Calculus

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squaring number

平方数

the square root

平方根

plane curve

平面曲线

sequence

序列

parabola

抛物线

convergence

收敛

scatterplot

散点图

integer

整数

equation

方程

rational function

有理函数

polar coordinates

极坐标

proportionality

比例

horizontal shifts

水平移动

asymptote

渐近线

vector

矢量向量 One definition of a vector is that of a carrier — it might be an insect like a mosquito that carries and transmits a bacterium or virus, or it might be some agent that carries genetically engineered DNA into a cell. Considering that the Latin word vector comes from the word vehere, which means "to carry," it's not surprising that the current use of the word also "carries" the same meaning. In fact, in computers, a vector is a method used to propagate a computer virus. However, the word vector is also used in various scientific areas, including mathematics, where it indicates something possessing both size and direction; and aeronautics, where it indicates a projectile's course.

integral

积分 Something that is integral is very important or necessary. If you are an integral part of the team, it means that the team cannot function without you. An integral part is necessary to complete the whole. In this sense, the word essential is a near synonym. In mathematics, there are integrals of functions and equations. Integral is from Middle English, from Medieval Latin integralis "making up a whole," from Latin integer "untouched, entire."

absolute value function

绝对值函数

independent variable

自变量

derivative

衍生导数 Alert: shifting parts of speech! As a noun, a derivative is kind of financial agreement or deal. As an adjective, though, derivative describes something that borrows heavily from something else that came before it. The economic meltdown of the last decade is due largely to the mismanagement of derivatives, which are deals based on the outcome of other deals. A movie plot might be described as derivative if it steals from another film — say, if it lifts the tornado, the witch, and the dancing scarecrow from The Wizard of Oz.

implicit differentiation

隐函数微分

nonnegative numbers

非负数

secant

a straight line that intersects a curve at two or more points 割线

conic section = conic

圆锥曲线

vertical shifts

垂直移动

vertical line

垂直线

composite functions

复合函数 f(g(x))

functions

复数形式就是函数

polynomials

多项式

the degree of the polynomial

多项式次数

odd function

奇函数

definite integral

定积分

logarithms

对数

opposite adjacent hypotenuse

对边邻边斜边

slope

A linear function with positive slope whose graph passes through the origin is called a proportionality relationship. If you find yourself on a slippery slope, watch out: you could be sliding down a hill. Land that is not level is called a slope. Slope can also be a verb, as in land that slopes down to sea level. Language experts believe the word slope came from the Middle English word aslope, an adverb that means "at an angle." The word has a noun form you can use for something that is at an angle — on a slope — like a steep hill or the ramp in a parking garage. It also has a verb form that can be used to describe something that slants, like someone's signature with letters that slope to the left or an angled haircut that slopes to cover one eye. 斜斜率

quotient

A rational function is a quotient or ratio f(x)=p(x)/q(x), where p and q are polynomials. When you add two numbers the answer is called the sum. When you divide two numbers the answer is called the quotient. The quotient of six divided by two is three. Quotient comes from Latin and means "how many times." That makes a lot of sense: if you divide one number by a second, you are figuring out "how many times" the second number goes into the first. Outside of math, use of the word is restricted: the IQ test is short for "Intelligence Quotient," and very rarely you might hear someone ask, "What's my quotient of cupcakes?" when they mean "What is my share?" 商商数

encounter

A variety of important types of functions are frequently encountered in calculus. We identify and briefly describe them here. If you run into that cute guy (or girl) from the local deli when you're at the grocery store and you stop to chat, you've just had an encounter, which is a casual meeting, often resulting by chance. When you encounter the word encounter, context will tell you if it's acting as a verb or a noun. The sentence "When Spencer and Susanna encounter a bear on the trail, they stand very still" illustrates the verb form. "The encounter in the subway left her wishing she had stayed at home" shows the noun form. Whether acting as a verb or a noun, the word carries the connotation of "chance meeting." You don't plan an encounter; it just happens.

abacus

An abacus is an ancient tool used for calculating that remains popular in some places even today. Some sort of counter (beads, beans, stones) is moved in a groove or on a wire to represent the different numbers in the equation. Abacus is a Latin word from a Greek word abax, which meant "counting table." The original abaci were created in sand. The plural abacuses can also be used. In architecture, an abacus can also refer to a flat slab that sits on top of the broad part of a pillar or column (called the capital) to help support a beam (called an architrave) that rests across several pillars. Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus),[1] is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

positive x-axis

An angle in the xy-plane is said to be in standard position if its vertex lies at the origin and its initial ray lies along the positive x-axis. Angles measured counter-clockwise from the positive x-axis are assigned positive measures; angles measured clock-wise are assigned negative measures.

