The document discusses reduction formulae, which expresses a given integral as the sum of a function and a known integral. It provides an example of using integration by parts to derive the reduction formula for the integral of cos(nx)dx and evaluates the integral of cos(5x)dx. It also shows how to find the reduction formula for the integral of cot(nx)dx and states that this formula can be used to find the value of I6, the integral of cot(6x)dx.
The document presents information on partial differentiation including:
- Partial differentiation involves a function with more than one independent variable and partial derivatives.
- Notation for partial derivatives is presented.
- Methods for computing first and higher order partial derivatives are explained with examples.
- The concepts of homogeneous functions and the chain rule for partial differentiation are defined.
The definite integral is used to find the area of a region bounded by curves. Specifically, the area A between two curves f(x) and g(x) on the interval [a,b] is given by the definite integral:
A = ∫ab [f(x) - g(x)] dx
To calculate the area, one finds the points where the two curves intersect, then evaluates the integral of the top curve minus the bottom curve between the bounds a and b. For examples where the region is bounded along the y-axis, the integral is evaluated with respect to y. The definite integral provides a way to precisely calculate the area of a region defined by curves.
This document presents reduction formulas for integrals of sinnx and cosnx (where n is greater than or equal to 2). It derives the reduction formulas by repeatedly applying integration by parts. For sinnx, the reduction formula expresses In (the integral of sinnx) in terms of In-1 and In-2. For cosnx, the reduction formula expresses In in terms of In-1 and In-2. The document provides detailed step-by-step working to arrive at each reduction formula.
(1) The document discusses inner product spaces and related linear algebra concepts such as orthogonal vectors and bases, Gram-Schmidt process, orthogonal complements, and orthogonal projections.
(2) Key topics covered include defining inner products and their properties, finding orthogonal vectors and constructing orthogonal bases, using Gram-Schmidt process to orthogonalize a set of vectors, defining and finding orthogonal complements of subspaces, and computing orthogonal projections of vectors.
(3) Examples are provided to demonstrate computing orthogonal bases, orthogonal complements, and orthogonal projections in inner product spaces.
This document provides an overview of different types of differential equations. It defines ordinary and partial differential equations, and explains that ordinary differential equations contain only ordinary derivatives while partial differential equations contain partial derivatives. It also defines key concepts like the order of a differential equation as the order of the highest derivative, and the degree as the power of the highest order derivative. The document then describes various types of first order differential equations including separable variables, homogeneous, linear, and exact equations. Examples are provided for each type.
The document discusses second order derivatives. The second derivative of a function is the derivative of the first derivative. It can tell us whether a function is concave up or down at a point. If the second derivative is zero at a point, it does not tell us the slope. The point where a function changes from concave up to down is called the point of inflection. The second derivative test can determine if a point is a local minimum or maximum.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
The document discusses higher order differential equations. It defines nth order differential equations and describes their general forms. For homogeneous equations, the general solution method involves making an operator form, constructing an auxiliary equation, solving for roots, and finding the complementary solution. For non-homogeneous equations, the method of undetermined coefficients is used to find a particular solution and the general solution is the sum of the complementary and particular solutions. Examples are provided to illustrate the solution methods.
The document presents information on partial differentiation including:
- Partial differentiation involves a function with more than one independent variable and partial derivatives.
- Notation for partial derivatives is presented.
- Methods for computing first and higher order partial derivatives are explained with examples.
- The concepts of homogeneous functions and the chain rule for partial differentiation are defined.
The definite integral is used to find the area of a region bounded by curves. Specifically, the area A between two curves f(x) and g(x) on the interval [a,b] is given by the definite integral:
A = ∫ab [f(x) - g(x)] dx
To calculate the area, one finds the points where the two curves intersect, then evaluates the integral of the top curve minus the bottom curve between the bounds a and b. For examples where the region is bounded along the y-axis, the integral is evaluated with respect to y. The definite integral provides a way to precisely calculate the area of a region defined by curves.
This document presents reduction formulas for integrals of sinnx and cosnx (where n is greater than or equal to 2). It derives the reduction formulas by repeatedly applying integration by parts. For sinnx, the reduction formula expresses In (the integral of sinnx) in terms of In-1 and In-2. For cosnx, the reduction formula expresses In in terms of In-1 and In-2. The document provides detailed step-by-step working to arrive at each reduction formula.
(1) The document discusses inner product spaces and related linear algebra concepts such as orthogonal vectors and bases, Gram-Schmidt process, orthogonal complements, and orthogonal projections.
