This document discusses chapter 5 of an introductory mathematical analysis textbook. The chapter covers various topics related to mathematics of finance, including compound interest, present value, interest compounded continuously, annuities, and amortization of loans. It provides examples and solutions for various types of problems involving these topics, such as calculating compound interest, present value, effective interest rates, sums of geometric series, present value of annuities, and amortizing loans. The chapter objectives are to solve problems requiring logarithms, involving time value of money, with continuously compounded interest, and to introduce ordinary annuities, annuities due, and amortization of loans.
O documento descreve preliminares sobre o limite inferior de Cramér-Rao, incluindo:
1) Amostra aleatória i.i.d com função de densidade f(x,θ) onde θ é o parâmetro a ser estimado;
2) Definições de expectativa matemática e covariância;
3) Regras de derivação para funções compostas e logarítmicas.
1. The document discusses inverse trigonometric functions such as arcsin, arccos, and arctan.
2. It derives the derivatives of these inverse functions using the Inverse Function Theorem and properties of trigonometric functions.
3. The derivatives are derived to be 1/(√(1-x^2)) for arcsin, 1/√(1-x^2) for arccos, and 1/(1+x^2) for arctan.
This document discusses chapter 5 of an introductory mathematical analysis textbook. The chapter covers various topics related to mathematics of finance, including compound interest, present value, interest compounded continuously, annuities, and amortization of loans. It provides examples and solutions for various types of problems involving these topics, such as calculating compound interest, present value, effective interest rates, sums of geometric series, present value of annuities, and amortizing loans. The chapter objectives are to solve problems requiring logarithms, involving time value of money, with continuously compounded interest, and to introduce ordinary annuities, annuities due, and amortization of loans.
O documento descreve preliminares sobre o limite inferior de Cramér-Rao, incluindo:
1) Amostra aleatória i.i.d com função de densidade f(x,θ) onde θ é o parâmetro a ser estimado;
2) Definições de expectativa matemática e covariância;
3) Regras de derivação para funções compostas e logarítmicas.
1. The document discusses inverse trigonometric functions such as arcsin, arccos, and arctan.
2. It derives the derivatives of these inverse functions using the Inverse Function Theorem and properties of trigonometric functions.
3. The derivatives are derived to be 1/(√(1-x^2)) for arcsin, 1/√(1-x^2) for arccos, and 1/(1+x^2) for arctan.
Vectors are directed line segments used to represent quantities with both magnitude and direction. They have an initial point and terminal point. The document provides examples of calculating the component form and magnitude of vectors, as well as the standard operations of vector addition and scalar multiplication. It also discusses the dot product and using it to determine the angle between two vectors.
The document defines various sets of numbers and binary operations. It then provides examples of binary operations on sets of numbers, such as addition and multiplication on sets of natural numbers, integers, rational numbers, real numbers, and complex numbers. The document also defines properties of binary operations such as commutativity, associativity, identity elements, and inverse elements. It provides problems and solutions showing examples of binary operations and verifying their properties.
1. The document describes techniques for integrating trigonometric functions using trigonometric substitution and identities involving sine, cosine, tangent, and secant.
2. Trigonometric substitution involves redefining the variable in terms of a trigonometric function, unlike traditional substitution which defines a new variable.
3. The techniques are demonstrated through examples such as finding antiderivatives of √9-x^2/x^2 and √x^2+4/x^2.
This document contains exercises related to inverse functions and their properties. It includes 53 multi-part exercises involving determining if functions are inverses, composing functions, finding inverse functions, and evaluating derivatives of inverse functions. The exercises involve algebraic manipulation and graphical analysis of functions and their inverses.
The document discusses the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse operations. It provides examples of using the theorem to evaluate definite integrals and find the average value of functions. The Second Fundamental Theorem of Calculus describes integrals as accumulation functions and relates the integral and derivative. The Net Change Theorem applies the Fundamental Theorem to relate the net change in a function to its integral over an interval. Examples demonstrate using these theorems to solve problems in physics, chemistry, and particle motion.
The document describes the expansion of sin nθ and cos nθ in powers of sinθ and cosθ using De Moivre's theorem and the binomial theorem. It shows that cos nθ can be expressed as the sum of terms involving nC0cosnθ, nC2cosn-2θsin2θ, etc. and sin nθ can be expressed as the sum of terms involving nC1cosn-1θsinθ, nC3cosn-3θsin3θ, etc. The expansions are obtained by equating the real and imaginary parts of (cosθ + i sinθ)n.
The document describes how complex numbers can be represented geometrically using an Argand diagram. The Argand diagram plots the real component of a complex number along the x-axis and the imaginary component along the y-axis, allowing any complex number to be represented as a point in the diagram. The document also explains how every complex number can be expressed in terms of its modulus and argument, where the modulus is the distance from the origin to the point and the argument is the angle relative to the positive real axis. An example of finding the modulus and argument of a complex number is provided.
