Calculus 1
Calculus 101:
single and mutiple variables
Today I’m going to start the study of calculus is the sixth addition of it is written by several writers end is named that single end a multi valuable calculus starting from preface to calculus is absolutely one of the greatest achievements of the human intellect it is is filed by problems in astronomy end developed by Newton end 11 its 300 years ago
The content of evidence visions of how calculus can be taught is flexible enough to accommodate different types of students the chapter 1 discusses the library of functions chapter 2 is about the derivative chapter 3 is short cuts to differentiation 4 is using the derivative to the five is cut, key concept to definite integral chapter 6 is, constructing anti derivative chapter 7 is integration chapter 8 is using the definite integral chapter 9 issequences and series chapter 10 is approximating Functions Chapter 11 is differential equations chapter 12’s functions of several variables chapter 13 is fundamental tool vectors chapter 14 is different in functions of several variables chapter 15 if theoptimization
在逻辑学里面有谓词，函数和命题，他们3个的关系?
为什么函数可以背谱表被表格背公式背描述，所表述呢
首先我们确定的是一个x轴和一个万州这样我们就可以用图表和表格来描述，x轴表示有一个变量y轴表示另外一个变量，所以二元函数可以用一个二元的，表格.
换句话说一条直线可以用来背表达一个变量，上面的每一个点则是变量的不同取值，但是，直线因为没有一个，没有一个起点一个终点所以还需要另外一条与他相交的线来确定一个一个，一个标法标志一个一个尺度.
Chirp rate：调制斜率。
Mathematics is chapter 1 in mathematics the function is used to represent the dependency of a quantity up on another
The definition of a function is as follows function is the rule that takes certain numbers as inputs end assigns to each a definite output number .
The set of input number is called the domain of the function and as a set of resulting output numbers is called the range of the function.
The include is called independent variable end the outcome is called dependent variable some quantities are discrete end others are continuous
The rule of four tables graphs, formulas, words.
Function is linear is if slope/ rate of change is the same as every。
Intercept on vertical axis.
Difference quotients and delta notation
Delta: change in

A linear function has the form
y = f (x) = b + mx.
Its graph is a line such that
• m is the slope, or rate of change of y with respect to x.
• b is the vertical intercept, or value of y when x is zero.
Formulas such as f (x) = b + mx, in which the constants m and b can take on various values, give a family of functions
constants b and m are parameters.
There are increasing/ decreasing/ monotonic functions.
y proportional to x if y kx.
y inversely proportional to x if …
总结：没啥，初中内容。
Chapter 1.2 exponentional functions
bn exponent=power= n b=base
concavity: a curve is said to be concave up if we move from left to right, the curve BENDs UPWARDS
we say that a function is exponential if the function is as followed
f(x) = P0 ax
P0 is the initial quantity. a is the factor by which f(x) changes.a is the base
when a>1, we have exponential growth.
when a<1,we have exponential decay.
Half time and doubling time of a exponential function is the time needed to be reduced to a half or to be doubled.
Exponential functions with base e (ruler’s number)
any exponential growth function could be written as in:
f(x)= P0 e kx
Ekx= (Ex)k Ax/Ex = a/ex if a= Eb,than ax = (Eb)x= Ebx
for a>1 and k>0
exponential decay:
Q=Q0 a t = Q0 e -kt where 00.
P and Q are growing/decaying at a continuous rate of k.
Continuous rate is further discussed in chapter 11
Chapter 1.3 Shifts and stretches.
multiplying a function/ shrinking a function.
move to right w if the x is replaced with (x-w)
moved to up z if the y is replaced with (y-z)
Odd and even function:symmetry
Inverse function
the independent variable is the dependent variable in f-1(x) and vice versa.
a function can be inverse if and only if at any horizontal line it is intersected only once.
If the x- and y-axes have the same scales, the graph of f−1 is the reflec- tion of the graph of f about the line y = x.
沿xx对称： reflection of the graph about the linexx
logarithms to base 10 and to base e.
The logarithm to base of 10 of x is the power of 10 we need to get to x.
The natural logarithm written in lnx is the power of e needed to get to x.]
both lnx and logx have vertical asympotees at x= 0 (垂直渐近线）
property of logarithm calculations:
降一个维度。