(2) Key topics covered include defining inner products and their properties, finding orthogonal vectors and constructing orthogonal bases, using Gram-Schmidt process to orthogonalize a set of vectors, defining and finding orthogonal complements of subspaces, and computing orthogonal projections of vectors.
(3) Examples are provided to demonstrate computing orthogonal bases, orthogonal complements, and orthogonal projections in inner product spaces.
This document provides an overview of different types of differential equations. It defines ordinary and partial differential equations, and explains that ordinary differential equations contain only ordinary derivatives while partial differential equations contain partial derivatives. It also defines key concepts like the order of a differential equation as the order of the highest derivative, and the degree as the power of the highest order derivative. The document then describes various types of first order differential equations including separable variables, homogeneous, linear, and exact equations. Examples are provided for each type.
The document discusses second order derivatives. The second derivative of a function is the derivative of the first derivative. It can tell us whether a function is concave up or down at a point. If the second derivative is zero at a point, it does not tell us the slope. The point where a function changes from concave up to down is called the point of inflection. The second derivative test can determine if a point is a local minimum or maximum.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
The document discusses higher order differential equations. It defines nth order differential equations and describes their general forms. For homogeneous equations, the general solution method involves making an operator form, constructing an auxiliary equation, solving for roots, and finding the complementary solution. For non-homogeneous equations, the method of undetermined coefficients is used to find a particular solution and the general solution is the sum of the complementary and particular solutions. Examples are provided to illustrate the solution methods.
This document discusses the topic of differentiation. It begins by defining differentiation and listing some fundamental rules, such as the chain rule and differentiation of constants. It then discusses geometrically what the derivative represents at a point and lists several types of differentiable functions. The document goes on to explain differentiation using substitution, of implicit functions, and of parametric functions. It also covers successive differentiation, Leibnitz's theorem, and differentiation of special function types. The document provides an overview of differentiation concepts and rules.
Application of definite integrals,we will explore some of the many application of definite integral by using it to calculate areas between two curves, volumes, length of curves, and several other application.
This document defines and explains partial derivatives. It begins by defining a partial derivative as the rate of change of a function with respect to one variable, holding other variables fixed. It then covers notation, calculating partial derivatives, interpreting them geometrically and as rates of change, higher derivatives, and applications to partial differential equations.
This document discusses differentiation and derivatives. It defines differentiation as finding the average rate of change of one variable with respect to another. It then discusses various methods of finding derivatives, including the direct method using derivative rules, as well as discussing specific rules like the power rule, product rule, quotient rule, chain rule, and rules for derivatives of trigonometric, exponential, and logarithmic functions.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
This document discusses several topics related to calculus including:
1) Derivatives of position, velocity, and acceleration and how they relate to each other.
2) An example problem calculating velocity from a position function.
3) The Mean Value Theorem and how to apply it to find critical points of a function.
4) How the first and second derivatives of a function relate to critical points, maxima, minima, and points of inflection or concavity.
5) Related rates problems and how to set them up using derivatives and relationships between variables.
The document summarizes key concepts and applications of integration. It discusses:
1) Important historical figures like Archimedes, Gauss, Leibniz and Newton who contributed to the development of integration and calculus.
2) Engineering applications of integration like in the design of the Petronas Towers and Sydney Opera House.
3) The integration by parts formula and examples of using it to evaluate integrals of composite functions.
This document discusses using least squares approximation to fit linear and polynomial models to data. It introduces the concept of best approximation in a subspace and shows that the orthogonal projection of a vector onto the subspace is the best approximation. The document then applies these concepts to derive the normal equations and the least squares solution for linear and polynomial regression models. Examples are provided to illustrate fitting lines and curves to data using least squares.
Partial derivatives are used to calculate the rate of change of a function of two or more variables with respect to one variable, while holding the other variables constant. The partial derivative of z with respect to x, denoted ∂z/∂x, is defined as the limit of the difference quotient as Δx approaches 0, while holding y constant. Similarly, the partial derivative of z with respect to y, denoted ∂z/∂y, is defined as the limit of the difference quotient as Δy approaches 0, while holding x constant. Notations for higher order partial derivatives are also introduced. An example problem finds the first and second order partial derivatives of the function z=x^2y^3+6
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
This document discusses antiderivatives and indefinite integrals. It begins by introducing the concept of an antiderivative, which is a function whose derivative is a known function. It then defines the indefinite integral as representing the set of all antiderivatives. Several properties of antiderivatives and indefinite integrals are presented, including: the constant of integration; basic integration rules like power, exponential, and logarithmic rules; and notation used to represent indefinite integrals. Examples are provided to illustrate key concepts and properties.
