Exponential Functions
数学英语词汇大全
数学英语词汇数学mathematics, maths(BrE), math(AmE)公理axiom定理theorem计算calculation运算operation证明prove假设hypothesis, hypothes(pl.)命题proposition算术arithmetic加plus(prep.), add(v.), addition(n.)被加数augend, summand加数addend和sum减minus(prep.), subtract(v.), subtraction(n.) 被减数minuend减数subtrahend差remainder乘times(prep.), multiply(v.), multiplication(n.) 被乘数multiplicand, faciend乘数multiplicator积product除divided by(prep.), divide(v.), division(n.) 被除数dividend除数divisor商quotient等于equals, is equal to, is equivalent to大于is greater than小于is lesser than大于等于is equal or greater than小于等于is equal or lesser than运算符operator数字digit数number自然数natural number整数integer小数decimal小数点decimal point分数fraction分子numerator分母denominator比ratio正positive负negative零null, zero, nought, nil十进制decimal system二进制binary system十六进制hexadecimal system权weight, significance进位carry截尾truncation四舍五入round下舍入round down上舍入round up有效数字significant digit无效数字insignificant digit代数algebra公式formula, formulae(pl.)单项式monomial多项式polynomial, multinomial系数coefficient未知数unknown, x-factor, y-factor, z-factor 等式，方程式equation一次方程simple equation二次方程quadratic equation三次方程cubic equation四次方程quartic equation不等式inequation阶乘factorial对数logarithm指数，幂exponent乘方power二次方，平方square三次方，立方cube四次方the power of four, the fourth power n次方the power of n, the nth power开方evolution, extraction二次方根，平方根square root三次方根，立方根cube root四次方根the root of four, the fourth root n次方根the root of n, the nth root集合aggregate元素element空集void子集subset交集intersection并集union映射mapping函数function定义域domain, field of definition 值域range常量constant变量variable单调性monotonicity奇偶性parity周期性periodicity图象image数列，级数series微积分calculus微分differential导数derivative极限limit无穷大infinite(a.) infinity(n.)无穷小infinitesimal积分integral定积分definite integral不定积分indefinite integral 有理数rational number无理数irrational number 实数real number虚数imaginary number矩阵matrix行列式determinant几何geometry点point线line面plane体solid线段segment射线radial平行parallel相交intersect角angle角度degree弧度radian锐角acute angle直角right angle钝角obtuse angle平角straight angle周角perigon底base边side高height三角形triangle锐角三角形acute triangle直角三角形right triangle直角边leg斜边hypotenuse勾股定理Pythagorean theorem 钝角三角形obtuse triangle不等边三角形scalene triangle 等腰三角形isosceles triangle 等边三角形equilateral triangle 四边形quadrilateral平行四边形parallelogram矩形rectangle长length宽width菱形rhomb, rhombus, rhombi(pl.), diamond 正方形square梯形trapezoid直角梯形right trapezoid等腰梯形isosceles trapezoid五边形pentagon六边形hexagon七边形heptagon八边形octagon九边形enneagon十边形decagon十一边形hendecagon十二边形dodecagon多边形polygon正多边形equilateral polygon圆circle圆心centre(BrE), center(AmE)半径radius直径diameter圆周率pi弧arc半圆semicircle扇形sector环ring椭圆ellipse圆周circumference周长perimeter面积area轨迹locus, loca(pl.)相似similar全等congruent四面体tetrahedron五面体pentahedron六面体hexahedron平行六面体parallepiped 立方体cube七面体heptahedron八面体octahedron九面体enneahedron十面体decahedron十一面体hendecahedron十二面体dodecahedron 二十面体icosahedron多面体polyhedron棱锥pyramid棱柱prism棱台frustum of a prism旋转rotation轴axis圆锥cone圆柱cylinder圆台frustum of a cone球sphere半球hemisphere底面undersurface表面积surface area体积volume空间space坐标系coordinates坐标轴x-axis, y-axis, z-axis 横坐标x-coordinate纵坐标y-coordinate原点origin双曲线hyperbola抛物线parabola三角trigonometry正弦sine余弦cosine正切tangent余切cotangent正割secant余割cosecant反正弦arc sine反余弦arc cosine反正切arc tangent反余切arc cotangent反正割arc secant反余割arc cosecant相位phase周期period振幅amplitude内心incentre(BrE), incenter(AmE) 外心excentre(BrE), excenter(AmE)旁心escentre(BrE), escenter(AmE)垂心orthocentre(BrE), orthocenter(AmE)重心barycentre(BrE), barycenter(AmE)内切圆inscribed circle外切圆circumcircle统计statistics平均数average加权平均数weighted average方差variance标准差root-mean-square deviation, standard deviation 比例propotion百分比percent百分点percentage百分位数percentile排列permutation概率，或然率probability分布distribution正态分布normal distribution非正态分布abnormal distribution图表graph条形统计图bar graph柱形统计图histogram折线统计图broken line graph 曲线统计图curve diagram扇形统计图pie diagram数学专业英语词汇代数部分1.有关数*算add，plus加?subtract减?difference差??multiply, times乘?product积?divide除?divisible可被整除的? divided evenly被整除? dividend被除数，红利? divisor因子，除数? quotient商?remainder余数??factorial阶乘?power乘方?radical sign, root sign根号?round to四舍五入?to the nearest四舍五入2.有关集合union并集?proper subset真子集?solution set解集??3.?有关代数式、方程和不等式algebraic term代数项?like terms, similar terms同类项? numerical coefficient数字系数?literal coefficient字母系数??inequality不等式?triangle inequality三角不等式??range值域??original equation原方程?equivalent equation同解方程，等价方程? linear equation线性方程(e.g. 5?x?+6=22)?4.?有关分数和小数proper fraction真分数?improper fraction假分数?mixed number带分数? simple fraction简分数? numerator分子? denominator分母?quarter四分之一?decimal fraction纯小数? infinite decimal无穷小数recurring decimal循环小数? tenths unit十分位??5.基本数学概念?? arithmetic mean算术平均值? weighted average加权平均值? geometric mean几何平均数? exponent指数，幂?base乘幂的底数,底边?cube立方数，立方体? square root平方根?cube root立方根??digit数字?constant常数?variable变量??inverse function反函数?linear一次的，线性的?factorization因式分解?absolute value绝对值，e.g.｜-32｜=32?round off四舍五入?6.?有关数论?natural number自然数?positive number正数?negative number负数?odd integer, odd number奇数?even integer, even number偶数?integer, whole number整数?positive whole number正整数?negative whole number负整数??consecutive number连续整数?real number, rational number实数,有理数?irrational（number）无理数??inverse倒数?prime number质数e.g. 2,3,5,7,11,13,15……注意：所有的质数(2除外)都是奇数，但奇数不一定是质数reciprocal倒数?? multiple倍数?(prime) factor (质)因子?ordinary scale, decimal scale十进制? nonnegative非负的??tens十位?units个位??mode众数?median中数??7.?数列arithmetic progression(sequence)等差数列? geometric progression(sequence)等比数列??8.?其它?approximate近似?(anti)clockwise (逆)顺时针方向?cardinal基数?ordinal序数?direct proportion正比?distinct不同的?estimation估计，近似?parenthes括号?proportion比例?permutation排列?table表格?trigonometric function三角函数? unit单位,位?几何部分1.所有的角alternate angle内错角? corresponding angle同位角? vertical angle对顶角?central angle圆心角?interior angle内角?exterior angle外角? supplementary angles补角? adjacent angle邻角?acute angle锐角?obtuse angle钝角?right angle直角?round angle周角?straight angle平角?included angle夹角??2.?所有的三角形equilateral triangle等边三角形? scalene triangle不等边三角形? isosceles triangle等腰三角形?right triangle直角三角形?oblique斜三角形?inscribed triangle内接三角形??3.?有关收敛的平面图形，除三角形外? semicircle半圆?concentric circles同心圆? quadrilateral四边形?pentagon五边形?hexagon六边形?heptagon七边形?octagon八边形?nonagon九边形?decagon十边形?polygon多边形?parallelogram平行四边形? equilateral等边形?plane平面?square正方形，平方?rectangle长方形?regular polygon正多边形?rhombus菱形?trapezoid梯形??4.?其它平面图形arc弧?line, straight line直线?line segment线段?parallel lines平行线?segment of a circle弧形??5.?有关立体图形cube立方体，立方数?rectangular solid长方体?regular solid/regular polyhedron正多面体? circular cylinder圆柱体?cone圆锥?sphere球体?solid立体的??6.?有关图形上的附属物altitude高?depth深度?side边长?circumference, perimeter周长?radian弧度?surface area表面积?volume体积?arm直角三角形的股?cross section横截面?center of a circle圆心?chord弦?radius半径?angle bisector角平分线?diagonal对角线?diameter直径?edge棱?face of a solid立体的面?hypotenuse斜边?included side夹边?leg三角形的直角边?median of a triangle三角形的中线?base底边，底数（e.g. 2的5次方，2就是底数）? opposite直角三角形中的对边?midpoint中点?endpoint端点?vertex (复数形式vertices)顶点? tangent切线的?transversal截线?intercept截距??7.?有关坐标??coordinate system坐标系? rectangular coordinate直角坐标系? origin原点?abscissa横坐标?ordinate纵坐标?number line数轴?quadrant象限?slope斜率?8.?其它plane geometry平面几何? trigonometry三角学?bisect平分?circumscribe外切?inscribe内切?intersect相交? perpendicular垂直? pythagorean theorem勾股定理? congruent全等的? multilateral多边的?其它??1.?单位类?cent美分?penny一美分硬币?nickel 5美分硬币?dime一角硬币?dozen打（12个）?score廿(20个)?Centigrade摄氏?Fahrenheit华氏?quart夸脱?gallon加仑(1 gallon = 4 quart)? yard码?meter米?micron微米?inch英寸?foot英尺?minute分(角度的度量单位，60分=1度)? square measure平方单位制?cubic meter立方米?pint品脱(干量或液量的单位)??2.?有关文字叙述题，主要是有关商业intercalary year(leap year)闰年(366天)? depreciation折旧?down payment直接付款?discount打折?margin利润?profit利润?interest利息?simple interest单利?dividend红利?decrease to减少到?decrease by减少了?increase to增加到?increase by增加了?denote表示?list price标价?markup涨价?per capita每人?ratio比率?retail price零售价?tie打Chapter onefunction notation方程符号函数符号quadratic functions二次函数quadratic equations二次方程式二次等式chapter twoEquivalent algebraic expressions等价代数表达式rational expression有理式有理表达式horizontal and vertical translation of functions函数的水平和垂直的平移reflections of functions函数的倒映映射chapter threeExponential functions指数函数exponential decay指数式衰减exponent指数properties of exponential functions指数函数的特性chapter fourTrigonometry三角学Reciprocal trigonometric ratios倒数三角函数比Trigonometric functions三角函数Discrete functions离散函数。
Exponential Functions(指数函数)
B
Transformations
• Horizontal translation: g(x)=bx+c • Shifts the graph to the left if c > 0 • Shifts the graph to the right if c < 0
y 2x
y 2( x 3) y 2( x 4)
n
The Number e - Definition
The table shows 1 A ( 1 ) the values of n as n gets increasingly large.