vertex

clockwise counterclockwise

finite

Calling something finite means it has an end or finishing point. Preparing for a standardized test might be unpleasant, but you have to remember that the work is finite; you won't be doing it forever. Most people are far more familiar with the word finite when they see it inside the word infinite, or without end. Finite can be used for conceptual things like time, "We have to get out of here, we only have a finite amount of time," and for more tangible things like beans or dirt, "We have to be careful with the cooking, we only have a finite amount of fuel." You might want to think of things being finito — a word that looks a lot like finite — to remember that it means, with an end. fin= end/limit 末尾界限 fin来自拉丁动词finis，意思是end末尾或limit界限

radian

Angles are measured in degrees or radians. 弧度

multiplication

Any function constructed from polynomials using algebraic operations (addition, subtraction, multiplication, division, and taking roots) lies within the class of algebraic functions. All rational functions are algebraic, but also included are more complicated functions. 乘法

algebraic functions

subtraction

addition

division

definite

Definite is an adjective describing something that is known for certain. For example, there is no more definite way to get into trouble with a police officer than speeding in front of the police station with a broken taillight. Definite is an adjective describing something that is known for certain. For example, there is no more definite way to get into trouble with a police officer than speeding in front of the police station with a broken taillight. definite integral 定积分 indefinite integral 不定积分

context

Essentially, the definition says that the values of ƒ(x) are close to the number L whenever x is close to x0（on either side of x0）.This definition is "informal" because phrases like arbitrarily close and sufficiently close are imprecise; their meaning depends on the context. (To a machinist manufacturing a piston, close may mean within a few thousandths of an inch. To an astronomer studying distant galaxies, close may mean within a few thousand light-years.) Nevertheless, the definition is clear enough to enable us to recognize and evaluate limits of specific functions. We will need the precise definition of Section 2.3, however, when we set out to prove theorems about limits. Here are several more examples exploring the idea of limits. Context means the setting of a word or event. If your friend is furious at you for calling her your worst enemy, remind her that the context of those remarks was Opposite Day. Context comes from the Latin for how something is made. It was first used to talk about writing, as in "the beautiful phrase occurs in the context of the concluding paragraph." We use it now to talk about any circumstance in which something happens. You might say that you can't understand what happens without looking at the context. When someone takes your words but makes it sound like you meant something else, they've taken your words out of context. 环境语境上下文

sufficient

Essentially, the definition says that the values of ƒ(x) are close to the number L whenever x is close to x0（on either side of x0）.This definition is "informal" because phrases like arbitrarily close and sufficiently close are imprecise; their meaning depends on the context. (To a machinist manufacturing a piston, close may mean within a few thousandths of an inch. To an astronomer studying distant galaxies, close may mean within a few thousand light-years.) Nevertheless, the definition is clear enough to enable us to recognize and evaluate limits of specific functions. We will need the precise definition of Section 2.3, however, when we set out to prove theorems about limits. Here are several more examples exploring the idea of limits. If you have a sufficient amount of something, it's enough — not too much, not too little, just right. Goldilocks would be pleased. Sufficient comes from a Latin verb meaning "to meet the need." If something is sufficient it has met, or satisfied, a need. Enough is often used as a synonym for sufficient, and when something is not sufficient, it is too little to take care of what's needed. Sufficient can, however, also suggest just enough and not an abundance, as in "the money was sufficient for groceries, but we needed more to fill the gas tank." 充足的足够的刚好的？

arbitrary

Essentially, the definition says that the values of ƒ(x) are close to the number L whenever x is close to x0（on either side of x0）.This definition is "informal" because phrases like arbitrarily close and sufficiently close are imprecise; their meaning depends on the context. (To a machinist manufacturing a piston, close may mean within a few thousandths of an inch. To an astronomer studying distant galaxies, close may mean within a few thousand light-years.) Nevertheless, the definition is clear enough to enable us to recognize and evaluate limits of specific functions. We will need the precise definition of Section 2.3, however, when we set out to prove theorems about limits. Here are several more examples exploring the idea of limits. Something that's arbitrary seems like it's chosen at random instead of following a consistent rule. Team members would dislike their coach using a totally arbitrary method to pick starting players. Even though arbitrary comes from a word meaning "judge" (arbiter), that doesn't mean judges are always fair. Calling a decision-maker arbitrary is usually a negative thing, suggesting the person is making rules based on whim rather than justice. A coach who selects starting players arbitrarily isn't strictly applying a rule; he could just be picking names out of a hat.