The document defines the derivative of a function and discusses:
- The definition of the derivative as the limit of the slope between two points as they approach each other.
- Notation used to represent derivatives, including f'(x), dy/dx, and df/dx.
- How the graph of a function's derivative f' relates to the graph of the original function f - where f' is positive/negative/zero corresponds to parts of f that are increasing/decreasing/at an extremum.
- How to graph f given a graph of its derivative f' by sketching the curve that matches the behavior of f' at each point.
- One-sided derivatives at endpoints of functions defined
Increasing and decreasing functions ap calc sec 3.3Ron Eick
The document discusses increasing and decreasing functions and the first derivative test. It defines that a function is increasing if the derivative is positive, decreasing if the derivative is negative, and constant if the derivative is zero. It provides examples of finding the intervals where a function is increasing or decreasing by identifying critical numbers and testing points in each interval. The document also summarizes the first derivative test, stating that a critical point is an extremum if the derivative changes sign there, and whether it is a maximum or minimum depends on if the derivative changes from negative to positive or positive to negative.
Series solutions at ordinary point and regular singular pointvaibhav tailor
The document discusses series solutions for second order linear differential equations near ordinary and regular singular points.
It defines an ordinary point as a point where the functions p(x) and q(x) in the normalized form of the differential equation are analytic. Near an ordinary point, there exist two linearly independent power series solutions of the form Σcn(x-a)n that converge within the radii of convergence of p(x) and q(x).
It also discusses finding series solutions near a regular singular point x0=0, where the limits of p(x) and q(x) as x approaches 0 exist. An initial guess of a power series solution with exponent r is made, and the
Lesson 7-8: Derivatives and Rates of Change, The Derivative as a functionMatthew Leingang
The derivative is one of the fundamental quantities in calculus, partly because it is ubiquitous in nature. We give examples of it coming about, a few calculations, and ways information about the function an imply information about the derivative
L19 increasing & decreasing functionsJames Tagara
This document discusses analysis of functions including derivatives, extrema, and graphing. It defines key concepts such as increasing and decreasing functions, concavity, points of inflection, stationary points, and relative maxima and minima. It presents Rolle's theorem and the mean value theorem. Examples demonstrate finding critical points and determining the behavior of functions based on the signs of the first and second derivatives. The first and second derivative tests are introduced to identify relative extrema at critical points.
Second order homogeneous linear differential equations Viraj Patel
1) The document discusses second order linear homogeneous differential equations, which have the general form P(x)y'' + Q(x)y' + R(x)y = 0.
2) It describes methods for finding the general solution including reduction of order, and discusses the solutions when the coefficients are constants.
3) The general solution depends on the nature of the roots of the auxiliary equation: distinct real roots, repeated real roots, or complex roots.
This document discusses using integration by parts to evaluate integrals involving the product of two functions. Specifically, it shows how to integrate xsinx and integrals involving natural logarithms, such as ln x, by letting one function be u and the other dv, then applying the integration by parts formula.
Polar coordinates represent points in a plane using an ordered pair (r, θ) where r is the distance from a fixed point called the pole (or origin) and θ is the angle between the line from the pole to the point and a reference axis called the polar axis. Some key properties:
- Points (r, θ) and (-r, θ + π) represent the same location.
- Polar equations like r = a describe circles with radius a centered at the pole.
- Equations can be converted between Cartesian (x, y) and polar (r, θ) coordinates using trigonometric relationships.
- Graphs of polar equations r = f(θ) consist of all points satisfying
The document discusses concepts related to partial differentiation and its applications. It covers topics like tangent planes, linear approximations, differentials, Taylor expansions, maxima and minima problems, and the Lagrange method. Specifically, it defines the tangent plane to a surface at a point using partial derivatives, describes how to find the linear approximation of functions, and explains how to find maximum and minimum values of functions using critical points and the second derivative test.
1. The document contains definitions, examples, and properties of infinite sequences and series. It includes limits of specific sequences as well as proofs of the convergence or divergence of sequences using various tests.
2. Key topics covered include the limit definition of convergence of a sequence, tests for convergence such as direct comparison test, limit comparison test, ratio test, root test, alternating series test, and tests involving logarithms.
3. Various examples calculate the limits of sequences directly or use standard tests to determine convergence or divergence of sequences.