n
nபைடு நூலகம்
n
1 1 n
n
1 2
2 2.25
5
10 100
2.48832
Compound Interest 复利
• The formula for compound interest:
r A(t ) P 1 n
nt
Where n is the number of times per year interest is being compounded and r is the annual rate.
2.59374246 2.704813829
As n , the approximate value of e (to 9 decimal places) is ≈ 2.718281827
1000
10,000 100,000 1,000,000
2.716923932
2.718145927 2.718268237 2.718280469
y a x (a 1)
Money and the Exponential Function
Money and the Exponential Function Algebra 2Goals:1. Write and graph exponential functions of the form ()x f x a b =⋅ (3.15)2. Use exponential equations to solve problems. Solve by graphing, substitution. (3.17) 3. Use exponential equations of the form ()(1)x f x r =+ where r is given as a rate of growth or decay to solve problems. (4.03)4. Find the Annual Percentage Rate (APR) for a given compounded rate.Materials and Equipment needed by each student: 1. Copy of the student handout 2. Calculator3. Paper and pencil for note taking.Activity One: Compound InterestA friend borrows $300 from you for maximum of thirty days. You charge your friend either1% per day or 30% (once) for entire 30 days for the use of your money. Compare the amounts your friend owes you.1. Define compound interest as interest gained on the balance (amount borrowed and past interest).2. For the 30%: Calculate $300(10.30)$390.00+=3. On compounded interest: Evaluate 30$300(10.01)$404.35+=4. Clearly the borrower would like the 30% and the lender would like 1% per day.5. What is the full 30-day rate that is the same as 1% per day for 30 days? Evaluate:30(10.01) 1.3478+= which shows a rate of 34.78% for the full 30 days.Activity Two: Savings AccountLindsey’s grandmother gave her $1000 on the day she was born. Lindsey’s parents put the money into a savings account that earns 6.4% interest each year.a. How much money will Lindsey have in the account on her 18th birthday if the interest is compounded annually, quarterly, monthly, daily, hourly, each minute, each second, continuously?b. In banking, the term Annual Percentage Rate (APR) is used to give the value of the annual simple interest rate that is equal to the compounded rate of an account. What is the APR for 6.4% each of the compounding schemes above?1. Discuss the meaning of compounding using the formula 01ntr A A n ⎛⎫=+ ⎪⎝⎭, where0A =the initial investment, r =the annual rate, n =number of times compounded inthe year, and t =number of years. Discuss the values of each of these variables in this problem.2. Use the 2nd Enter aspect of the calculator to re-enter the equation just used and then by editing find the value for each of the 18 year compounding methods.3. To find the APR, find the value of 1nr n ⎛⎫+ ⎪⎝⎭. This value will determine the amountthat the principal is multiplied by each year or by subtracting 1, gives the annual rateThis result is probably not reliable.Activity Two point Seven: What is e?One dollar is invested at 100% interest per year for a year. How much money is in the account at the end of the year? What value does the amount in the account approach as the number of compounding periods increases?1. One dollar invested at 100% interest per year for a year would yield 1$1(11)$2+=.The value of 2.718281828e = which helps us see the definition of 1lim 1nn e n →∞⎛⎫=+ ⎪⎝⎭.Follow Up: Several ApplicationsThe following questions from the Algebra 2 Indicators prepared by the NC Department of Public Instruction.1. The number of airline passengers increased from 465.6 million in 1990 to 614.3 million in 1998.∙ What was the average annual growth rate (percent) for the 1990 to 1998 period? ∙ If that rate remains constant after 1998, how many airline passengers can be expected in 2005?∙ Give the algebraic model for this growth.2. At the end of four years (t ), a savings account paying 5.35% annually ( r )compounded continuously had a balance (B ) of $3096.56. What was the initial deposit (P )? (Use B = Pe rt ) If the initial deposit had been in an account compounded annually, how much less interest would have been earned? 3. Solve 2350200r e = for r . Justify each step. 4. Solve 8960 1.075A =⋅ for A . Justify each step. 5. Solve 663492.165x =⋅for x . Justify each step.Each of these problems is simplified by using the graphing calculator.1. In this question, the two points can become ordered pairs with the x -values in L1 and the y -values in L2. Use 0 for 1990 and 8 for 1998. Using ExpReg from the Stat Calc menu, the calculator produces the function 465.6(1.03525)x y =. Since x -values are years, we know that each year the number of riders is multiplied by 1.03525. This creates an annual growth rate of3.525% each year. To find the number of passengers in 2005, find the value of Y1(15) since 2005 is 15 years from 1990. This result is 782.9, which is the number of passengers expected in 2005. The algebraic model is 465.6(1.03525)x y =.2. When we substitute the values given into the equation, the result is0.053543096.56Be = . Solve this by finding the value of 0.05354e .0.053543096.563096.56 1.2386226553096.561.2386226552500.00Be B BB==⋅== With annual compounding, the equation would be 40.0535250013079.491P ⎛⎫=+= ⎪⎝⎭This will result is $17.07 less than with continuous compounding.3. Graph 350y = and 2200r y e =on the same window. Use the intersect option on the calculator. A window that shows the intersection is 05,0400x y <<<<. The point of intersection occurs at (0.2798,350). Therefore, 0.2798r =4. By finding the value of 81.075and substituting this value into the equation,8960 1.075960 1.7834778269601.783477826538.274A A AA=⋅=⋅== 5. In this question, graph two functions 663y = and 492.165x y =⋅. A window that shows the intersection is 05,0800x y <<<<. The point of intersection occurs at (3.37246,663). Therefore, the solution is 3.37246x =.Student HandoutMoney and the Exponential FunctionAlgebra 21. A friend borrows $300 from you for maximum of thirty days. You charge your friend 1% per day or 30% for entire 30 days for the use of your money. Compare the amounts your friend owes you.2. Lindsey’s grandmother gave her $1000 on the day she was born. Lindsey’s parents put the money into a savings account that earns 6.4% interest each year.a. How much money will Lindsey have in the account on her 18th birthday if the interest is compounded annually, quarterly, monthly, daily, hourly, each minute, each second, continuously?b. In banking, the term Annual Percentage Rate (APR) is used to give the value of the annual simple interest rate that is equal to the compounded rate of an account. What is the APR for 6.4% each of the compounding schemes2.7. One dollar is invested at 100% per year for a year. How much money is in the account at the end of the year? What value does the amount in the account approach as the number of compounding periods increases?Follow UpMoney and the Exponential FunctionAlgebra 2The following questions from the Algebra 2 Indicators prepared by the NC Department of Public Instruction.1.The number of airline passengers increased from 465.6 million in 1990 to614.3 million in 1998.∙What was the average annual growth rate (percent) for the 1990 to 1998 period?∙If that rate remains constant after 1998, how many airline passengers can be expected in 2005?∙Give the algebraic model for this growth.2.At the end of four years (t), a savings account paying 5.35% annually ( r )compounded continuously had a balance (B) of $3096.56. What was the initial deposit (P)? (Use B = Pe rt) If the initial deposit had been in an account compounded annually, how much less interest would have been earned?3.Solve 2= for r. Justify each step.350200r e4.Solve 8=⋅ for A. Justify each step.A960 1.0755.Solve 663492.165x=⋅for x. Justify each step.。
