最新-AP微积分CALCULUS知识点总结

A DERIV ATIVE FUNCTION
1. The derivative function or simply the derivative is defined as
)(x f '＝y '＝x
x f x x f x y x x ∆-∆+=∆∆→∆→∆)()(lim lim
00
2. Find the derivative function a) Find y ∆,
b) Find the average rate of change x y ∆∆, c) Find the limit x
y x ∆∆→∆0
lim .
3. Geometric significance
Consider a general function y=f(x), a fixed point A(a,f(a)) and a variable point B(x,f(x)). The slope of chord AB=a
x a f x f --)()(.
Now as B →A, x →a and the slope of chord AB →slope of tangent at A. So, a
x a f x f a x --→)()(lim is )(a f '.
Thus, we can know the derivative at x=a is the slope of the tangent at x=a.
4. Rules
)(x f
)(x f '
C(a constant) 0
n x
1-n nx
x sin
x cos
x cos
x sin -
x tan
x
x 22cos 1sec =
x arcsin
2
-11x
5. The chain rule
If )(u f y = where )(x u u = then
dx
du du dy dx dy =. )()(x g e x f = )()()(x g e x f x g '=' )(ln )(x g x f = )()
()(x g x g x f '=
' )
(ln )()
(ln )()
()()(x u x v x u x v e e x u x f x v ===,
])()
()()(ln )([)()
(ln )(x u x u x v x u x v e
x f x u x v '+'='
6. Inverse function, Parametric function and Implicit function Inverse function:
dy dx dx dy 1=
, ])([1
)(1'
='-x f x f , i.e., x y arcsin =, y x sin =
Parametric function:
dt
dx dt
dy dx dy =
, i.e., )(t y ϕ=,)(t x ψ=→)(1
x t -=ψ, )]([1
x y -=ψϕ
)
()
(t t dt dx dt dy dx dt dt dy dx dy ψϕ''=
== Implicit function: 0))(,(=x y x F , 0))(,(=x f x F .
0-2
2
2
=+a y x ,
t
a y t a x sin cos ==, t ]2,0[π∈
t t
a t a dx dy x y cot sin cos )(-=-=='
7. High derivative
x
x f x x f dx y d x f x ∆'-∆+'==''→∆)
()(lim )(0
22 t
a t a t dt dx dt y d dx y d x y x y x 3
2sin 1
sin csc ])([)(-=-='='=''='' x
x f x x f x f n n x n ∆-∆+=--→∆)()(lim )()
1()1(0
)
( y=sinx )2sin(cos π
+
=='x x y , )2
2sin()2cos(π
π⨯+=+=''x x y )2
sin()
(π
⨯+=n x y
n
B APPLICATIONS OF DIFFERENTIAL CALCULUS
1. Monotonicity
a) If S is an interval of real numbers and f(x) is defined for all x in S,
then ：
f(x) is increasing on S ⇔ 0)(≥'x f for all x in S, and f(x) is decreasing on S ⇔0)(≤'x f for all x in S. b)
c) Find the monotone interval ●
● Find domain of the function, ●
● Find )(x f ', and x which make 0)(='x f , ● Draw sign diagram, find the monotone interval. 2. Maxima/Minima, Horizontal inflection, Stationary point
C INTEGRAL
1. The idea of definite integral
We define the unique number between all lower and upper sums as
⎰b
a
dx x f )( and call it “the definite integral of )(x f from a to b ”,
i.e., ∑∑⎰=-=∆〈〈∆n
i i n i b
a i x x f dx x f x x f 1
1
)()()( where n
a
b x -=
∆.
We note that as ∞→n , ∑⎰-=→∆1
0)()(n i b
a i dx x f x x f and
⎰∑→∆=b
a
n
i i
dx x f x x f )()(1
We write ⎰∑=∆=∞
→b
a n
i i n dx x f x x f )()(lim 1.
If 0)(≥x f for all x on [a,b] then
⎰b
a
dx x f )( is the shaded area.
2. Properties of definite integrals
⎰⎰-=-b
a b a dx x f dx x f )()]([
⎰⎰=b
a
b
a
dx x f c dx x cf )()(, c is any constant
⎰⎰⎰=+c
a
b a
c b
dx x f dx x f dx x f )()()(
⎰⎰⎰+=+b
a b a b a dx x g dx x f dx x g x f )()()]()([
)()()()(a F b F x F dx x f b
a
b
a
-==⎰, where ⎰=dx x f x F )()(
⎰-=a a
dx x f 0)(（f(x) odd ）,⎰⎰-=a a
a
dx x f dx x f 0
)(2)((f(x)even)
If 0)(≥x f on b x a ≤≤ then
⎰≥b
a
dx x f 0)(
If )()(x g x f ≥ on b x a ≤≤ then
⎰⎰≥b a
b
a
dx x g dx x f )()(
The average value of a function on an interval [a,b]
⎰-=b
a ave
dx x f a
b f )(1
3. The infinite integral
If )()(x f x F =', then
⎰+=C x F dx x f )()(
Formulas:⎰++=
+C x n dx x n n
11
1, C a a dx a x x
+=⎰ln 1 ⎰+-=C
x inxdx cos s ,
⎰+=C
x xdx sin cos ,
C x xdx +-=⎰cos ln tan ,⎰+=C x xdx sin ln cot
C x x
dx +=-⎰arcsin 12
(12
<x ), C x x dx +=+⎰arctan 12 U Substitution
⎰'dx x g x g f )())(( substitution u=g(x) ⎰du u f )(
Integration by Parts
⎰⎰-=vdu uv udv。