The document discusses reduction formulae, which expresses a given integral as the sum of a function and a known integral. It provides an example of using integration by parts to derive the reduction formula for the integral ∫cosnxdx. This formula expresses the integral In in terms of the integral In-2. The document then applies this formula to evaluate the integral ∫cos5xdx. It also derives the reduction formula for the integral ∫cotnxdx and states that this formula can be used to find the value of I6.
This document discusses the topic of differentiation. It begins by defining differentiation and listing some fundamental rules, such as the chain rule and differentiation of constants. It then discusses geometrically what the derivative represents at a point and lists several types of differentiable functions. The document goes on to explain differentiation using substitution, of implicit functions, and of parametric functions. It also covers successive differentiation, Leibnitz's theorem, and differentiation of special function types. The document provides an overview of differentiation concepts and rules.
Application of definite integrals,we will explore some of the many application of definite integral by using it to calculate areas between two curves, volumes, length of curves, and several other application.
This document defines and explains partial derivatives. It begins by defining a partial derivative as the rate of change of a function with respect to one variable, holding other variables fixed. It then covers notation, calculating partial derivatives, interpreting them geometrically and as rates of change, higher derivatives, and applications to partial differential equations.
This document discusses differentiation and derivatives. It defines differentiation as finding the average rate of change of one variable with respect to another. It then discusses various methods of finding derivatives, including the direct method using derivative rules, as well as discussing specific rules like the power rule, product rule, quotient rule, chain rule, and rules for derivatives of trigonometric, exponential, and logarithmic functions.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
This document discusses several topics related to calculus including:
1) Derivatives of position, velocity, and acceleration and how they relate to each other.
2) An example problem calculating velocity from a position function.
3) The Mean Value Theorem and how to apply it to find critical points of a function.
4) How the first and second derivatives of a function relate to critical points, maxima, minima, and points of inflection or concavity.
5) Related rates problems and how to set them up using derivatives and relationships between variables.
The document summarizes key concepts and applications of integration. It discusses:
1) Important historical figures like Archimedes, Gauss, Leibniz and Newton who contributed to the development of integration and calculus.
2) Engineering applications of integration like in the design of the Petronas Towers and Sydney Opera House.
3) The integration by parts formula and examples of using it to evaluate integrals of composite functions.
This document discusses using least squares approximation to fit linear and polynomial models to data. It introduces the concept of best approximation in a subspace and shows that the orthogonal projection of a vector onto the subspace is the best approximation. The document then applies these concepts to derive the normal equations and the least squares solution for linear and polynomial regression models. Examples are provided to illustrate fitting lines and curves to data using least squares.
Partial derivatives are used to calculate the rate of change of a function of two or more variables with respect to one variable, while holding the other variables constant. The partial derivative of z with respect to x, denoted ∂z/∂x, is defined as the limit of the difference quotient as Δx approaches 0, while holding y constant. Similarly, the partial derivative of z with respect to y, denoted ∂z/∂y, is defined as the limit of the difference quotient as Δy approaches 0, while holding x constant. Notations for higher order partial derivatives are also introduced. An example problem finds the first and second order partial derivatives of the function z=x^2y^3+6
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
This document discusses antiderivatives and indefinite integrals. It begins by introducing the concept of an antiderivative, which is a function whose derivative is a known function. It then defines the indefinite integral as representing the set of all antiderivatives. Several properties of antiderivatives and indefinite integrals are presented, including: the constant of integration; basic integration rules like power, exponential, and logarithmic rules; and notation used to represent indefinite integrals. Examples are provided to illustrate key concepts and properties.
The document defines the derivative of a function and discusses:
- The definition of the derivative as the limit of the slope between two points as they approach each other.
- Notation used to represent derivatives, including f'(x), dy/dx, and df/dx.
- How the graph of a function's derivative f' relates to the graph of the original function f - where f' is positive/negative/zero corresponds to parts of f that are increasing/decreasing/at an extremum.
- How to graph f given a graph of its derivative f' by sketching the curve that matches the behavior of f' at each point.
- One-sided derivatives at endpoints of functions defined
Increasing and decreasing functions ap calc sec 3.3Ron Eick
The document discusses increasing and decreasing functions and the first derivative test. It defines that a function is increasing if the derivative is positive, decreasing if the derivative is negative, and constant if the derivative is zero. It provides examples of finding the intervals where a function is increasing or decreasing by identifying critical numbers and testing points in each interval. The document also summarizes the first derivative test, stating that a critical point is an extremum if the derivative changes sign there, and whether it is a maximum or minimum depends on if the derivative changes from negative to positive or positive to negative.
