AP微积分CALCULUS知识点总结

A DERIVATIVE 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
C(a constant)
5. The chain rule
If )(u f y = where )(x u u = then dx
du du dy dx dy =. )
(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=→)(1x t -=ψ, )]([1x y -=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∈
7. High derivative
y=sinx )2sin(cos π+=='x x y , )22sin()2cos(π
π⨯+=+=''x x y
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) 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 11
0)()()( where n
a
b x -=∆.
We note that as ∞→n , ∑
⎰-=→∆1
)()(n i b
a i dx x f x x f and
We write ⎰∑
=∆=∞→b
a n
i i n dx x f x x f )()(lim 1
.
If
)(≥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 c dx x cf )()(, c is any constant
)()()()(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] 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。