Change of variables in double integralsTarun Gehlot
1. The document discusses change of variables for double integrals, introducing the Jacobian determinant which relates the differentials of the original and transformed variables.
2. It provides an example of using a change of variables (u=x-y, v=x+y) to evaluate an integral over a parallelogram region.
3. Polar coordinates are also discussed as a common change of variables technique for double integrals, with an example evaluating an integral over a circular region in polar coordinates.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
Lesson 4 derivative of inverse hyperbolic functionsLawrence De Vera
1. The document discusses differentiation formulas for inverse hyperbolic functions including sinh-1(x), cosh-1(x), tanh-1(x), and coth-1(x).
2. Examples are provided to demonstrate taking the derivative of inverse hyperbolic functions and simplifying the results.
3. A series of exercises asks the reader to take the derivative of various functions involving inverse hyperbolic and related transcendental functions.
The document defines trigonometric functions using right triangles and the unit circle. It lists properties of the trig functions including domain, range, period, formulas, and identities. It also covers inverse trig functions, the laws of sines, cosines, and tangents, and the unit circle.
Vectors are directed line segments used to represent quantities with both magnitude and direction. They have an initial point and terminal point. The document provides examples of calculating the component form and magnitude of vectors, as well as the standard operations of vector addition and scalar multiplication. It also discusses the dot product and using it to determine the angle between two vectors.
The document defines various sets of numbers and binary operations. It then provides examples of binary operations on sets of numbers, such as addition and multiplication on sets of natural numbers, integers, rational numbers, real numbers, and complex numbers. The document also defines properties of binary operations such as commutativity, associativity, identity elements, and inverse elements. It provides problems and solutions showing examples of binary operations and verifying their properties.
1. The document describes techniques for integrating trigonometric functions using trigonometric substitution and identities involving sine, cosine, tangent, and secant.
2. Trigonometric substitution involves redefining the variable in terms of a trigonometric function, unlike traditional substitution which defines a new variable.
3. The techniques are demonstrated through examples such as finding antiderivatives of √9-x^2/x^2 and √x^2+4/x^2.
This document contains exercises related to inverse functions and their properties. It includes 53 multi-part exercises involving determining if functions are inverses, composing functions, finding inverse functions, and evaluating derivatives of inverse functions. The exercises involve algebraic manipulation and graphical analysis of functions and their inverses.
The document discusses the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse operations. It provides examples of using the theorem to evaluate definite integrals and find the average value of functions. The Second Fundamental Theorem of Calculus describes integrals as accumulation functions and relates the integral and derivative. The Net Change Theorem applies the Fundamental Theorem to relate the net change in a function to its integral over an interval. Examples demonstrate using these theorems to solve problems in physics, chemistry, and particle motion.
The document describes the expansion of sin nθ and cos nθ in powers of sinθ and cosθ using De Moivre's theorem and the binomial theorem. It shows that cos nθ can be expressed as the sum of terms involving nC0cosnθ, nC2cosn-2θsin2θ, etc. and sin nθ can be expressed as the sum of terms involving nC1cosn-1θsinθ, nC3cosn-3θsin3θ, etc. The expansions are obtained by equating the real and imaginary parts of (cosθ + i sinθ)n.
The document describes how complex numbers can be represented geometrically using an Argand diagram. The Argand diagram plots the real component of a complex number along the x-axis and the imaginary component along the y-axis, allowing any complex number to be represented as a point in the diagram. The document also explains how every complex number can be expressed in terms of its modulus and argument, where the modulus is the distance from the origin to the point and the argument is the angle relative to the positive real axis. An example of finding the modulus and argument of a complex number is provided.
Change of variables in double integralsTarun Gehlot
1. The document discusses change of variables for double integrals, introducing the Jacobian determinant which relates the differentials of the original and transformed variables.
2. It provides an example of using a change of variables (u=x-y, v=x+y) to evaluate an integral over a parallelogram region.
3. Polar coordinates are also discussed as a common change of variables technique for double integrals, with an example evaluating an integral over a circular region in polar coordinates.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
Lesson 4 derivative of inverse hyperbolic functionsLawrence De Vera
1. The document discusses differentiation formulas for inverse hyperbolic functions including sinh-1(x), cosh-1(x), tanh-1(x), and coth-1(x).
2. Examples are provided to demonstrate taking the derivative of inverse hyperbolic functions and simplifying the results.
3. A series of exercises asks the reader to take the derivative of various functions involving inverse hyperbolic and related transcendental functions.
The document defines trigonometric functions using right triangles and the unit circle. It lists properties of the trig functions including domain, range, period, formulas, and identities. It also covers inverse trig functions, the laws of sines, cosines, and tangents, and the unit circle.