Series solutions at ordinary point and regular singular pointvaibhav tailor
The document discusses series solutions for second order linear differential equations near ordinary and regular singular points.
It defines an ordinary point as a point where the functions p(x) and q(x) in the normalized form of the differential equation are analytic. Near an ordinary point, there exist two linearly independent power series solutions of the form Σcn(x-a)n that converge within the radii of convergence of p(x) and q(x).
It also discusses finding series solutions near a regular singular point x0=0, where the limits of p(x) and q(x) as x approaches 0 exist. An initial guess of a power series solution with exponent r is made, and the
Lesson 7-8: Derivatives and Rates of Change, The Derivative as a functionMatthew Leingang
The derivative is one of the fundamental quantities in calculus, partly because it is ubiquitous in nature. We give examples of it coming about, a few calculations, and ways information about the function an imply information about the derivative
L19 increasing & decreasing functionsJames Tagara
This document discusses analysis of functions including derivatives, extrema, and graphing. It defines key concepts such as increasing and decreasing functions, concavity, points of inflection, stationary points, and relative maxima and minima. It presents Rolle's theorem and the mean value theorem. Examples demonstrate finding critical points and determining the behavior of functions based on the signs of the first and second derivatives. The first and second derivative tests are introduced to identify relative extrema at critical points.
Second order homogeneous linear differential equations Viraj Patel
1) The document discusses second order linear homogeneous differential equations, which have the general form P(x)y'' + Q(x)y' + R(x)y = 0.
2) It describes methods for finding the general solution including reduction of order, and discusses the solutions when the coefficients are constants.
3) The general solution depends on the nature of the roots of the auxiliary equation: distinct real roots, repeated real roots, or complex roots.
This document discusses using integration by parts to evaluate integrals involving the product of two functions. Specifically, it shows how to integrate xsinx and integrals involving natural logarithms, such as ln x, by letting one function be u and the other dv, then applying the integration by parts formula.
Polar coordinates represent points in a plane using an ordered pair (r, θ) where r is the distance from a fixed point called the pole (or origin) and θ is the angle between the line from the pole to the point and a reference axis called the polar axis. Some key properties:
- Points (r, θ) and (-r, θ + π) represent the same location.
- Polar equations like r = a describe circles with radius a centered at the pole.
- Equations can be converted between Cartesian (x, y) and polar (r, θ) coordinates using trigonometric relationships.
- Graphs of polar equations r = f(θ) consist of all points satisfying
The document discusses concepts related to partial differentiation and its applications. It covers topics like tangent planes, linear approximations, differentials, Taylor expansions, maxima and minima problems, and the Lagrange method. Specifically, it defines the tangent plane to a surface at a point using partial derivatives, describes how to find the linear approximation of functions, and explains how to find maximum and minimum values of functions using critical points and the second derivative test.
1. The document contains definitions, examples, and properties of infinite sequences and series. It includes limits of specific sequences as well as proofs of the convergence or divergence of sequences using various tests.
2. Key topics covered include the limit definition of convergence of a sequence, tests for convergence such as direct comparison test, limit comparison test, ratio test, root test, alternating series test, and tests involving logarithms.
3. Various examples calculate the limits of sequences directly or use standard tests to determine convergence or divergence of sequences.
The document discusses reduction formulae, which expresses a given integral as the sum of a function and a known integral. It provides an example of using integration by parts to derive the reduction formula for the integral ∫cosnxdx. This formula expresses the integral In in terms of the integral In-2. The document then applies this formula to evaluate the integral ∫cos5xdx. It also derives the reduction formula for the integral ∫cotnxdx and states that this formula can be used to find the value of I6.
The document discusses reduction formulae, which expresses a given integral as the sum of a function and a known integral. It provides an example of using integration by parts to derive the reduction formula for the integral of cos(nx)dx and evaluates the integral of cos(5x)dx. It also shows how to find the reduction formula for the integral of cot(nx)dx and states that this formula can be used to find the value of I6, the integral of cot(6x)dx.
This document defines and provides examples of sequences of real numbers. It begins by defining a sequence as a set of numbers written in a definite order. Examples are provided to illustrate bounded, increasing, decreasing, convergent, and divergent sequences. The limit of a sequence is defined as the number L that the terms of the sequence approach as n becomes large. A sequence is convergent if its limit exists and divergent otherwise. Bounded sequences are those for which the terms are all less than some positive number M, but bounded sequences may still diverge. Recursively defined sequences are also discussed.
