2023年微积分知识点总结.doc

A DERIVATIVE FUNCTION
1. The derivative function or simply the derivative is defined as
＝＝
2. Find the derivative function
a) Find ,
b) Find the average rate of change ,
c) Find the limit .
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=.
Now as BA, xa and the slope of chord ABslope of tangent at A.
So, is .
Thus, we can know the derivative at x=a is the slope of the tangent at x=a.

4. Rules

C(a constant)
0

5. The chain rule
If where then .

,
6. Inverse function, Parametric function and Implicit function
Inverse function:, ,
i.e., ,
Parametric function:,
i.e., ,→,

Implicit function: , .
, , t

7. High derivative

y=sinx ,

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 for all x in S, and
f(x) is decreasing on S for all x in S.
b) Find the monotone interval
l Find domain of the function,
l Find , and x which make ,
l 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 and call it “the definite integral of from a to b”,

i.e., where .
We note that as , and

We write .
If for all x on [a,b] then is the shaded area.

2. Properties of definite integrals

, c is any constant

, where
（f(x) odd）,(f(x)even)
If on then
If on then
The average value of a function on an interval [a,b]

3. The infinite integral
If , then
Formulas:,
,,,
(),
U Substitution
substitution u=g(x)
Integration by Parts