《微积分英文版》课件.ppt

CHAPTER 7 TRANSCENDENTAL FUNCTIONS 7.1 Natural Logarithm Function The natural logarith function, denoted by ln, is defined by The domaian of the natural logarithm function is the set of positive real numbers. Geometric Meaning Consider the graph of f(x) = 1/x. The ln(x) represents the area under f(x) between 1 and x. Use First Fundamental Theorem to find the Derivative of the Natural Logarithm Function Properties of natural logarithms If a and b are positive numbers and r is any rational number, then A) ln 1 = 0 B) ln ab = ln a + ln b C) ln(a/b) = ln a – ln b D) 6.2 Inverse Functions Their Derivatives If f is strictly monotonic on its domain, f has an inverse. 反函数的求导法则 例. 求反三角函数及指数函数的导数. 2) 设 7.3 The Natural Exponential Function The inverse of ln is called the natural exponential function and is denoted by exp. Thus x = exp y, and y = ln x. The letter “e” denotest he unique positive real number such than ln e = 1. The natural exponential function and the natural logarithmic function are inverses of each other. Properties that apply to inverse functions apply to these 2 functions. The derivative of the natural exponential function is itself Integrate the following: 6.4 General Exponential Logarithmic Functions Logarithms could have a base other than e. 7.5 Exponential Growth Decay Functions modeling exponential growth (or decay) are of this form: 7.6 1st-Order Linear Differential Equations Sometimes it is not possible to separate an equation such that all expressions with x and dx are on one side and y and dy are on the other. General form of a first-order linear differential equation: 一、一阶线性微分方程 2. 解非齐次方程 例. 解方程 例. 求方程 7.7 Approximations for Differential Equations Slope fields: Consider a first-order differential equation of the form y’ = f(x,y) At the point (x,y) the slope of a solution is given by f(x,y). Example:y’ = 3xy, at (2,4), y’=24, at (-2,1), y’=-6; at (0,5), y’ = 0; at (2,0), y’=0, etc. If all the slopes (y’) were graphed on a coord