Este documento describe la evolución de Ethernet y las tecnologías de transporte por conmutación de paquetes necesarias para soportar redes públicas troncales. Ethernet nativo tiene limitaciones para este uso debido a su falta de mecanismos de gestión, control y calidad de servicio. Las mejoras como Carrier Ethernet proporcionan escalabilidad, disponibilidad, protección y calidad de servicio superior requeridas por los proveedores. Tecnologías como MPLS y PBT permiten adaptar Ethernet para transporte de red, superando las limitaciones originales y permitiendo
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
The document discusses reduction formulae, which expresses a given integral as the sum of a function and a known integral. It provides an example of using integration by parts to derive the reduction formula for the integral ∫cosnxdx. This formula expresses the integral In in terms of the integral In-2. The document then applies this formula to evaluate the integral ∫cos5xdx. It also derives the reduction formula for the integral ∫cotnxdx and states that this formula can be used to find the value of I6.
The average value of a function f(x) over an interval (a,b) can be approximated as:
f(x) = (f(x1) + f(x2) + ... + f(xn))/n, where x1, x2, ..., xn are values in the interval.
The Fourier coefficients for a periodic function f(x) are:
a0 = (1/π) ∫ f(x) dx
an = (2/π) ∫ f(x) cos(nx) dx
bn = (2/π) ∫ f(x) sin(nx) dx
The Fourier series expansion of
The document discusses solving inequalities and graphs. It provides an example of solving the inequality x^2 ≤ 1/(x+2). It also discusses the oblique asymptote of a hyperbola. Additionally, it shows that the graph y=x is increasing for all x≥0. It proves that the sum of the first n positive integers is greater than or equal to the integral of x from 0 to n. Finally, it uses mathematical induction to show an inequality relating the sum of the first n positive integers to 4n+3/6.
The document discusses solving inequalities and graphs. It provides an example of solving the inequality x^2 ≤ 1/(x+2). It also discusses properties of the graph of y=x, showing that it is increasing for x≥0. It uses this to prove inequalities relating sums and integrals. Finally, it introduces a proof by mathematical induction to show an inequality relating sums and fractions is true for all integers n≥1.
X2 T08 01 inequalities and graphs (2010)Nigel Simmons
The document discusses solving inequalities and graphs. It provides an example of solving the inequality x^2 ≤ 1/(x+2). It also discusses the oblique asymptote of a hyperbola. Additionally, it shows that the graph y=x is increasing for all x≥0. It proves that the sum of the first n positive integers is greater than or equal to the integral of x from 0 to n. Finally, it uses mathematical induction to show an inequality relating the sum of the first n positive integers to 4n+3/6.
The line x + y = 7 passes through the points (-2, 4) and (7, 7).
Therefore, the center of the circle must lie on this line.
Let the center be (h, k). Then:
h + k = 7
Using the circle equation:
(x - h)2 + (y - k)2 = r2
Plugging the points A(-2, 4) and B(7, 7) into the circle equation:
(-2 - h)2 + (4 - k)2 = r2 ........(1)
(7 - h)2 + (7 - k)2 = r2........(2)
Solving
This document is a powerpoint presentation that serves as a pre-calculus review quiz covering topics like:
- Definitions of even and odd functions
- Identifying graphs of simple functions like f(x)=x^2
- Logarithmic and trigonometric identities and properties
- Trigonometric function values for common angles
- Algebraic identities for expanding expressions like (a+b)^3
The presentation is interactive, asking questions and providing feedback on slides to test the user's knowledge of important pre-calculus concepts needed for success in calculus. It concludes by stating if the user knows this material, they are prepared to begin studying calculus.
1. The document lists various trigonometric formulae including definitions of radians, trigonometric ratios, domains and ranges, allied angle relations, sum and difference formulae, and solutions to trigonometric equations.
2. Key formulae include the definitions of radians as 180°/π and degrees as π/180 radians, trigonometric ratios in terms of sine and cosine, and multiple angle formulae for sine, cosine, and tangent of doubled angles.
