ap微積分bc相關知識點整理

　　1. 函數的定義、函數的圖像、分段函數、絕對值函數、定義域和值域等;

　　2. 函數的運算及復合函數，函數圖像的對稱性;

　　3. x的n次冪的函數、反比例函數、多項式函數、有理函數、三角函數的定義、性質和圖像分析;

　　4. 反函數和反三角函數的圖像和性質;

　　5. 指數函數和對數函數;

　　6. 參數方程(只是Calculus BC所要求的內容)

　　這些基礎內容的講解將主要以做題帶動講解的方式，通過一定數量的例題引導，加速學生對基礎知識的回憶，為后面的微積分學習打下一定的堅實基礎。

　　1. 函數的基本知識

　　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. 函數的運算

　　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

　　(f .g)(x)=f(g(x))

　　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. 常見的函數

　　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, such that

　　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. 反函數

　　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. 指數函數、對數函數

　　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. 參數方程

　　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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