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AP 微积分核心词汇
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AP 微积分核心公式
1.1 Function and Limit
1. 五种基本初等函数图像性质
Power: y = x
n
Exponential: y = a
x
Logarithmic: y = log
a
x
Trigonometric: y = sin x, cos x, tan x, cot x, sec x, csc x
Inverse trigonometric: y = sin
-1
x, cos
-1
x, tan
-1
x, cot
-1
x, sec
-1
x, csc
-1
x
2. 四种表达函数的解析式
Standard: y = f(x)
Parametric: { x = f(t), y = g(t) }
Polar: r = f(θ)
Vector: r(t) = ⟨f(t), g(t)⟩
3. 三个重要极限
lim
x→0
sin x
/
x
= 1
lim
x→∞
(1 +
1
/
x
)
x
= e
lim
x→∞
a
m
x
m
+ ...
/
b
n
x
n
+ ...
= {
a
m
/
b
n
(if m=n), 0 (if n>m), ∞ (if n
1.2 Derivatives
1. 导数定义式
f'(x
0
) = lim
Δx→0
f(x
0
+Δx) - f(x
0
)
/
Δx
f'(x
0
) = lim
x→x
0
f(x) - f(x
0
)
/
x-x
0
2. 求导公式和法则
(x
n
)' = nx
n-1
(a
x
)' = a
x
ln a
(sin x)' = cos x
(cos x)' = -sin x
(tan x)' = sec
2
x
(cot x)' = -csc
2
x
(sec x)' = sec x tan x
(csc x)' = -csc x cot x
(ln x)' =
1
/
x
(log
a
x)' =
1
/
x ln a
(arcsin x)' =
1
/
√(1-x
2
)
(arctan x)' =
1
/
1+x
2
Chain rule:
d
/
dx
[f(g(x))] = f'(g(x)) * g'(x)
1.3 Integrals
1. 不定积分定义式
∫f(x)dx = F(x) + C
2. 求不定积分的四种方法
Formulas: ∫x
n
dx =
x
n+1
/
n+1
+ C
∫sin x dx = -cos x + C
∫cos x dx = sin x + C
∫sec
2
x dx = tan x + C
U-substitution
Partial fractions
Integration by parts: ∫u dv = uv - ∫v du
1.4 Series
1. 级数的定义与收敛性
Series: ∑a
n
= a
1
+ a
2
+ ...
Partial Sum: S
n
= a
1
+ a
2
+ ... + a
n
Convergence: If lim
n→∞
S
n
exists, the series converges.
2. 判定级数收敛性的三大审敛法
Ratio Test: lim
n→∞
|
a
n+1
/
a
n
| = ρ
Integral Test
Comparison Test
4. 幂级数和泰勒级数
Power Series: ∑c
n
(x-a)
n
Taylor Series: f(x) = f(a) + f'(a)(x-a) +
f''(a)
/
2!
(x-a)
2
+ ...
Maclaurin Series: Taylor series at a=0.