3. Trigonometric functions are also defined over their domains, with ranges between -1 and 1 except for cosecant, secant, and cotangent. Basic trigonometric identities and relations between quadrants are also provided.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...EduSkills OECD
Andreas Schleicher, Director of Education and Skills at the OECD presents at the launch of PISA 2022 Volume III - Creative Minds, Creative Schools on 18 June 2024.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
This presentation was provided by Racquel Jemison, Ph.D., Christina MacLaughlin, Ph.D., and Paulomi Majumder. Ph.D., all of the American Chemical Society, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
How to Manage Reception Report in Odoo 17Celine George
A business may deal with both sales and purchases occasionally. They buy things from vendors and then sell them to their customers. Such dealings can be confusing at times. Because multiple clients may inquire about the same product at the same time, after purchasing those products, customers must be assigned to them. Odoo has a tool called Reception Report that can be used to complete this assignment. By enabling this, a reception report comes automatically after confirming a receipt, from which we can assign products to orders.
2. Reduction Formula
Reduction (or recurrence) formulae expresses a given integral as
the sum of a function and a known integral.
Integration by parts often used to find the formula.
3. Reduction Formula
Reduction (or recurrence) formulae expresses a given integral as
the sum of a function and a known integral.
Integration by parts often used to find the formula.
e.g. (i) (1987)
n 1I
2
Given that I n cos xdx, prove that I n
n
n2
0 n
2
where n is an integer and n 2, hence evaluate cos 5 xdx
0
5.
2
I n cos n xdx
0
2
cos n 1 x cos xdx
0
6.
2
I n cos n xdx
0
2 u cos n 1 x v sin x
cos n 1 x cos xdx du n 1 cos n 2 x sin xdx dv cos xdx
0
7.
2
I n cos n xdx
0
2 u cos n 1 x v sin x
cos n 1 x cos xdx du n 1 cos n 2 x sin xdx dv cos xdx
0
cos n 1 x sin x 0 n 1 cos n 2 x sin 2 xdx
2
2
0
8.
2
I n cos n xdx
0
2 u cos n 1 x v sin x
cos n 1 x cos xdx du n 1 cos n 2 x sin xdx dv cos xdx
0
cos n 1 x sin x 0 n 1 cos n 2 x sin 2 xdx
2
2
0
cos n 1 sin cos n 1 0 sin 0 n 1 cos n 2 x1 cos 2 x dx
2
2 2
0
9.
2
I n cos n xdx
0
2 u cos n 1 x v sin x
cos n 1 x cos xdx du n 1 cos n 2 x sin xdx dv cos xdx
0
cos n 1 x sin x 0 n 1 cos n 2 x sin 2 xdx
2
2
0
cos n 1 sin cos n 1 0 sin 0 n 1 cos n 2 x1 cos 2 x dx
2
2 2
0
2 2
n 1 cos n 2 xdx n 1 cos n xdx
0 0
10.
2
I n cos n xdx
0
2 u cos n 1 x v sin x
cos n 1 x cos xdx du n 1 cos n 2 x sin xdx dv cos xdx
0
cos n 1 x sin x 0 n 1 cos n 2 x sin 2 xdx
2
2
0
cos n 1 sin cos n 1 0 sin 0 n 1 cos n 2 x1 cos 2 x dx
2
2 2
0
2 2
n 1 cos n 2 xdx n 1 cos n xdx
0 0
n 1I n 2 n 1I n
11.
2
I n cos n xdx
0
2 u cos n 1 x v sin x
cos n 1 x cos xdx du n 1 cos n 2 x sin xdx dv cos xdx
0
cos n 1 x sin x 0 n 1 cos n 2 x sin 2 xdx
2
2
0
cos n 1 sin cos n 1 0 sin 0 n 1 cos n 2 x1 cos 2 x dx
2
2 2
0
2 2
n 1 cos n 2 xdx n 1 cos n xdx
0 0
n 1I n 2 n 1I n
nI n n 1I n 2
12.