数学英语词汇大全
数学英语词汇数学mathematics, maths（BrE），math(AmE）公理axiom定理theorem计算calculation运算operation证明prove假设hypothesis，hypotheses（pl.）命题proposition算术arithmetic加plus(prep。
)，add(v。
），addition(n.)被加数augend，summand加数addend和sum减minus(prep。
), subtract（v.）, subtraction（n.)被减数minuend减数subtrahend差remainder乘times（prep.）, multiply（v。
)，multiplication（n。
）被乘数multiplicand，faciend乘数multiplicator积product除divided by（prep.)，divide(v.), division(n.)被除数dividend除数divisor商quotient等于equals，is equal to，is equivalent to大于is greater than小于is lesser than大于等于is equal or greater than小于等于is equal or lesser than运算符operator数字digit数number自然数natural number整数integer小数decimal小数点decimal point分数fraction分子numerator分母denominator比ratio负negative零null, zero，nought, nil十进制decimal system二进制binary system十六进制hexadecimal system权weight, significance进位carry截尾truncation四舍五入round下舍入round down上舍入round up有效数字significant digit无效数字insignificant digit代数algebra公式formula, formulae（pl.)单项式monomial多项式polynomial，multinomial系数coefficient未知数unknown, x-factor，y-factor, z—factor 等式，方程式equation一次方程simple equation二次方程quadratic equation三次方程cubic equation四次方程quartic equation不等式inequation阶乘factorial对数logarithm指数，幂exponent乘方power二次方，平方square三次方，立方cube四次方the power of four, the fourth powern次方the power of n，the nth power开方evolution，extraction二次方根，平方根square root三次方根，立方根cube root四次方根the root of four, the fourth rootn次方根the root of n, the nth root集合aggregate元素element空集void子集subset交集intersection补集complement映射mapping函数function定义域domain，field of definition 值域range常量constant变量variable单调性monotonicity奇偶性parity周期性periodicity图象image数列，级数series微积分calculus微分differential导数derivative极限limit无穷大infinite(a。
B16 – Derivatives of exponential functions：B16–指数函数求导共18页文档
M acin to sh P IC T im ag e fo rm at
(3) Differentiate is n o t s u p p o rte d
(1) Use GDC to draw the tangent lines at various x values
(2) Record the slopes of the tangent lines on a table. (3) Prepare a scatter plot from the table of values. (4) Describe scatter plot
Investigate lim h0 (4h – 1)/h Investigate lim h0 (eh – 1)/h
numerically with a table of
numerically with a table of
values
values
x
y
-0.00010 1.38620
5
11.10.2019 Calculus - Santowski
EXPLORATION – PART 3
Now let’s use graphing technology:
Use the TI-89 to directly and immediately prepare a graph of the derivative of y = bx.
M a c in to s h P IC T im a g e fo rm a t
数学专业英语常用词汇
数学专业英语词汇代数部分1. 有关数*算add，plus 加?subtract 减?difference 差??multiply, times 乘?product 积?divide 除?divisible 可被整除的?divided evenly被整除? dividend 被除数，红利?divisor 因子，除数?quotient 商?remainder余数??factorial 阶乘?power 乘方?radical sign, root sign 根号? round to四舍五入?to the nearest 四舍五入2. 有关集合union 并集?proper subset 真子集?solution set 解集??3.有关代数式、方程和不等式algebraic term 代数项?like terms, similar terms同类项?numerical coefficient 数字系数?literal coefficient 字母系数??inequality 不等式?triangle inequality 三角不等式??range 值域??original equation 原方程?equivalent equation 同解方程，等价方程?linear equation 线性方程(e.g.5?x?+6=22)?4.有关分数和小数proper fraction真分数?improper fraction 假分数?mixed number 带分数?vulgar fraction，common fraction 普通分数?simple fraction简分数?complex fraction繁分数??numerator 分子?denominator 分母?(least) common denominator（最小）公分母?quarter 四分之一?decimal fraction 纯小数?infinite decimal 无穷小数recurring decimal循环小数?tenths unit 十分位??5. 基本数学概念arithmetic mean 算术平均值?weighted average 加权平均值?geometric mean 几何平均数?exponent 指数，幂?base 乘幂的底数,底边?cube 立方数，立方体?square root平方根?cube root 立方根??common logarithm 常用对数?digit 数字?constant 常数?variable 变量??inverse function反函数? complementary function 余函数?linear 一次的，线性的?factorization 因式分解?absolute value绝对值，e.g.｜-32｜=32? round off四舍五入 ?6.有关数论natural number 自然数?positive number 正数?negative number 负数?odd integer, odd number 奇数?even integer, even number 偶数? integer, whole number 整数?positive whole number 正整数?negative whole number 负整数?? consecutive number 连续整数?real number, rational number 实数,有理数?irrational（number）无理数??inverse 倒数?composite number 合数 e.g.4,6,8,9,10,12,14,15……?prime number 质数 e.g.2,3,5,7,11,13,15……注意：所有的质数(2除外)都是奇数，但奇数不一定是质数reciprocal 倒数??common divisor 公约数?multiple 倍数?(least)common multiple (最小)公倍数??(prime) factor (质)因子?common factor 公因子??ordinary scale, decimal scale 十进制?nonnegative 非负的??tens 十位?units 个位??mode众数?median 中数??common ratio 公比??7.数列arithmetic progression(sequence) 等差数列?geometric progression(sequence) 等比数列??approximate 近似?(anti)clockwise (逆) 顺时针方向? cardinal 基数?ordinal 序数?direct proportion 正比?distinct 不同的?estimation 估计，近似? parentheses 括号?proportion 比例?permutation 排列?combination 组合?table 表格?trigonometric function 三角函数? unit 单位,位?几何部分1. 所有的角alternate angle 内错角? corresponding angle 同位角? vertical angle对顶角?central angle圆心角?interior angle 内角?exterior angle 外角? supplementary angles补角? complementary angle余角? adjacent angle 邻角?acute angle 锐角?obtuse angle 钝角?right angle 直角?round angle周角?straight angle 平角?included angle夹角??2.所有的三角形equilateral triangle 等边三角形? scalene triangle不等边三角形? isosceles triangle等腰三角形? right triangle 直角三角形? oblique 斜三角形?inscribed triangle 内接三角形??3.有关收敛的平面图形，除三角形外semicircle 半圆?concentric circles 同心圆? quadrilateral四边形?pentagon 五边形?hexagon 六边形?heptagon 七边形?octagon 八边形?nonagon 九边形?decagon 十边形?polygon多边形?parallelogram 平行四边形? equilateral 等边形?plane 平面? square 正方形，平方?rectangle 长方形?regular polygon 正多边形?rhombus 菱形?trapezoid梯形??4.其它平面图形arc 弧?line, straight line 直线?line segment 线段?parallel lines 平行线?segment of a circle 弧形??5.有关立体图形cube 立方体，立方数?rectangular solid 长方体?regular solid/regular polyhedron 正多面体?circular cylinder 圆柱体?cone圆锥?sphere 球体?solid 立体的??6.有关图形上的附属物altitude 高?depth 深度?side 边长?circumference, perimeter 周长? radian弧度?surface area 表面积?volume 体积?arm 直角三角形的股?cross section 横截面?center of a circle 圆心?chord 弦?radius 半径?angle bisector 角平分线?diagonal 对角线?diameter 直径?edge 棱?face of a solid 立体的面? hypotenuse 斜边?included side夹边?leg三角形的直角边?median of a triangle 三角形的中线?base 底边，底数（e.g. 2的5次方，2就是底数）?opposite直角三角形中的对边? midpoint 中点? endpoint 端点?vertex (复数形式vertices)顶点? tangent 切线的?transversal截线?intercept 截距??7.有关坐标coordinate system 坐标系? rectangular coordinate 直角坐标系? origin 原点?abscissa横坐标?ordinate纵坐标?number line 数轴?quadrant 象限?slope斜率?complex plane 复平面??8.其它plane geometry 平面几何? trigonometry 三角学?bisect 平分?circumscribe 外切?inscribe 内切?intersect相交?perpendicular 垂直?pythagorean theorem勾股定理? congruent 全等的?multilateral 多边的?1.单位类cent 美分?penny 一美分硬币 ?nickel 5美分硬币?dime 一角硬币?dozen 打（12个）?score 廿(20个)?Centigrade 摄氏?Fahrenheit 华氏?quart 夸脱?gallon 加仑(1 gallon = 4 quart)?yard 码?meter 米?micron 微米?inch 英寸?foot 英尺?minute 分(角度的度量单位，60分=1度)? square measure 平方单位制? cubic meter 立方米?pint 品脱(干量或液量的单位)??2.有关文字叙述题，主要是有关商业intercalary year(leap year) 闰年(366天)?common year 平年(365天)? depreciation 折旧?down payment 直接付款?discount 打折?margin 利润?profit 利润?interest 利息?simple interest 单利?compounded interest 复利?dividend 红利?decrease to 减少到?decrease by 减少了?increase to 增加到?increase by 增加了?denote 表示?list price 标价?markup 涨价?per capita 每人?ratio 比率?retail price 零售价?tie 打Chapter onefunction notation方程符号函数符号quadratic functions 二次函数quadratic equations 二次方程式二次等式chapter twoEquivalent algebraic expressions 等价代数表达式rational expression 有理式有理表达式horizontal and vertical translation of functions 函数的水平和垂直的平移reflections of functions 函数的倒映映射chapter threeExponential functions 指数函数exponential decay 指数式衰减exponent 指数properties of exponential functions 指数函数的特性chapter fourTrigonometry 三角学Reciprocal trigonometric ratios 倒数三角函数比Trigonometric functions 三角函数Discrete functions 离散函数数学 mathematics, maths(BrE), math(AmE) 公理 axiom定理 theorem计算 calculation运算 operation证明 prove假设 hypothesis, hypotheses(pl.)命题 proposition算术 arithmetic加 plus(prep.), add(v.), addition(n.)被加数 augend, summand加数 addend和 sum减 minus(prep.), subtract(v.), subtraction(n.)被减数 minuend减数 subtrahend差 remainder乘 times(prep.), multiply(v.), multiplication(n.)