2.7 Differentiation: Glossary

Costas Loizou; Shuwei Chen; Deak Kirkham

2.7 Differentiation: Glossary

The pronunciation guide given in these glossaries draws on the symbols International Phonetic Alphabet. We have made two adaptations to the IPA system. Firstly, we use slanted lines as opposed to square brackets to enclose pronunciation material. We do this as the slanted line convention is more common in language teaching contexts. Secondly, we use the symbol <r> to denote the voiced alveolar approximant sound which appears in British English at the beginning of the each word in the phrase 'red rhombus'; the IPA would instead use an inverted 'r' symbols namely <ɹ>. We have done this to avoid unnecessary complication to the visuals of the glossaries.

In addition to these conventions, please also note that the pronunciations given are indicative of contemporary Standard British English pronunciation and should not be read as normative, nor as a target pronunciation that you must aim to produce. The pronunciation of English varies widely across World Englishes; part of becoming fluent in the academic English of mathematics is developing an awareness of, and sensitivity to, the variation in pronunciation that you will encounter.

Word	Pronunciation	Part of speech词性	Related terminology	Chinese meaning	Example sentence 1	Example sentence 2
calculus	/ˈkælkjʊləs/	n.		微积分	Calculus is the study of change and motion. 微积分研究变化与运动。	We learn limits, derivatives, and integrals in calculus. 我们在微积分中学习极限、导数和积分。
differentiation	/ˌdɪfərenʃiˈeɪʃən/	n.	differentiate v.求导；derivative n. 导数（求导得到的结果）；partial derivative 偏导数	求导，求微分（的过程）	Differentiation of \(x^3\) with respect to \(x\) gives \(3x^2\). 对 \(x^3\) 关于 \(x\) 求导得到\(3x^2\)。	Apply differentiation by the product rule to \(uv\). 对 \(uv\) 使用乘积法则求导。
limit	/ˈlɪmɪt/	n.		极限	The limit of a function describes its behavior near a point. 函数的极限描述其在某一点附近的行为。	As \(x\) approaches zero, the limit of \(\frac{\sin x}{x}\) is \(1\). 当 \(x\) 趋近于0时， \(\frac{\sin x}{x}\) 的极限是\(1\)。
rate of change	/reɪt əv tʃeɪndʒ/	n. phr.		变化率	instantaneous rate of change 瞬时变化率	rate of change of a function 函数的变化率
relative change	/ˈrɛlətɪv tʃeɪndʒ/	n. phr.		相对变化	\(\frac{Δf}{Δx}\) depicts the relative change of \(f\) with respect to \(x\). \(\frac{Δf}{Δx}\) 刻画了 \(f\) 相对于 \(x\) 的变化率。	relative change in / of... …中的相对变化，或, …的相对变化量
decrease	/ˈdɪkriːs/	n.	decrease v. /dɪˈkriːs/	减少	The function is decreasing. 该函数单调递减。	The function \(f(x)\) has a decrease on the interval \([1, 3]\). 函数 \(f(x)\) 在区间 \([1,3]\) 上是减函数。
increase	/ˈɪnkriːs/	n.	increase v. /ɪnˈkriːs/	增加	The function is increasing. 该函数单调递增。	Where \(f'(x) \gt 0\), the function exhibits an increase. 当 \(f'(x) \gt 0\) 时，函数呈上升趋势。
monotonic	/ˌmɒnəˈtɒnɪk/	adj.		单调的	A function \(f(x)\) is monotonic on \([a,b]\) if \(x_1 \lt x_2\) implies \(f(x_1) \leq f(x_2)\). 若对任意 \(x_1 \lt x_2\) 都有 \(f(x_1) \leq f(x_2)\)，则函数 \(f(x)\) 在区间 \([a,b]\) 上单调。	Every bounded monotonic sequence \(\{a_n\}\) must converge to a real limit. 每个有界单调数列 \(\{a_n\}\) 必定收敛到一个实数极限。