domain

Functions; Domain and Range If you have a place that's all your own, somewhere real or in cyberspace that has your name all over it — literally or figuratively — it's your domain. And if you have something you really excel at, that is your domain too. You own it. A domain used to mean only land owned by wealthy people, such as lords in the 15th century, but modern usage is much wider, or has a broader domain in the English language. On the Internet, a domain is a space with a specific address, but a domain also can have a physical address, like a home. It can be a specialty, too, as in, "The main domain of the art school was sculpture," or "She was so good at math that algebra became her domain." 域

reciprocal

If the variable y is proportional to the reciprocal 1/x, then sometimes it is said that y is inversely proportional to x, because 1/x is the multiplicative inverse of x. Reciprocal describes something that's the same on both sides. If you and your sister are in a big fight on a long car trip, you might resolve it through a reciprocal agreement that you'll stop poking her and she'll stop reading road signs out loud. The word mutual is a near synonym in most uses: reciprocal/mutual friendship, describing, a relationship in which two people feel the same way about each other, or do or give similar things to each other. If you tell someone you like them and they say, "The feelings are reciprocal," that means they like you too. In math, a reciprocal is a number that when multiplied by a given number gives one as a product. 相互的互相的倒数

coefficient

In math and science, a coefficient is a constant term related to the properties of a product. In the equation that measures friction, for example, the number that always stays the same is the coefficient. In plain English, coefficient means "joining together to produce a result." Sometimes people use the word to talk about social phenomena, like the coefficient factors of vanity and self-loathing in a celebrity's alcoholic demise. But mostly you'll encounter it in math and science. In algebra, the coefficient is the number that you multiply a variable by, like the 4 in 4x=y. In chemistry, when you see a number in front of a chemical like 2H2o, you're looking at the coefficient. 率；系数

prescribe

In the previous example we determined how close x must be to a particular value to ensure that the outputs ƒ(x) of some function lie within a prescribed interval about a limit value L. To show that the limit of ƒ(x) as x->x0 actually equals L, we must be able to show that the gap between f(x) and L can be made less than any prescribed error, no matter hoe small, by holding x close enough to x0. To prescribe is make orders or give directions for something to be done. These days, the word is mainly used by doctors who prescribe medications to take. Doctors do a lot of prescribing: they prescribe drugs, rest, exercise, and getting rid of bad habits like smoking. When a doctor prescribes something, he or she is saying, "You need to do this. You should do it." That's the most common use of prescribe, but it pops up anywhere someone is advising or ordering someone to do something. All laws and rules prescribe things — they tell you what to do. 开药方指示

denominator

In words, the Sum Rule says that the limit of a sum is the sum of the limits. Similarly, the next rules say that the limit of a difference is the difference of the limits; the limit of a constant times a function is the constant times the limit of the function; the limit of a product is the product of the limits; the limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0); the limit of a positive integer power (or root) of a function is the integer power (or root) of the limit (provided that the root of the limit is a real number). 分母

infinite

Infinite describes things that are endless, like the universe, or your uncle's corny jokes. Finite means "something with an end," and when you add the prefix, in- meaning "not," you get infinite: something that never, ever ends. If someone has read every single book about pyramids, you might say he as an infinite knowledge of ancient Egyptian culture, even though that's an exaggeration. He will sure stop talking about them at some point. Right?

theorem

It is reasonable that the properties in Theorem 1 are true (although these intuitive arguments do not constitute proofs). If x is sufficiently close to c, then ƒ(x) is close to L and g(x) is close to M, from our informal definition of a limit. A theorem is a proposition or statement that can be proven to be true every time. In mathematics, if you plug in the numbers, you can show a theorem is true. Just as a theory is an idea that can be supported or disproved, a theorem is also an idea, but it's one that has been proven and can be demonstrated again and again if used properly. In math class, you might have learned some theorems. One example is the Pythagorean theorem, which can be represented as A squared plus B squared equals C squared. Although it's usually used in math, theorems can be laws, rules, formulas, or even logical deductions.

oscillate

It oscillates too much to have a limit: ƒ(x) has no limit as because the function'svalues oscillate between +1 and -1 in every open interval containing 0. The value do not stay close to any one number as x->0 On a hot day, you'll be happy to have a fan that can oscillate, meaning it moves back and forth in a steady motion. The verb oscillate can be traced back to the Latin word oscillum, meaning "swing," so it makes sense that oscillate is used to describe an object like a fan or a pendulum that swings from side to side. The word also can be used to describe a different kind of motion — the wavering of someone who is going back and forth between conflicting beliefs or actions. If you've ever had trouble making up your mind about something, you probably know what it feels like to oscillate — back and forth from one decision and to another and then back again. And again. And again. 摆动

interval

Notice that as the power n gets larger, the curves tend to flatten toward the x-axis on the interval （-1,1）, and also rise more steeply for x>1 or x<-1. A clock breaks time down into intervals of seconds, minutes, and hours. An interval is a distinct measure of time or the physical or temporal distance between two things. When you are driving down the highway at 60 mph, you'll see distance markers at intervals of .1 miles. That means that every 1/10th of a mile, you will see one of these markers. Do the math and you'll see that these signs flash by you at intervals of 6 seconds. If your coach tells you to try interval training, he is telling you to do something like run for three minutes, lift weights for two, then run again for three. If he tells you do this again and again, you might want to end this interval of your life. 间隔 (数学)区间