被乘数 multiplicand, faciend乘数 multiplicator积 product除 divided by(prep.), divide(v.), division(n.)被除数 dividend除数 divisor商 quotient等于 equals, is equal to, is equivalent to大于 is greater than小于 is lesser than大于等于 is equal or greater than小于等于 is equal or lesser than运算符 operator数字 digit数 number自然数 natural number整数 integer小数 decimal小数点 decimal point分数 fraction分子 numerator分母 denominator比 ratio正 positive负 negative零 null, zero, nought, nil十进制 decimal system二进制 binary system十六进制 hexadecimal system权 weight, significance进位 carry截尾 truncation 四舍五入 round下舍入 round down上舍入 round up有效数字 significant digit无效数字 insignificant digit代数 algebra公式 formula, formulae(pl.)单项式 monomial多项式 polynomial, multinomial系数 coefficient未知数 unknown, x-factor, y-factor, z-factor等式，方程式 equation一次方程 simple equation二次方程 quadratic equation三次方程 cubic equation四次方程 quartic equation不等式 inequation阶乘 factorial对数 logarithm指数，幂 exponent乘方 power二次方，平方 square三次方，立方 cube四次方 the power of four, the fourth powern次方 the power of n, the nth power开方 evolution, extraction二次方根，平方根 square root三次方根，立方根 cube root四次方根 the root of four, the fourth rootn次方根 the root of n, the nth root集合 aggregate元素 element空集 void子集 subset交集 intersection并集 union补集 complement映射 mapping函数 function定义域 domain, field of definition值域 range常量 constant变量 variable单调性 monotonicity奇偶性 parity周期性 periodicity图象 image数列，级数 series微积分 calculus微分 differential 导数 derivative极限 limit无穷大 infinite(a.) infinity(n.) 无穷小 infinitesimal积分 integral定积分 definite integral不定积分 indefinite integral有理数 rational number无理数 irrational number实数 real number虚数 imaginary number复数 complex number矩阵 matrix行列式 determinant几何 geometry点 point线 line面 plane体 solid线段 segment射线 radial平行 parallel相交 intersect角 angle角度 degree弧度 radian锐角 acute angle直角 right angle钝角 obtuse angle平角 straight angle周角 perigon底 base边 side高 height三角形 triangle锐角三角形 acute triangle直角三角形 right triangle直角边 leg斜边 hypotenuse勾股定理 Pythagorean theorem钝角三角形 obtuse triangle不等边三角形 scalene triangle等腰三角形 isosceles triangle等边三角形 equilateral triangle四边形 quadrilateral平行四边形 parallelogram矩形 rectangle长 length宽 width菱形 rhomb, rhombus, rhombi(pl.), diamond正方形 square 梯形 trapezoid直角梯形 right trapezoid等腰梯形 isosceles trapezoid 五边形 pentagon六边形 hexagon七边形 heptagon八边形 octagon九边形 enneagon十边形 decagon十一边形 hendecagon十二边形 dodecagon多边形 polygon正多边形 equilateral polygon 圆 circle圆心 centre(BrE), center(AmE) 半径 radius直径 diameter圆周率 pi弧 arc半圆 semicircle扇形 sector环 ring椭圆 ellipse圆周 circumference周长 perimeter面积 area轨迹 locus, loca(pl.)相似 similar全等 congruent四面体 tetrahedron五面体 pentahedron六面体 hexahedron平行六面体 parallelepiped 立方体 cube七面体 heptahedron八面体 octahedron九面体 enneahedron十面体 decahedron十一面体 hendecahedron十二面体 dodecahedron二十面体 icosahedron多面体 polyhedron棱锥 pyramid棱柱 prism棱台 frustum of a prism 旋转 rotation轴 axis圆锥 cone圆柱 cylinder圆台 frustum of a cone球 sphere半球 hemisphere 底面 undersurface表面积 surface area体积 volume空间 space坐标系 coordinates坐标轴 x-axis, y-axis, z-axis 横坐标 x-coordinate纵坐标 y-coordinate原点 origin双曲线 hyperbola抛物线 parabola三角 trigonometry正弦 sine余弦 cosine正切 tangent余切 cotangent正割 secant余割 cosecant反正弦 arc sine反余弦 arc cosine反正切 arc tangent反余切 arc cotangent反正割 arc secant反余割 arc cosecant相位 phase周期 period振幅 amplitude内心 incentre(BrE), incenter(AmE)外心 excentre(BrE), excenter(AmE)旁心 escentre(BrE), escenter(AmE)垂心 orthocentre(BrE),orthocenter(AmE)重心 barycentre(BrE), barycenter(AmE)内切圆 inscribed circle外切圆 circumcircle统计 statistics平均数 average加权平均数 weighted average方差 variance标准差 root-mean-square deviation, standard deviation比例 propotion百分比 percent百分点 percentage百分位数 percentile排列 permutation组合 combination概率，或然率 probability分布 distribution正态分布 normal distribution非正态分布 abnormal distribution图表 graph条形统计图 bar graph 柱形统计图 histogram折线统计图 broken line graph 曲线统计图 curve diagram扇形统计图 pie diagram。
数学专业英语常用词汇
数学专业英语词汇代数部分1. 有关数*算add，plus 加?subtract 减?difference 差??multiply, times 乘?product 积?divide 除?divisible 可被整除的?divided evenly被整除?dividend 被除数，红利?divisor 因子，除数?quotient 商?remainder余数??factorial 阶乘?power 乘方?radical sign, root sign 根号? round to四舍五入?to the nearest 四舍五入2. 有关集合union 并集?proper subset 真子集?solution set 解集??3.有关代数式、方程和不等式algebraic term 代数项?like terms, similar terms同类项? numerical coefficient 数字系数? literal coefficient 字母系数?? inequality 不等式?triangle inequality 三角不等式?? range 值域??original equation 原方程?equivalent equation 同解方程，等价方程? linear equation 线性方程(e.g. 5?x?+6=22)?4.有关分数和小数proper fraction真分数?improper fraction 假分数?mixed number 带分数?vulgar fraction，common fraction 普通分数?simple fraction简分数?complex fraction繁分数??numerator 分子?denominator 分母?(least) common denominator（最小）公分母?quarter 四分之一?decimal fraction 纯小数?infinite decimal 无穷小数recurring decimal循环小数?tenths unit 十分位??5. 基本数学概念arithmetic mean 算术平均值?weighted average 加权平均值?geometric mean 几何平均数?exponent 指数，幂?base 乘幂的底数,底边?cube 立方数，立方体?square root平方根?cube root 立方根??common logarithm 常用对数?digit 数字?constant 常数?variable 变量??inverse function反函数?complementary function 余函数?linear 一次的，线性的?factorization 因式分解?absolute value绝对值，e.g.｜-32｜=32? round off四舍五入 ?6.有关数论natural number 自然数?positive number 正数?negative number 负数?odd integer, odd number 奇数?even integer, even number 偶数? integer, whole number 整数?positive whole number 正整数?negative whole number 负整数?? consecutive number 连续整数?real number, rational number 实数,有理数?irrational（number）无理数??inverse 倒数?composite number 合数 e.g. 4,6,8,9,10,12,14,15……?prime number 质数 e.g. 2,3,5,7,11,13,15……注意：所有的质数(2除外)都是奇数，但奇数不一定是质数reciprocal 倒数??common divisor 公约数?multiple 倍数?(least)common multiple (最小)公倍数?? (prime) factor (质)因子?common factor 公因子??ordinary scale, decimal scale 十进制? nonnegative 非负的??tens 十位?units 个位??mode众数?median 中数??common ratio 公比?? 7.数列arithmetic progression(sequence) 等差数列?geometric progression(sequence) 等比数列??approximate 近似?(anti)clockwise (逆) 顺时针方向? cardinal 基数?ordinal 序数?direct proportion 正比?distinct 不同的?estimation 估计，近似?parentheses 括号?proportion 比例?permutation 排列?combination 组合?table 表格?trigonometric function 三角函数?unit 单位,位?几何部分1. 所有的角alternate angle 内错角? corresponding angle 同位角?vertical angle对顶角?central angle圆心角?interior angle 内角?exterior angle 外角?supplementary angles补角? complementary angle余角?adjacent angle 邻角?acute angle 锐角?obtuse angle 钝角?right angle 直角?round angle周角?straight angle 平角?included angle夹角??2.所有的三角形equilateral triangle 等边三角形? scalene triangle不等边三角形? isosceles triangle等腰三角形? right triangle 直角三角形? oblique 斜三角形?inscribed triangle 内接三角形??3.有关收敛的平面图形，除三角形外semicircle 半圆?concentric circles 同心圆? quadrilateral四边形?pentagon 五边形?hexagon 六边形?heptagon 七边形?octagon 八边形?nonagon 九边形?decagon 十边形?polygon多边形?parallelogram 平行四边形? equilateral 等边形?plane 平面?square 正方形，平方?rectangle 长方形?regular polygon 正多边形? rhombus 菱形?trapezoid梯形??4.其它平面图形arc 弧?line, straight line 直线?line segment 线段?parallel lines 平行线?segment of a circle 弧形?? 5.有关立体图形cube 立方体，立方数?rectangular solid 长方体?regular solid/regular polyhedron 正多面体?circular cylinder 圆柱体?cone圆锥?sphere 球体?solid 立体的??6.有关图形上的附属物altitude 高?depth 深度?side 边长?circumference, perimeter 周长?radian弧度?surface area 表面积?volume 体积?arm 直角三角形的股?cross section 横截面?center of a circle 圆心?chord 弦?radius 半径?angle bisector 角平分线?diagonal 对角线?diameter 直径?edge 棱?face of a solid 立体的面? hypotenuse 斜边?included side夹边?leg三角形的直角边?median of a triangle 三角形的中线? base 底边，底数（e.g. 2的5次方，2就是底数）?opposite直角三角形中的对边? midpoint 中点?endpoint 端点?vertex (复数形式vertices)顶点? tangent 切线的?transversal截线?intercept 截距??7.有关坐标coordinate system 坐标系? rectangular coordinate 直角坐标系? origin 原点?abscissa横坐标?ordinate纵坐标?number line 数轴?quadrant 象限?slope斜率?complex plane 复平面??8.其它plane geometry 平面几何? trigonometry 三角学?bisect 平分?circumscribe 外切?inscribe 内切?intersect相交?perpendicular 垂直?pythagorean theorem勾股定理? congruent 全等的?multilateral 多边的?1.单位类cent 美分?penny 一美分硬币 ?nickel 5美分硬币?dime 一角硬币?dozen 打（12个）?score 廿(20个)? Centigrade 摄氏?Fahrenheit 华氏?quart 夸脱?gallon 加仑(1 gallon = 4 quart)?yard 码?meter 米?micron 微米?inch 英寸?foot 英尺?minute 分(角度的度量单位，60分=1度)? square measure 平方单位制?cubic meter 立方米?pint 品脱(干量或液量的单位)??2.