摘要：AP微积分中用到的高中数学知识 微积分中用到的高中知识主要是函数相关知识，主要有以下几方面内容： 1. 函数的定义、函数的图像、分段函数、绝对值函数、定义域和值域等; 2. 函数的运算及复合函数，函数图像的对称性; 3. x的n次幂的函数、反比例函数、多项式函数、有理函数、三角函数的定义、性质和图像分析; 4. 反函数和反三角函数的
AP微积分中用到的高中数学知识
微积分中用到的高中知识主要是函数相关知识，主要有以下几方面内容：
1. 函数的定义、函数的图像、分段函数、绝对值函数、定义域和值域等;
2. 函数的运算及复合函数，函数图像的对称性;
3. x的n次幂的函数、反比例函数、多项式函数、有理函数、三角函数的定义、性质和图像分析;
4. 反函数和反三角函数的图像和性质;
5. 指数函数和对数函数;
6. 参数方程(只是Calculus BC所要求的内容)
1. 函数的基本知识
1.1. Definition
If a variable y depends on a variable x in such a way that each value of x determines exactly one value of y, then we say that y is a function of x.
1.2. The vertical line test:
A curve in the xy-plane is the graph of some function f if and only if no vertical line intersects the curve more than once.
1.3. The absolute value function
2. 函数的运算
2.1. Composition of f with g
Given functions f and g, the composition of f with g, denoted by f ο g, is the function defined by
The donation of f o g is defined to consist of all x in the domain of g for which g(x) is in the domain of f.
2.2. Symmetry Tests
a) A plane curve is symmetric about the y-axis if and only if replacing x by –x in its equation produces an equivalent equation.
b) A plane curve is symmetric about the x-axis if and only if replacing y by –y in its equation produces an equivalent equation.
c) A plane curve is symmetric about the origin if and only if replacing x by –x and y by –y in its equation produces an equivalent equation
3. 常见的函数
3.1. Inverse function
A variable is said to be inversely proportional to a variable x if there is a positive constant k, called the constant of proportionality.
3.2. Polynomials
A polynomial in x is a function that is expressible as a sum of finitely many terms of the form cxn, wherec is a constant and n is a nonnegative integar.
3.3. Rational function
A function that can be expressed as a ratio of two polynomials is called a rational function.
4. 反函数
4.1. Inverse function
If the function f and g satisfy the two conditions：
g(f(x))=x for every x in the domain of f
f(g(x))=y for every y in the domain of g
then we say that f is an inverse of g and g is an inverse of f or that f and g are inverse functions.
4.2. The Horizontal Line Test
A function has an inverse function if and only if its graph is cut at most once by any horizontal line.
5. 指数函数、对数函数
5.1. A function of the form f(x)=bx, where b>0, is called an exponential function with base b.
5.2. The basic characteristic of exponential function
5.3. The basic characteristic of logarithmic function
5.4. If b>0 and b≠1, then bx and logbx are inverse functions.
6. 参数方程
6.1. Definition
Suppose that a particle moves along a curve C in the xy-plane in such a way that its x- and y- coordinates, as functions of time, are
x=f(t), y=g(t)
We call these the parametric equations of motion for the particle and refer to C as the trajectory of the particle or the graphs of the equations. The variable t is called the parameter for the equations.
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