5. لماذا؟
تحتوي كثير من الصيغ
ص ً
على وحيدات حد، فمثال
صيغة قوة محرك
السيارة بالحصنة هي
432
ق = ك )ــــــــــــــ(3 ؛
ع
6. حيث تمثل: ق قوة المحرك
بالحصان، ك كتلة السيارة
بركابها، ع سرعتها بعد
مسيرها مسافة ربع ميل .
من الواضح أن قوة المحرك
بالحصان تزداد كلما ازدادت
السرعة .
7. وحيدات الحد: تكون وحيدة الحد
عددا، أو متغيرا، أو حاصل ضرب
ص ً
ص ً
عدد في متغير واحد أو أكثر بأسس
صحيحة غير سالبة. وتتكون من حد
واحد فقط .
8. ع
فمثال الحد: ك )ــــــــــــ( في صيغة
ص ً
432
حساب قوة محرك السيارة، هو
وحيدة حد .
أما العبارة التي تتضمن القسمة على
متغير مثل: أب
ــــــــــ، فليست وحيدة
جـ
حد .
3
9. الثابت: هو وحيدة حد تمثل عددا حقيقيا.
ص ً
ص ً
ووحيدة الحد 3 س هي مثال على عبارة
خطية؛ لن أس المتغير س فيها 1، أما
وحيدة الحد 2س2 فليست عبارة خطية؛
لن الس عدد موجب أكبر من 1 .
10. مثال1
تمييز وحيدات الحد
حدد إذا كانت العبارات التية وحيدة حد،
اكتب ،"نعم،" أو ،"ل،"، وفسر إجابتك:
أ( 01
نعم؛ العدد 01 ثابت، لذا فهو
وحيدة حد .
11. مثال1
تمييز وحيدات الحد
حدد إذا كانت العبارات التية وحيدة حد،
اكتب ،"نعم،" أو ،"ل،"، وفسر إجابتك:
ب( ف+42
ل؛ تتضمن هذه العبارة عملية جمع،
لذا فهي تحتوي على أكثر من حد .
12. مثال1
تمييز وحيدات الحد
حدد إذا كانت العبارات التية وحيدة حد،
اكتب ،"نعم،" أو ،"ل،"، وفسر إجابتك:
جـ( هـ
2
نعم؛ تمثل هذه العبارة حاصل ضرب
المتغير في نفسه .
13. مثال1
تمييز وحيدات الحد
حدد إذا كانت العبارات التية وحيدة حد،
اكتب ،"نعم،" أو ،"ل،"، وفسر إجابتك:
د( ل
نعم؛ المتغيرات المنفردة
وحيدات حد .
14. تحقق من فهمك
1أ( – س+5
ل، تتضمن هذه العبارة عملية جمع
لذا فهي تحتوي على أكثر من حد
15. تحقق من فهمك
1 ب( 32أ ب ج د
2
نعم، هذا حاصل ضرب عدد
ومتغيرات
16. تحقق من فهمك
سصع
1جـ( ــــــــــــــــــ
2
2
نعم، هذا حاصل ضرب
متغيرات ، في ثابت في المقام
17. تحقق من فهمك
مف
1د( ــــــــــــــــــ
ن
ل، تمثل هذه العبارة حاصل
ضرب وقسمة أكثر من متغير
18. تذكر أن العبارة التي على الصورة س
والتي تعبر عن نتيجة ضرب س في
نفسها ن مرة تسمى قوة .
سُ
ويطلق على س الساس، وعلى ن
الس. وقد تستعمل كلمة قوة لتعني
الس أحيانا .
ص ً
ن
19.
20. ويمكنك إيجاد حاصل ضرب القوى في
المثالين التيين بتطبيق تعريف القوة،
انظر نمط السسس في المثالين التيين:
42. مثال4
قوة حاصل الضرب
هندسة: عبر عن مساحة الدائرة على صورة وحيدة حد
المساحة = ط نق
2( 2
= ط )2 س ص
2
مساحة الدائرة
2
عوض عن نق ب 2 س ص
= ط )22 س2 ص4(
قوة حاصل الضرب
= 4 س2 ص4 ط
بسط
إذن، مساحة الدائرة يتساوي 4 س2 ص4 ط وحدة مربعة .
43. يتحقق من فهمك
4أ( عبر عن مساحة المربع الذي طول
ضلعه 3 س ص2 على صورة وحيدة حد .
المساحة=)3س ص2()3س
ص2(=)3×3()س×س(
4
)ص2×ص2(=9س2 ص
45. مفهوم أساسي
يتبسيط العبارات
لتبسيط وحيدة حد، اكتب عبارة مكافئة لها على أن:
يظهر كل متغير على صورة أساس مرة واحدة فقط . ل يتتضمن العبارة قوة قوة .- يتكون جميع الكسور في أبسط صورة .