有关文字叙述题，主要是有关商业intercalary year(leap year) 闰年(366天)? common year 平年(365天)?depreciation 折旧?down payment 直接付款?discount 打折?margin 利润?profit 利润?interest 利息?simple interest 单利?compounded interest 复利?dividend 红利?decrease to 减少到?decrease by 减少了?increase to 增加到?increase by 增加了?denote 表示?list price 标价?markup 涨价?per capita 每人?ratio 比率?retail price 零售价?tie 打Chapter onefunction notation方程符号函数符号quadratic functions 二次函数quadratic equations 二次方程式二次等式chapter twoEquivalent algebraic expressions 等价代数表达式rational expression 有理式有理表达式horizontal and vertical translation of functions 函数的水平和垂直的平移reflections of functions 函数的倒映映射chapter threeExponential functions 指数函数exponential decay 指数式衰减exponent 指数properties of exponential functions 指数函数的特性chapter fourTrigonometry 三角学Reciprocal trigonometric ratios 倒数三角函数比Trigonometric functions 三角函数Discrete functions 离散函数数学 mathematics, maths(BrE), math(AmE) 公理 axiom定理 theorem计算 calculation运算 operation证明 prove假设 hypothesis, hypotheses(pl.)命题 proposition算术 arithmetic加 plus(prep.), add(v.), addition(n.) 被加数 augend, summand加数 addend和 sum减minus(prep.), subtract(v.), subtraction(n.)被减数 minuend减数 subtrahend差 remainder乘times(prep.), multiply(v.), multiplication(n.)被乘数 multiplicand, faciend乘数 multiplicator积 product除divided by(prep.), divide(v.), division(n.)被除数 dividend除数 divisor商 quotient等于 equals, is equal to, is equivalent to大于 is greater than小于 is lesser than大于等于 is equal or greater than小于等于 is equal or lesser than运算符 operator数字 digit数 number自然数 natural number整数 integer小数 decimal小数点 decimal point分数 fraction分子 numerator分母 denominator比 ratio正 positive负 negative零 null, zero, nought, nil十进制 decimal system二进制 binary system十六进制 hexadecimal system权 weight, significance进位 carry截尾 truncation四舍五入 round下舍入 round down上舍入 round up有效数字 significant digit 无效数字 insignificant digit代数 algebra公式 formula, formulae(pl.)单项式 monomial多项式 polynomial, multinomial系数 coefficient未知数unknown, x-factor, y-factor, z-factor等式，方程式 equation一次方程 simple equation二次方程 quadratic equation三次方程 cubic equation四次方程 quartic equation不等式 inequation阶乘 factorial对数 logarithm指数，幂 exponent乘方 power二次方，平方 square三次方，立方 cube四次方the power of four, the fourth powern次方 the power of n, the nth power 开方 evolution, extraction二次方根，平方根 square root三次方根，立方根 cube root四次方根 the root of four, the fourth rootn次方根 the root of n, the nth root 集合 aggregate元素 element空集 void子集 subset交集 intersection并集 union补集 complement映射 mapping函数 function定义域 domain, field of definition值域 range常量 constant变量 variable单调性 monotonicity奇偶性 parity周期性 periodicity图象 image数列，级数 series微积分 calculus微分 differential导数 derivative极限 limit无穷大 infinite(a.) infinity(n.) 无穷小 infinitesimal积分 integral定积分 definite integral不定积分 indefinite integral有理数 rational number无理数 irrational number实数 real number虚数 imaginary number复数 complex number矩阵 matrix行列式 determinant几何 geometry点 point线 line面 plane体 solid线段 segment射线 radial平行 parallel相交 intersect角 angle角度 degree弧度 radian锐角 acute angle直角 right angle钝角 obtuse angle平角 straight angle周角 perigon 底 base边 side高 height三角形 triangle锐角三角形 acute triangle直角三角形 right triangle直角边 leg斜边 hypotenuse勾股定理 Pythagorean theorem钝角三角形 obtuse triangle不等边三角形 scalene triangle等腰三角形 isosceles triangle等边三角形 equilateral triangle四边形 quadrilateral平行四边形 parallelogram矩形 rectangle长 length宽 width菱形rhomb, rhombus, rhombi(pl.), diamond正方形 square梯形 trapezoid直角梯形 right trapezoid等腰梯形 isosceles trapezoid五边形 pentagon六边形 hexagon七边形 heptagon八边形 octagon九边形 enneagon十边形 decagon十一边形 hendecagon十二边形 dodecagon多边形 polygon正多边形 equilateral polygon圆 circle圆心 centre(BrE), center(AmE)半径 radius直径 diameter圆周率 pi弧 arc半圆 semicircle扇形 sector环 ring椭圆 ellipse圆周 circumference周长 perimeter面积 area轨迹 locus, loca(pl.)相似 similar全等 congruent四面体 tetrahedron五面体 pentahedron六面体 hexahedron平行六面体 parallelepiped立方体 cube七面体 heptahedron八面体 octahedron九面体 enneahedron十面体 decahedron十一面体 hendecahedron十二面体 dodecahedron二十面体 icosahedron多面体 polyhedron棱锥 pyramid棱柱 prism棱台 frustum of a prism旋转 rotation轴 axis圆锥 cone圆柱 cylinder圆台 frustum of a cone球 sphere半球 hemisphere底面 undersurface表面积 surface area体积 volume空间 space坐标系 coordinates坐标轴 x-axis, y-axis, z-axis 横坐标 x-coordinate 纵坐标 y-coordinate原点 origin双曲线 hyperbola抛物线 parabola三角 trigonometry正弦 sine余弦 cosine正切 tangent余切 cotangent正割 secant余割 cosecant反正弦 arc sine反余弦 arc cosine反正切 arc tangent反余切 arc cotangent反正割 arc secant反余割 arc cosecant相位 phase周期 period振幅 amplitude内心 incentre(BrE), incenter(AmE)外心 excentre(BrE), excenter(AmE)旁心 escentre(BrE), escenter(AmE)垂心 orthocentre(BrE), orthocenter(AmE) 重心 barycentre(BrE), barycenter(AmE)内切圆 inscribed circle外切圆 circumcircle统计 statistics平均数 average加权平均数 weighted average方差 variance标准差root-mean-square deviation, standard deviation比例 propotion百分比 percent百分点 percentage百分位数 percentile排列 permutation组合 combination概率，或然率 probability分布 distribution正态分布 normal distribution非正态分布 abnormal distribution 图表 graph条形统计图 bar graph柱形统计图 histogram折线统计图 broken line graph曲线统计图 curve diagram扇形统计图 pie diagram。
函数英语知识点归纳总结
函数英语知识点归纳总结Mathematical Functions1. Definition of a FunctionA function is a relation between a set of inputs (the domain) and a set of possible outputs (the codomain), such that each input is related to exactly one output. The notation f(x) is commonly used to represent the output of a function f when given the input x.2. Domain and RangeThe domain of a function is the set of all possible inputs, while the range is the set of all possible outputs. It is important to determine the domain and range of a function in order to fully understand its behavior.3. Function NotationFunctions are often represented using specific notation, such as f(x), g(x), or h(x), to indicate the relationship between the input variable x and the corresponding output.4. Linear FunctionsA linear function is a type of function that can be represented by a straight line on a graph. It has the form f(x) = mx + b, where m is the slope of the line and b is the y-intercept.5. Quadratic FunctionsQuadratic functions are functions of the form f(x) = ax^2 + bx + c, where a, b, and c are constants. These functions are characterized by their parabolic shape when graphed.6. Exponential FunctionsExponential functions have the form f(x) = ab^x, where a and b are constants and b is the base of the exponential function. These functions grow or decay at a constant rate.7. Trigonometric FunctionsTrigonometric functions such as sine, cosine, and tangent are used to model periodic phenomena and are defined based on the ratios of sides of right-angled triangles.8. Composite FunctionsA composite function is a function that is formed by applying one function to the output of another function. It is denoted as (g ∘ f)(x), where g and f are functions.Computer Science Functions1. Function Declaration and DefinitionIn computer science, a function is a block of code that performs a specific task and can be called or invoked from other parts of a program. A function is declared using a specific syntax and is defined with a block of code that specifies its behavior.2. Parameters and ArgumentsFunctions can take input parameters, which are variables that are passed into the function when it is called. These parameters are used to customize the behavior of the function. Arguments are the actual values that are passed as input when the function is called.3. Return ValuesFunctions in computer science can return a value as a result of their execution. This returned value can be used in other parts of the program. Some functions may not return any value and are called void functions.4. Function InvocationFunctions are called or invoked by using their name followed by parentheses, with any necessary arguments included inside