rapidly	/ˈræpɪdli/	adv.		快速地	Exponential functions increase more rapidly than linear functions. 指数函数比线性函数增长得更快。	The sequence decreases rapidly. 数列迅速减小。
tend to	/tɛnd tuː/	phr.		趋于	The difference of the two points tends to zero. 这两点之间的差趋于零.	The sequence tends to a limit. 数列趋于一个极限。
slope	/sləʊp/	n.		斜率	The slope of the tangent line at \(x = 3\) is \(5\). \(x = 3\) 处切线的斜率为 \(5\)。	Calculate the slope between two points. 求两点间斜率。
tangent	/ˈtændʒənt/	n.		切线；正切	Draw the tangent line to the circle at point \(P\). 在点 \(P\) 处画圆的切线。	calculate the tangent of the angle 计算角的正切
curvature	/ˈkɜːvətʃə/	n.		曲率（描述曲线弯曲程度的量）	the curvature of the original function 原函数的曲率	The curvature of the curve at \(x=0\) is equal to \(2\). 在 \(x=0\) 处，曲率为 \(2\)。
stationary point	/ˈsteɪʃənəri pɔɪnt/	n. phr.		驻点	The function has a stationary point at \(x = 2\). 这个函数在 \(x = 2\) 处有一个驻点。	Find the stationary point of the function. 求函数的驻点。
inflection point	/ɪnˈflekʃn pɔɪnt/	n. phr.		拐点	At an inflection point the second derivative changes sign. 在拐点处二阶导数改变符号。	The curve \(y=x^3\) has an inflection point at \((0,0)\). 曲线\(y=x^3\)在\((0,0)\)处有拐点。
maximum	/ˈmæksɪməm/	adj./n.	简写 max；maxima 极大值（单数）；maximise v. 最大化	最大的；极大值（复数）	local maximum 局部最大值； global maximum 全局最大值	The function reaches its global maximum at \(x = 3\). 该函数在 \(x = 3\) 处达到全局最大值。
minimum	/ˈmɪnɪməm/	adj./n.	简写 min；minima 极小值（单数）；minimise v. 最小化	最小的；极小值（复数）	local minimum 局部最⼩值； global minimum 全局最⼩值	The function has a local minimum at \(x = 2\). 该函数在 \(x = 2\) 处有一个局部最小值。
implicit differentiation	/ɪmˈplɪsɪt dɪˌfɜːrenʃɪˈeɪʃn/	n. phr.		隐函数求导	We use implicit differentiation to differentiate \(x^2+y^2=1\). 我们用隐函数求导对\(x^2+y^2=1\)求导.	For \(xy=1\) implicit differentiation gives \(y+x \frac{dy}{dx}=0\). 对\(xy=1\)隐函数求导得到\(y+x \frac{dy}{dx}=0\).
explicit differentiation	/ɪkˈsplɪsɪt ˌdɪfərenʃiˈeɪʃn/	n. phr.		显式求导	We apply explicit differentiation to find the derivative of \(y\) with respect to \(x\). 我们用显式求导来求 \(y\) 关于 \(x\) 的导数。	Explicit differentiation is used when \(y\) is given as an explicit function of \(x\). 当 \(y\) 以 \(x\) 的显式函数形式给出时使用显式求导。
power rule	/ˈpaʊə ruːl/	n. phr.		幂函数求导法则	The derivative of a single-term function like \(x^n\) can be found easily using the power rule. 对于像\(x^n\)的单项式函数，可以用幂法则轻松求导。	If \(f(x)=x^5\), then by the power rule, \(f'(x)=5 x^4\). 如果\(f(x)=x^5\)，利用幂函数求导法则，\(f'(x)=5 x^4\)。
product rule	/ˈprɒdʌkt ruːl/	n. phr.		乘积法则	We apply the product rule to differentiate \(x^2 \sin x\). 我们用乘积法则对\(x^2 \sin x\)求导。	We apply the product rule to differentiate \(x^2 \sin x\). 我们用乘积法则对\(x^2 \sin x\)求导。
quotient rule	/ˈkwəʊʃənt ruːl/	n. phr.		商法则，除法法则	By the quotient rule we differentiate \(f(x)=\frac{\sin x}{x}\). 利用商法则对\(f(x)=\frac{\sin x}{x}\)求导。	Apply the quotient rule by differentiating numerator and denominator separately. 分别对分子和分母求导来应用商法则。
chain rule	/tʃeɪn ruːl/	n. phr.		链式法则	To find the derivative of a composite function, we can apply the chain rule. 要求复合函数的导数，我们可以使用链式法则。	Combine the chain rule with the product rule to differentiate \(x^2 \sin (x^3)\). 对\(x^2 \sin (x^3)\) 求导时结合链式法则和乘积法则。