pebble

Pebbles are the small, round stones you might find on a beach. If you want a pet rock, a smooth pebble would be a good choice. A pebble beach might not be quite as soft underfoot as a sandy one, but pebbles tend to be smooth and rounded — unlike jagged-edged rocks. A beach covered with smooth pebbles is known as a "shingle beach." The origin of the word pebble is a mystery, although some suspect a connection to the Latin papula, "pustule, pimple, or swelling." Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus),[1] is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

identity function

The function f(x)=x where m=1 and b=0 is called identity function. 恒等函数

symmetry asymmetry symmetric symmetrical

The graphs of even and odd functions have characteristic symmetry properties. Things that have symmetry are balanced, with each side reflecting the other. A human body has such complex symmetry, from eyes, ears, and nostrils to arms, legs, and feet, that even a minor injury can make a body look unbalanced. Snowflakes and butterflies often have a remarkable natural symmetry, with patterns on one side matched by those on the other. Objects that have identical or very similar parts lying at equal distances from a central point or line or plane — that is, objects that have symmetry — often work better. Symmetry helps boats stay upright in water. In design, symmetry is a balancing of objects, as when two candlesticks on a fireplace mantel are at equal distances from the center of the mantel. A lack of symmetry — i.e., asymmetry — might mean putting both candlesticks together at one end of the mantel.

radius

The temperature at which water boils depends on the elevation above sea level (the boiling point drops as you ascend). The interest paid on a cash investment depends on the length of time the investment is held. The area of a circle depends on the radius of the circle. The distance an object travels at constant speed along a straight-line path depends on the elapsed time. If you're a detective working a crime investigation, you might be told to "search over a one-mile radius," meaning, scope out everything that's within one mile of the crime scene. The radius of a circle is the distance from its center to the circumference, and if you are working on your geometry homework, or designing anything circular, you'll be writing down a little "r" quite frequently, to stand for "radius." It's also the name of one of the bones in your forearm 半径

trigonometric functions

These are functions that are not algebraic. They include the trigonometric, inverse trigonometric,exponential,and logarithmic functions,and many other functions as well. A particular example of a transcendental function is a catenary. Its graph has the shape of a cable, like a telephone line or electric cable, strung from one support to another and hanging freely under its own weight. 三角函数

Inverse functions

exponential functions

transcendental functions

inequality

These inequalities will be useful in the next chapter. 不等式

constant constants

Think of something or someone that does not change as constant. A classmate's constant drumming on the table with his fingers could be a constant source of annoyance. Constant derives from Latin verb meaning "to stand with," so something constant is continually standing with you and not wavering. You may be thankful for the constant companionship of your dog but not necessarily for your teacher's constant homework assignments. In math and science, a constant is a number that is fixed and known, unlike a variable which changes with the context. That idea crosses over to real life. If a friend is a constant in your life, that means they have always been with you and there for you. constants 常数

scale reflect

To scale the graph of a function y = f(x) is to stretch or compress it, vertically or horizontally. This is accomplished by multiplying the function f,or the independent variable x, by an appropriate constant c. Reflections across the coordinate axes are special cases where c = -1.

perpendicular

Two lines that form a T are perpendicular to each other. They meet at a right angle. A person who is standing is perpendicular to the earth. Perpendicular is used to describe lines, angles and direction. In geometry a perpendicular angle is 90 degrees, a perfect L. On a compass, East and North are perpendicular to each other. The term can be used more generally to describe any steep angle. You might talk about a ski slope that is nearly perpendicular. That's impossible; gravity would make you fall off a 90-degree angle. But if it's close enough, no one's really measuring.

periodic

When an angle of measure and an angle of measure are in standard position, their terminal rays coincide. The two angles therefore have the same trigonometric function values, We describe this repeating behavior by saying that the six basic trigonometric functions are periodic. A function ƒ(x) is periodic if there is a positive number p such that ƒ(x + p) = ƒ(x) for every value of x. The smallest such value of p is the period of ƒ. 周期的定期的

power

幂 even power 偶数幂 odd power 奇数幂 The graphs of even and odd functions have characteristic symmetry properties. The names even and odd come from powers of x.

combining functions

the result of adding, subtracting, multiplying and/or dividing two or more functions 复合函数

first-order differential equations

一阶常微分方程式

products

乘积

quadratic functions cubic functions

二次函数三次函数

partial derivatives

偏导数偏微分偏导函数

even function

偶函数

piecewise-defined function

分段函数

tangent

切线

monotonic functions

单调函数

the orgin

原点

parametric equations

参数方程

hyperbolic functions

双曲函数

antiderivative

反导数（原函数？）不定积分

dependent variable

因变量

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