the parentheses. This causes the function's code to be executed.5. RecursionRecursion is a programming technique in which a function calls itself to solve a smaller instance of the same problem. This allows for elegant and concise solutions to certain types of problems.6. Function LibrariesIn computer science, functions are often organized into libraries or modules, which contain reusable code that can be shared across different programs. This allows for code reusability and improved maintainability.7. Anonymous FunctionsSome programming languages support the concept of anonymous functions, which are functions that do not have a name and can be used as values or arguments to other functions.8. Higher-order FunctionsHigher-order functions are functions that operate on other functions by taking them as arguments or returning them as results. This allows for powerful abstractions and expressive code.In conclusion, functions are a fundamental concept in mathematics and computer science, and they are used to represent relationships and perform tasks in both fields. Byunderstanding the key concepts and important points related to functions, individuals can develop a deeper comprehension and appreciation for the role that functions play in these disciplines.。
exp是什么意思的缩写_exp是什么意思
exp是什么意思的缩写_exp是什么意思EXP它的英文全称是Exponential，而它也是Exponential的缩写。
下面是店铺给大家整理的exp是什么意思的缩写，供大家参阅!目录exp是什么意思的缩写Exponential指数函数exp的英文全称Exponentialexp英语例句1. The policy tried to check the exponential growth of public expenditure.该政策试图控制公共开支的迅猛增长。
2. Populations tend to grow at an exponential rate.人口趋向于以指数比率增长.3. The logarithmic function is defined as the inverse of the exponential function.对数函数是作为指数函数的反函数来定义的.4. The envelopes of the strain peaks is an exponential function.应变峰值的包络线是一个指数函数.5. The frictional dissipation will introduce an exponential decay.摩擦耗散可导致指数式的衰减.6. These measurements showed that the electronic heat well below Tc was dominated by an exponential dependence.这些测量结果表明,在远低于T c的温度下,电子比热是按指数规律变化的.7. The first exponential has been introduced to allow for variation of amplitude with attitude.前面的指数是为了考虑波动振幅随高度变化而引入的.8. These exponential or logarithmic relationship, that characterize harmonious growth with changing proportions, are termed " allometric ".作为比例有变化的协调增长这些指数或对数关系被标为“ 开度量” 关系.9. An exponential function is one in which the variable appears in an exponent.指数函数是指数中出现变量的函数.10. Im writing a book about trigonometric and exponential functions.我在写一本关于三角和指数功能的书.11. Relationships between embryo weight and incubation time were exponential regression.乌骨鸡胚胎的重量与孵化时间的关系呈指数式回归.12. The confidence level and uncertainty limit of even exponential of uniform R.均匀随机变数偶指数函数之均值不确定范围之信心准位.13. The Fisher - type statistic is very effective for testing outlier in exponential sample.在指数样本异常值的检验中,Fisher型统计量是十分有效的.14. For many years, processing power has increased at an exponential rate.多年以来, 处理能力以几何级数增长率增加了.15. The hydrolysis rate constant increases at a exponential function with temperature increase.温度升高,水解速率常数呈指数函数关系递增.Exp的双语例句1. May I know what kind of EXP material you have?我可以知道您有什么样的EXP材料吗 ?2. EXP are divided equally among all characters participating in battle.EXP将会平均分配给参加战斗的所有角色.3. The fast and accurate evaluation of transcendental functions ( e . g . exp, log, sin, tan ) is very important in the field of scientific computing.科学计算中的许多领域都需要快速而精确地计算超越函数,即exp 、log 、 sin 、 tan 等此类函数.4. It also discusses two relatively advance application software : schedule management software P 3 and contract software EXP.介绍了目前比较先进的两种项目管理应用软件,P3进度管理软件和EXP合同管理软件.5. The regulations of EXP expression under ethylene, ASA, 1 - MCP, LTC treatments also studied in the experiment.枇杷果实为材料,研究采后果实贮藏过程中EXP基因表达模式, 以及乙烯、 ASA 、 1-MCP和LTC处理对其质地变化的调控和对EXP基因表达的影响.6. Exp . 3 : same as Exp . 1 with new question.实验 3: 与实验1相同,但探究新问题.7. Rien ne vaut l'exp é rience de la douleur pour apprendre à aimer.没有比学习爱更痛苦的经历了.8. Exp : The SOM object model also relies on this two table model.SOM对象模型也依靠这种双表格模型.9. Then he exp 1 ained how he pull the cut together with Stitches.斯通大夫说完后就开始解释他将如何缝伤口.10. True Ca and P digestibility values are 23.41±1.83 % and41.44±6.59 %, respectively . Exp.豆粕钙的真消化率为23.41±1.83%, 磷的真消化率为41.44±6.59%.11. Exp : In addition to acting , you're also a passionate photographer?除了演戏, 你也是一个很热诚的摄影师?12. Now my exp is over 1200, but I never think that is enough.现在虽然我的经验值超过1200了, 但我仍不满足.13. Chinese Imp & Exp Company that register in PuDong Shanghai.汽车和工程机械的零部件进出口公司,注册于浦东新区.14. Still, its a cool single player exp, and the price is right.尽管如此, 刀剑还是款很酷的单机游戏, 价位也比较合理.15. You receive EXP after the battle is won even when KO'd.战斗胜利之后,即使是在KO状态的角色也可以得到经验.。
B16 – Derivatives of exponential functions：B16–指数函数求导
9/12/2019 Calculus - Santowski
(A) EXPLORATION – PART 1
You are now pursuing derivatives of other functions (not just power functions)
To begin our study of derivatives of exponential functions, let’s explore a bit first
(3h – table
1)/h of
values
x
y
-0.00010 0.69312
-0.00007 0.69313
-0.00003 0.69314
0.00000 undefined
0.00003 0.69316
0.00007 0.69316
0.00010 0.69317
To explore, we can rewrite ax in base e as e(lna)x
So if d/dx ex was ex, then d/dx e(lna)x must be e(lna)x times lna (by the chain rule)
And so: ln(2) = 0.693 And so: ln(3) = 1.0986 And so: ln(4) = 1.386
M a c in to s h P IC T im a g e fo rm a t
is n o t s u p p o rte d
9/12/2019
Calculus - Santowski
12
9/12/2019 Calculus - Santowski
exponential decay function -回复
exponential decay function -回复Exponential Decay Function: A Step-by-Step ExplanationIntroduction:Exponential decay is a mathematical concept used to describe the decrease of a quantity over time. It is an essential concept in various fields such as physics, biology, finance, and environmental sciences. In this article, we will explore the exponential decay function and break it down into simple steps.Step 1: Understanding Exponential DecayExponential decay occurs when a quantity decreases at a constant relative rate over equal intervals of time. It can be represented mathematically by the formula:N(t) = N₀* e^(-kt),Where N(t) represents the quantity at time t, N₀is the initial quantity, k is the decay constant, and e is Euler's number approximately equal to 2.71828.Step 2: Exploring the Decay ConstantThe decay constant (k) is a crucial parameter in the exponential decay function. It determines the rate at which the quantity decreases. It is often expressed as a negative value to ensure the quantity decreases, following the definition of decay. The greater the magnitude of k, the faster the decay.Step 3: Analyzing the Initial QuantityThe initial quantity (N₀) in the exponential decay function corresponds to the value of the quantity at the beginning of the observation period. It serves as a reference point to measure the decay over time. The value of N₀can vary depending on the context of the problem or experiment under study.Step 4: Time and DecayIn the exponential decay function, time (t) represents the duration for which the decay occurs. It is essential to ensure that the unit of time is consistent with the context of the problem. If the decay ismeasured in hours, time should be in hours as well.Step 5: Euler's Number (e)Euler's number (e) plays a significant role in exponential decay. It is a mathematical constant whose value is approximately 2.71828. The presence of e in the exponential decay function ensures that the quantity declines continually over time. Understanding the properties and significance of e is essential in grasping exponential decay fully.Step 6: Investigating the Graphical RepresentationCreating a graph of the exponential decay function can provide valuable insights into the decay process. By plotting