# Chinese vocables for Calculus 1

This list will be extended as the semester progresses (and students give more input).

Recommended ditionaries: cedict, www.nciku.com, Chinese Wikipedia.

## Chapter 0: Introduction

 English 中文 meaning example 1 straight line 直线 shortest connection between two points y = mx+b 3 slope 斜率 rate of increment m = dy/dx 4 vertical intercept value at x = 0 f(0) 5 function 函数 f:D→R : x→y f(x) = √x2-1) 6 domain (of a function) 定义城 where f(x) is well-defined 7 codomain 上城 where f generally maps to R for real functions 8 range 直城 all image points (0,1] for f(x) = 1/(1+x2) 9 rational numbers 有理数 Q, n/d where d≠0 0, ½, -1/3, 2/7, 1.1, … 10 real numbers 实数 R, all points on a line 0, ½, -1/3, 1.1, π, e, ln 2, √5), … 11 integers 整数 Z 0, ±1, ±2, ±3, ±4, ±5 12 graph of a function 函数的图形 plot of y = f(x) in the x-y- coordinate system 13 vertical line test 垂直线测试 see if every one x has only one y-value 14 even 偶 f(-x) = f(x), and D(f) symmetric x2+1/2-x4/7 15 odd 寄 f(-x) = -f(x), and D(f) symmetric x/2-2x3+7x5/3 16 zero (of a function) 零点 x such that f(x) = 0 17 polynom 多项式 a_0+a_1x+a2x2+...+anxn x2-x7 18 degree 度 n if an ≠ 0 19 power function 幂函数 Axn -x3/2 20 exponential laws 幂运算法则 rules for multiplying, dividing, and exponentiating powers anbn = (ab)n 21 rational function 有理函数 p(x)/q(x) (x-1)/(x2-3x+1) 22 composition (of functions) 复合函数 applying a second function to the value of a first function f[g(x)] = (f◦g)(x) 23 inverse function 反函数 f -1 that undoes f( ) arcsin(x) 24 injective 单射 each y at most one x f(x) = x3 25 horizontal line test 水平线测试 see if one y has at most one x 26 algebraic function 代数函数 inverses of polynoms √n√x)+1)/(x-5) 27 transcendental function 超越函数 everything else exp, log, sin, arcsin, … 28 exponential function 指数函数 solution of y' = ky Aexp(kx) 29 logarithm function 对数函数 inverse of an exponential function ln(x+1) 30 logarithm laws 对数法则 rules for computing logarithms ln(xn) = n*ln(x) 31 piecewise defined functions 分段函数 f with different algebraic terms depending on x sgn(x), abs(x) 32 absolute value, modulus 绝对值 abs(x)≥0, abs(xy) = abs(x)abs(y) abs(±5) = 5 33 integer function, floor function 高斯符号 integral part of a real number int(-π) = -4 34 modeling (functional dependence) 建模 determining a term for f 35 linear regression 线性回归 determining the best-fitting line through several points 36 statistical error 统计误差/ 随机误差 difference between measured values and regression model 37 systematic error 系统误差 measurement/ prediction always wrong to the same side 38 parametrized curve 参数化曲线 t→(x,y) x = sin(t), y = sin(2t)

## Chapter 1: Limits and Continuity

 English 中文 meaning Example 40 rate of change 变动率 potential change of y per change of x ∆y/∆x 41 average velocity 平均速度 related net change over a finite interval ∆x/∆t 42 instantaneous velocity / momentary velocity 瞬时速度 (瞬间速度) infinitesimal rate dx/dt 43 limit 极限 approached value when getting close to x0 limx→0 [sin(x)/x] = 1 44 convergent 收敛的 finite value is approached 45 divergent 发散的 no (one) finite value is approached limx→0 [cos(x)/x] = ±∞ 46 limit laws 极限法则 rules to compute limits effectively limx→a [f(x)±g(x)] = f(a)±g(a) 47 limit from the right/left 右(左)极限 only approaching from one side (left/below or right/above) limx→0- exp(1/x) = 0 48 Squeeze lemma/ sandwich theorem 夹挤定理 f<(x)≤g(xx→ f,h = L, then limx→ g = L limx→0 xsin(1/x) = 0 48b limit infinity/ unbounded growth/drop 无限极限 nicely diverging terms limx→0 1/x2 = ∞ 49 horizontal asymptote 水平渐近线 limx→∞ or -∞ f(x) exists limx→±∞ [1/(1+x2)] = 0 50 vertical asymptote 垂直渐近线 limx→a- f(x) = ±∞ or x→a+ limx→0± -1/x3 = ∓∞ 51 asymptote/ asymptotic 渐近 simpler term for f as x approaches ∞ (or -∞), difference must approach 0 2x+1+(x-1)/(x2+1) → 2x+1 52 indeterminate expressions 未定式 cases when the (easy) limit laws are not (directly) applicable 0/0, ∞/∞, 0∞, ∞-∞, 1∞, 00, ∞0 53 continuous function 连续函数 limx→a f(x) = f(a) 1/(1+x2) 54 discontinuity 不连续点 point where f is not continuous exp(1/x) at 0 55 removable discontinuity 可去不连续点 limx→a f(x) exists, but ≠f(a) sin(x)/x 56 finite jump 有限跳跃 limx→a- f(x) ≠ limx→a+ f(x), but both exist sgn(x) at 0 57 infinite jump 无限跳跃 only one limit is ±∞ exp(1/x) at 0 58 pole 极点 both sides go to ±∞ 1/x or 1/x2 at 0 59 essential singularity 本质奇点 limit does not exists (not even infinite) sin(1/x) at 0 60 nowhere continuous 无处连续 limx→a f(x)≠ f(a) or even not exists anywhere “double line” 61 intermediate value theorem 介值定理 a

## 2. Derivatives

 English 中文 meaning Example 64 Derivative 导数 limit of the difference quotient d/dx(sin(x)) = cos(x) 65 difference quotient 差异的智商 quotient of differences of related quantities ∆y/∆x 66 differential quotient limit of the difference quotient dy/dx 67 differential infinitesimal quantity dy = f '(x)(x-x0) 68 higher order derivative repeated differentiation f ''(x) 69 marginal {revenue/cost/tax/interest} rate 边际{收益/成本/税率/利率} price for the last item dt/di 71 sum rule 求和法则 (f±g)' = f '±g' (sin+cos)' = cos-sin 72 product/ Leibniz rule 乘积法则 (fg)' = f 'g+fg' exsin(x) = ex[sin(x)+cos(x)] 73 quotient rule 除法定则 (f/g)' = (f 'g-fg')/g2 [x/(x+1)]' = (x+1-x)/(x+1)2 74 chain rule 链式法则 f[g(x)]' = f '[g(x)]g'(x) sin(ex)' = excos(ex) 75 implicit differentiation 隐函数的导数 differentiation of an implicitly given function dx/dy = 1/(dy/dx) 76 logarithmic differentiation 用对数的异数 ？ differentiation after taking the logarithm on both sides (xx)' = xx(1+lnx) 77 related rates 有关费率 ? rates of change of functionally related quantities x2+y2 = R2, dx/dy = -y/x 78 error propagation 误差传播 influence of the observation error on computed values ∆(xy)/|xy| = ∆x/|x|+∆y/|y|

Remember that you may only use a (sorted) list without annotations during the exam.