ a f (x)dx
b
c
b
∫ a f (x)dx = ∫ a f (x)dx + ∫ c f (x)dx
a
∫a f (x)dx = 0
a
b
∫b f (x)dx = −∫a f (x)dx
4
Chapter4. Series
1. 级数的定义与收敛性 Definition :
∞
∑ (1) an = a1 + a2 + a3 + ..... + an + .....;m : Sn = an = a1 + a2 + a3 + ..... + an;
n=1
1
1 + x2
,
(arc
sec
x)′
=
|
x
|
1 , (arc cot x)′ = −
x2 −1
|x|
1 x2 −1
(2)Operation
:
(u
±
v)
'
=
u
'±
v
',
(uv)
'
=
u
'v
+
uv
',
(u v
)
'
=
u
'v − v2
uv
'
;
(3)Chain rule y = f [g(x)] ⇒ dy = dy ⋅ du dx du dx
1
x)x
=
e
x→∞
x
x→0
lim Pm (x) x→∞ Qn (x)
=
lim
x→∞
a0 xm + a1xm−1 b0 xn + b1xn−1
+⋯ + am−1x + am + ⋯ + bn−1x + bn
=
⎧ a0 ⎪⎪⎪⎨∞b0
⎪⎪0
(n = m)
(n > m) (n < m)
⎪⎩
Chapter2. Derivatives
∞
∞
If ∑bn converges, then ∑ an converges;
n=1
n=1
∞
∞
If ∑ an diverges, then ∑ bn diverges.
n=1
n=1
3. 四种重要的级数 Four series :
∑∞ 1 1 1 1
1
(1)Harmonic series : = + + + ....... + + .....(divergence);
∫ ∫ (4)Length of curve : L = x2 1+ (dy )2 dx = y2 1+ ( dx )2 dy
x1
dx
y1
dy
∫ or L = β x′2 (t) + y′2 (t)dt ( parametric equation) α
7. 微分方程
Differential equation :
2!
n!
Let f (x) = Pn (x) + Rn (x)
Pn (x) =
f (a) +
f
'(a)(x − a) + ... +
f
(n) (a) (x − a)n n!
Lagrange error bound
Rn (x) =
f (n+1) (ξ ) (x − a)n+1(ξ (n +1)!
2. 求不定积分的四种方法
∫ ∫ (1)Formulas : xndx = xn+1 + C, audu = 1 au + C
n +1
ln a
∫ sin xdx = − cos x + C, ∫ cos xdx = sin x + C,
∫ tan xdx = − ln | cos x | +C
∫ cot xdx = ln | sin x | +C, ∫ sec xdx = ln | sec x tan x | +C, ∫ csc xdx = − ln | csc x cot x | +C
AP 微积分公式大全
AP 微积分考试为闭卷考试，考试时也不给任何数学公式。因此，熟练掌握考试大 纲要求的微积分公式十分必要。为了帮助考生更快的掌握 AP 微积分的相关公式， 本文总结了 AP 微积分考试当中常用的重要公式，供大家参考学习。
Chapter1. Function and Limit
1. 5 种基本初等函数图像性质 Power y = xµ ; Exponential y = ax; Logarithmic : y = loga x Trigonometric : y = sin x / cos x / tan x / cot x / sec x / csc x Inverse trigonometricy = sin−1 x / cos−1 x / tan−1 x / cot−1 x / sec−1 x / csc−1 x