time on thex-axis and quantity on the y-axis, one can observe how the quantity decreases over time. The graph will show a rapidly decreasing curve, often starting high and gradually approaching zero.Step 7: Analyzing Half-LifeThe half-life of a substance or quantity is the time taken for half ofthe initial amount to decay. It is a significant concept in exponential decay. The formula to calculate the half-life is:t₁/₂= ln(2) / k,Where t₁/₂represents the half-life, and ln(2) denotes the natural logarithm of 2. The half-life provides crucial information about the rate of decay and is especially useful in fields such as radioactive decay and pharmaceuticals.Step 8: Real-Life ApplicationsExponential decay functions are widely utilized in various real-life scenarios. For instance, in radioactive decay, the decay constant determines the rate at which radioactive isotopes disintegrate. In finance, exponential decay is used to model the decrease in the present value of an investment over time. Environmental sciences employ exponential decay to study pollutant degradation or population decline over time.Conclusion:Exponential decay is a powerful mathematical concept used to describe the decline of a quantity over time. By understanding the steps involved in the exponential decay function, one can analyze and interpret the decay process accurately. Each component, from the decay constant to the initial quantity and the role of Euler's number, contributes to a comprehensive understanding of this fundamental mathematical model.。
数学指数 英语
数学指数英语Exponential Mathematics in EnglishMathematics is a fundamental discipline that underpins many aspects of our lives, from the natural sciences to finance and economics. One of the most important and versatile mathematical concepts is the exponential function, which has far-reaching applications in various fields. In this essay, we will explore the significance of exponential mathematics and its relevance in the English language.Exponential functions are characterized by a variable that is raised to a power, where the power itself is a variable. This unique property allows exponential functions to model a wide range of phenomena that exhibit rapid growth or decay, such as population growth, radioactive decay, and compound interest. In the context of English, the exponential function can be used to understand and analyze the evolution of language, the spread of ideas, and the dynamics of communication.One of the most striking examples of exponential growth in the English language is the rapid expansion of vocabulary. As newtechnologies, cultural trends, and societal changes emerge, the English language continuously incorporates new words and phrases to accommodate these developments. According to the Oxford English Dictionary, the English language has over 170,000 words currently in use, and this number is constantly increasing. The rate at which new words are added to the language can be modeled using exponential functions, as the growth of the vocabulary follows a pattern of accelerating expansion.Furthermore, the concept of exponential growth can be applied to the dissemination of ideas and information in the English-speaking world. In the digital age, the rapid spread of information through social media, online platforms, and global communication networks has led to a phenomenon known as "viral" content. When a piece of information or an idea resonates with a large audience, it can be shared and propagated exponentially, reaching millions of people in a matter of days or even hours. This exponential diffusion of information has profound implications for the way we consume, process, and share knowledge in the English-speaking community.Another area where exponential mathematics intersects with the English language is in the field of natural language processing (NLP). NLP is a branch of artificial intelligence that focuses on the interaction between computers and human language, with the goal of enabling machines to understand, interpret, and generate naturallanguage. Exponential functions play a crucial role in NLP algorithms, particularly in the modeling of language patterns, the analysis of text corpora, and the development of predictive models for language-based tasks.For example, the concept of perplexity, a measure of how well a probabilistic model predicts a sample of text, is closely related to the exponential function. The lower the perplexity, the better the model is at predicting the language patterns, and this metric is essential in the development of language models, machine translation systems, and other NLP applications.Moreover, the exponential function is deeply embedded in the structure of the English language itself. Many linguistic phenomena, such as the growth of vocabulary, the distribution of word frequencies, and the evolution of grammatical structures, can be described using exponential models. By understanding the exponential nature of these linguistic patterns, researchers and linguists can gain valuable insights into the underlying mechanisms that govern the development and dynamics of the English language.In conclusion, the exponential function is a powerful mathematical concept that has far-reaching applications in the English language. From the rapid expansion of vocabulary to the dissemination of ideas and the modeling of language patterns, the exponential functionplays a crucial role in our understanding and analysis of the English language. As we continue to explore the frontiers of language, the insights gained from the study of exponential mathematics will undoubtedly play a vital role in shaping our understanding of this complex and ever-evolving system of communication.。
B16 – Derivatives of exponential functions：B16–指数函数求导
0.00000 undefined
0.00003 1.00002
0.00007 1.00003
0.00010 1.00005
And we see the value of
And we see the value of
1.386 as an approximation of
1.000 as an approximation of
Factor
M a c in to s h P IC T im a g e fo rm a t
is n o t s u p p o rte d
Simplify (2x)(22)
Solve xe3x – 3xex = 0
M acin to sh P IC T
Evaluate the limit im age form at
Which we will now rewrite as
M a c in to s h P IC T im a g e fo rm a t
is n o t s u p p o rte d
And we will see one special derivative when the exponential base is e, then the derivative becomes:
M a c in to s h P IC T im a g e fo rm a t
is n o t s u p p o rte d
So work with it ……
7
9/5/2019 Calculus - Santowski
DERIVATIVE OF EXPONENTIAL FUNCTIONS
9/5/2019 Calculus - Santowski
B16 – Derivatives of exponential functions：B16–指数函数求导
4
7/23/2020 Calculus - Santowski
(B) EXPLORATION – PART 2
We will go back to our "first principles" - that being the idea that we can determine instantaneous rates of changes using tangent lines
(3) Differentiate
M acin to sh P IC T im ag e fo rm at
is n o t su p p o rted
(4) Differentiate
M a c in to s h P IC T im a g e fo rm a t
is n o t s u p p o rte d
(2) Develop an algebraic derivation of the derivative of an exponential function
(3) Apply the various differentiation rules to differentiate exponential functions
(1) Use GDC to draw the tangent lines at various x values (2) Record the slopes of the tangent lines on a table. (3) Prepare a scatter plot from the table of values. (4) Describe scatter plot
And we see the value of 1.386 And we see the value of 1.000
ExponentialfunctionvsPowerfunction：指数函数和幂函数
