Chinese vocables for Calculus 1

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

Recommended ditionaries: cedict,, Chinese Wikipedia.

Chapter 0: Introduction

1straight line直线shortest connection between two pointsy = mx+b
3slope斜率rate of incrementm = dy/dx
4vertical intercept value at x = 0f(0)
5function函数f:DR : xyf(x) = √x2-1)
6domain (of a function)定义城where f(x) is well-defined
7codomain上城where f generally maps toR for real functions
8range直城all image points(0,1] for f(x) = 1/(1+x2)
9rational numbers有理数Q, n/d where d≠00, ½, -1/3, 2/7, 1.1, …
10real numbers实数R, all points on a line0, ½, -1/3, 1.1, π, e, ln 2, √5), …
11integers整数Z0, ±1, ±2, ±3, ±4, ±5
12graph of a function函数的图形plot of y = f(x) in the x-y- coordinate system
13vertical line test垂直线测试see if every one x has only one y-value
14evenf(-x) = f(x), and D(f) symmetricx2+1/2-x4/7
15oddf(-x) = -f(x), and D(f) symmetricx/2-2x3+7x5/3
16zero (of a function)零点x such that f(x) = 0
18degreen if an ≠ 0
19power function幂函数Axn-x3/2
20exponential laws幂运算法则rules for multiplying, dividing, and exponentiating powersanbn = (ab)n
21rational function有理函数p(x)/q(x)(x-1)/(x2-3x+1)
22composition (of functions)复合函数applying a second function to the value of a first functionf[g(x)] = (fg)(x)
23inverse function反函数f -1 that undoes f( )arcsin(x)
24injective单射each y at most one xf(x) = x3
25horizontal line test水平线测试see if one y has at most one x
26algebraic function代数函数inverses of polynomsnx)+1)/(x-5)
27transcendental function超越函数everything elseexp, log, sin, arcsin, …
28exponential function指数函数solution of y' = kyAexp(kx)
29logarithm function对数函数inverse of an exponential functionln(x+1)
30logarithm laws对数法则rules for computing logarithmsln(xn) = n*ln(x)
31piecewise defined functions分段函数f with different algebraic terms depending on xsgn(x), abs(x)
32absolute value, modulus绝对值abs(x)≥0, abs(xy) = abs(x)abs(y)abs(±5) = 5
33integer function, floor function高斯符号integral part of a real numberint(-π) = -4
34modeling (functional dependence)建模determining a term for f
35linear regression线性回归determining the best-fitting line through several points
36statistical error统计误差/ 随机误差difference between measured values and regression model
37systematic error系统误差measurement/ prediction always wrong to the same side
38parametrized curve参数化曲线t→(x,y)x = sin(t), y = sin(2t)

Chapter 1: Limits and Continuity

40rate of change变动率potential change of y per change of xy/∆x
41average velocity平均速度related net change over a finite intervalx/∆t
42instantaneous velocity / momentary velocity瞬时速度 (瞬间速度)infinitesimal ratedx/dt
43limit极限approached value when getting close to x0limx→0 [sin(x)/x] = 1
44convergent收敛的finite value is approached
45divergent发散的no (one) finite value is approachedlimx→0 [cos(x)/x] = ±∞
46limit laws极限法则rules to compute limits effectivelylimxa [f(xg(x)] = f(ag(a)
47limit from the right/left右(左)极限only approaching from one side (left/below or right/above)limx→0- exp(1/x) = 0
48Squeeze lemma/ sandwich theorem夹挤定理f<(x)≤g(xx→ f,h = L, then limx g = Llimx→0 xsin(1/x) = 0
48blimit infinity/ unbounded growth/drop无限极限nicely diverging termslimx→0 1/x2 = ∞
49horizontal asymptote水平渐近线limx→∞ or -∞ f(x) existslimx→±∞ [1/(1+x2)] = 0
50vertical asymptote垂直渐近线limxa- f(x) = ±∞ or xa+limx→0± -1/x3 = ∓∞
51asymptote/ asymptotic渐近simpler term for f as x approaches ∞ (or -∞), difference must approach 02x+1+(x-1)/(x2+1) → 2x+1
52indeterminate expressions未定式cases when the (easy) limit laws are not (directly) applicable0/0, ∞/∞, 0∞, ∞-∞, 1, 00, ∞0
53continuous function连续函数limxa f(x) = f(a)1/(1+x2)
54discontinuity不连续点point where f is not continuousexp(1/x) at 0
55removable discontinuity可去不连续点limxa f(x) exists, but ≠f(a)sin(x)/x
56finite jump有限跳跃limxa- f(x) ≠ limxa+ f(x), but both existsgn(x) at 0
57infinite jump无限跳跃only one limit is ±∞exp(1/x) at 0
58pole极点both sides go to ±∞1/x or 1/x2 at 0
59essential singularity本质奇点limit does not exists (not even infinite)sin(1/x) at 0
60nowhere continuous无处连续limxa f(x)≠ f(a) or even not exists anywhere“double line”
61intermediate value theorem介值定理a<b, f(a)<f(b) then every y∈[f(a),f(b)] has an x∈[a,b], with f(x) = yf continuous
62tangent line切线line that touches in first order at a (f(a) = t(a), f '(a) = t'(a) )t(x) = 0 at a = 0 for f(x) = x2

2. Derivatives

64Derivative导数limit of the difference quotientd/dx(sin(x)) = cos(x)
65difference quotient差异的智商quotient of differences of related quantitiesy/∆x 
66differential quotient limit of the difference quotientdy/dx
67differential infinitesimal quantitydy = f '(x)(x-x0)
68higher order derivative repeated differentiationf ''(x)
69marginal {revenue/cost/tax/interest} rate边际{收益/成本/税率/利率}price for the last itemdt/di
71sum rule求和法则(f±g)' = f '±g'(sin+cos)' = cos-sin
72product/ Leibniz rule乘积法则(fg)' = f 'g+fg'exsin(x) = ex[sin(x)+cos(x)]
73quotient rule除法定则(f/g)' = (f 'g-fg')/g2[x/(x+1)]' = (x+1-x)/(x+1)2
74chain rule链式法则f[g(x)]' = f '[g(x)]g'(x)sin(ex)' = excos(ex)
75implicit differentiation隐函数的导数differentiation of an implicitly given functiondx/dy = 1/(dy/dx)
76logarithmic differentiation用对数的异数 ?differentiation after taking the logarithm on both sides(xx)' = xx(1+lnx)
77related rates有关费率 ?rates of change of functionally related quantitiesx2+y2 = R2, dx/dy = -y/x
78error 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.

3. Applications of derivatives

4. Integration

5. Applications of integration

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