Name: __________________________ Date: ____________________ Hr: ________Ch 11 WS #2 Exponential Functions vs. Power Function Power Functions and Exponential Functions look similar, but they are actually quite different.Compare Graphs:1. Compare the graphs of y = x2and y = 2xFill in each chart. Then graph each on the axes provided. Label your axes withappropriate numbers. List the domain and range of each graph below.2xD: ____________ R: ____________ D: ____________ R: ____________2. Compare the graphs of y = x3and y = 3xFill in each chart. Then graph each on the axes provided. Label your axes withappropriate numbers:3xD: ____________ R: ____________ D: ____________ R: ____________ 3. Given a graph of y = x n and y = n x, how could you tell which graph is which? Explain.4. Compare the graphs of y = x1/2and y = ( ½ )xFill in each chart. Then graph each on the axes provided. Label your axes withappropriate numbers:1/2xD: ____________ R: ____________ D: ____________ R: ____________ (c) How are the graphs the same? How are they different?5. Compare the graphs of y = 5∙x2 and y = 5∙2xFill in each chart. Then graph each on the axes provided. Label your axes withappropriate numbers:2xD: ____________ R: ____________ D: ____________ R: ____________ (c) How are the graphs the same? How are they different?Power Function: y = k∙x n where k & n are constantsExamples: y = x2y = -3x5y = 3 x 1/2 Exponential Function: y = k∙n x where k & n are constants Examples: y = 2x y = -3∙(5)x y = 3 ( ½ )x6. Look at the examples above. What is the difference between a power function and anexponential function? Explain.7. Look at all of the functions listed in questions 1, 2, 4, & 5. List all those that are…(a) Power Functions:(b) Exponential Functions:Graphs of Exponential Functions: Growth vs. Decay:Exponential functions are all of the form y = k∙n x. Often they are thought of as functions of time and thus written y = k∙n t. Complete the following graphs to try and figure out what the ‘k’ value and the ‘n’ value tell you.t tt tt tt t16. Sometimes exponential functions grow as time goes on, and sometimes they decay.Which values in the formula y = k∙n t tells you whether the function grows or decays: the ‘k’ value or the ‘n’ value? Explain.17. Challenge Question: How can you use the formula y = k∙n t to tell you the actual percentageof growth each day or year? How can you use the formula y = k∙n t to tell you the actual percentage of decay each day or year? Explain.18. In the formula y = k∙n t, what does the ‘k’ value tell you? Explain.19.The city leaders in Camelot are having adisagreement about population growth.They have graphed the population vs. timesince the year 1990 and got a graph thatcurves up as pictured at right.The mayor believes that the growth can bemodeled by a normal parabola while thecity manager insists it is an exponentialgraph.(a) Explore what will happen if the mayor is correct and population follows the formula:P = 3t2 + 30 where population is in thousandsand time is in years, year zero is 2005(b) Explore what will happen if the city manager is correct and population follows the formula:P = 30(1.30)t where population is in thousandsand time is in years, year zero is 2005(c) In terms of planning Camelot city facilities over the next 20 years, what will be thedifference between the mayor’s conjecture and the city manager’s conjecture aboutpopulation growth? Which growth model would be better supported by city services? (d) You are considering moving to Camelot. Would you be more interested if the populationwas going to follow the mayor’s formula and be a parabola? Or would you prefer it to follow the exponential model? Explain.20.Human population actually has followed an exponential growth model as opposed to theparabolic model. Why might this be a concern in terms of resources such as energy and food? Explain.。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
More Mathematical Modeling
Internet Technology
• The Internet is growing faster than all other technologies that have preceded it. • Radio existed for 38 years before it had 50 million listeners. • Television took 13 years to reach that mark. • The Internet crossed the line in just four years.
Exponential Functions
• Consider the following example, is this exponential?
x 5 10 y 0.5 1.5
15
4.5
20 13.5
Exponential Functions
• For a fixed change in x, y gets multiplied by a fixed amount. If the column is constant, then the relationship is exponential.
5
6
$121.55 + $6.08 = $127.63
$127.63 + $6.38 = $134.01
= $127.63 * 0.05 = $6.38
= $134.01 * 0.05 = $6.70
= $127.63 / $121.55 = 1.05
= $134.01 / $127.63 = 1.05
Exampቤተ መጻሕፍቲ ባይዱe:
In 2000, the U.S. population was 282 million. The U.S. population has been growing by about 0.8% each year. In this case, population A is growing by r % each year. After one year, population A will become »Population + Population Increased by Rate r »A + A*r »282 + 282*0.008 Using some algebra, we see that A + A*r = A(1 + r ). Notice that A is being multiplied by the quantity 1 + r, Numerically, 282(1 + 0.008).
48
24
Exponential Functions
• For a fixed change in x, y gets multiplied by a fixed amount. If the column is constant, then the relationship is exponential.
year Population by adding percent 282 =B2+B2*0.008 =B3+B3*0.008
Population by multiplying by growth factor 282
=C2*(1.008) =C3*(1.008)
2000
2001 2002
2003 …
x 0
1 2 3
y 192
96 48 24 192 / 96 0.5 96 / 48 0.5 48 / 24 0.5
Other Examples of Exponential Functions
• Populations tend to growth exponentially not linearly. • When an object cools (e.g., a pot of soup on the dinner table), the temperature decreases exponentially toward the ambient temperature. • Radioactive substances decay exponentially. • Viruses and even rumors tend to spread exponentially through a population (at first).
• Source: The Emerging Digital Economy, April 1998 report of the United States Department of Commerce.
Exponential Functions
• A function is called an exponential function if it has a constant growth factor. • This means that for a fixed change in x, y gets multiplied by a fixed amount. • Example: Money accumulating in a bank at a fixed rate of interest increases exponentially.
•
y = Abx
• where A is the initial value of y when x = 0, and b is that growth factor. • An example of the equation of the last relationship above is simply y = $100 (1.05)x.
Exponential Growth
• Exponential growth occurs when some quantity regularly increases by a fixed percentage. • The equation for an exponential relationship is given by
= $100.00 * 0.05 = $5.00 = $105.00 * 0.05 = $5.25 = $110.25 * 0.05 = $5.51 = $115.76 * 0.05 = $5.79 = $121.55 * 0.05 = $6.08
Constant Growth Factor
= $105.00 / $100.00 = 1.05 = $110.25 / $105.00 = 1.05 = $115.76 / $110.25 = 1.05 = $121.55 / $115.76 = 1.05
Bank Account
year
0 1 2 3 4
Amount
$100.00 $100.00 + $5.00 = $105.00 $105.00 + $5.25 = $110.25 $110.25 + $5.51 = $115.76 $115.76 + $5.79 = $121.55
Interest Earned
x
5 10 15
y
0.5 1.5 4.5 1.5 / 0.5 4.5 / 1.5 3 3 3
20 13.5 13.5 / 4.5
Exponential Functions
• Consider another example, is this exponential?
x 0 1 y 192 96
2
3
Exponential Decay: Example
• China’s one-child policy was implemented in 1978 with a goal of reducing China’s population to 700 million by 2050. China’s 2000 population is about 1.2 billion. Suppose that China’s population declines at a rate of 0.5% per year. Will this rate be sufficient to meet the original goal?
…
=B4+B4*0.008 …
…
=C4*(1.008) …
…
Exponential Functions
• If a quantity grows by a fixed percentage change, it grows exponentially. • Example: Bank Account
Internet Traffic
• In 1994, a mere 3 million people were connected to the Internet. • By the end of 1997, more than 100 million were using it. • Traffic on the Internet has doubled every 100 days.
– Suppose you deposit $100 into an account that earns 5% annual interest. – Interest is paid once at the end of year. – You do not make additional deposits or withdrawals. – What is the amount in the bank account after eight years?