2
I n cos n xdx
0
2 u cos n 1 x v sin x
cos n 1 x cos xdx du n 1 cos n 2 x sin xdx dv cos xdx
0
cos n 1 x sin x 0 n 1 cos n 2 x sin 2 xdx
2
2
0
cos n 1 sin cos n 1 0 sin 0 n 1 cos n 2 x1 cos 2 x dx
2
2 2
0
2 2
n 1 cos n 2 xdx n 1 cos n xdx
0 0
n 1I n 2 n 1I n
nI n n 1I n 2
In n 1I
n2
n
20. ii Given that I n cot n xdx, find I 6
I n cot n xdx
21. ii Given that I n cot n xdx, find I 6
I n cot n xdx
cot n 2 x cot 2 xdx
22. ii Given that I n cot n xdx, find I 6
I n cot n xdx
cot n 2 x cot 2 xdx
cot n 2 xcosec 2 x 1dx
cot n 2 xcosec 2 xdx cot n 2 xdx
23. ii Given that I n cot n xdx, find I 6
I n cot n xdx
cot n 2 x cot 2 xdx
cot n 2 xcosec 2 x 1dx
cot n 2 xcosec 2 xdx cot n 2 xdx u cot x
du cosec 2 xdx
24. ii Given that I n cot n xdx, find I 6
I n cot n xdx
cot n 2 x cot 2 xdx
cot n 2 xcosec 2 x 1dx
cot n 2 xcosec 2 xdx cot n 2 xdx u cot x
u n2
du I n 2 du cosec 2 xdx
25. ii Given that I n cot n xdx, find I 6
I n cot n xdx
cot n 2 x cot 2 xdx
cot n 2 xcosec 2 x 1dx
cot n 2 xcosec 2 xdx cot n 2 xdx u cot x
u n2
du I n 2 du cosec 2 xdx
1 n 1
u I n2
n 1
1
cot n 1 x I n 2
n 1
28. cot 6 xdx I 6
1 5
cot x I 4
5
1 1
cot 5 x cot 3 x I 2
5 3
29. cot 6 xdx I 6
1 5
cot x I 4
5
1 1
cot 5 x cot 3 x I 2
5 3
1 5 1 3
cot x cot x cot x I 0
5 3
30. cot 6 xdx I 6
1 5
cot x I 4
5
1 1
cot 5 x cot 3 x I 2
5 3
1 5 1 3
cot x cot x cot x I 0
5 3
1 5 1 3
cot x cot x cot x dx
5 3
31. cot 6 xdx I 6
1 5
cot x I 4
5
1 1
cot 5 x cot 3 x I 2
5 3
1 5 1 3
cot x cot x cot x I 0
5 3
1 5 1 3
cot x cot x cot x dx
5 3
1 5 1 3
cot x cot x cot x x c
5 3
32. (iii) (2004 Question 8b)
Let I n 4 tan n xdx and let J n 1 I 2 n for n 0,1, 2,
n
0
1
a ) Show that I n I n2
n 1
33. (iii) (2004 Question 8b)
Let I n 4 tan n xdx and let J n 1 I 2 n for n 0,1, 2,
n
0
1
a ) Show that I n I n2
n 1
I n I n2 4 tan n dx 4 tan n2 dx
0 0
34. (iii) (2004 Question 8b)
Let I n 4 tan n xdx and let J n 1 I 2 n for n 0,1, 2,
n
0
1
a ) Show that I n I n2
n 1
I n I n2 4 tan n dx 4 tan n2 dx
0 0
4 tan n x 1 tan 2 x dx
0
4 tan n x sec 2 xdx
0
35. (iii) (2004 Question 8b)
Let I n 4 tan n xdx and let J n 1 I 2 n for n 0,1, 2,
n
0
1
a ) Show that I n I n2
n 1
I n I n2 4 tan n dx 4 tan n2 dx
0 0
4 tan n x 1 tan 2 x dx
0
u tan x
4 tan n x sec 2 xdx du sec 2 xdx
0
36. (iii) (2004 Question 8b)
Let I n 4 tan n xdx and let J n 1 I 2 n for n 0,1, 2,
n
0
1
a ) Show that I n I n2
n 1
I n I n2 4 tan n dx 4 tan n2 dx
0 0
4 tan n x 1 tan 2 x dx
0
u tan x
4 tan n x sec 2 xdx du sec 2 xdx
0
when x 0, u 0
x ,u 1
4
37. (iii) (2004 Question 8b)
Let I n 4 tan n xdx and let J n 1 I 2 n for n 0,1, 2,
n
0
1
a ) Show that I n I n2
n 1
I n I n2 4 tan n dx 4 tan n2 dx
0 0
4 tan n x 1 tan 2 x dx
0
u tan x
4 tan n x sec 2 xdx du sec 2 xdx
0
1
u n du when x 0, u 0
0
x ,u 1
4
38. (iii) (2004 Question 8b)
Let I n 4 tan n xdx and let J n 1 I 2 n for n 0,1, 2,
n
0
1
a ) Show that I n I n2
n 1
I n I n2 4 tan n dx 4 tan n2 dx
0 0
4 tan n x 1 tan 2 x dx
0
u tan x
4 tan n x sec 2 xdx du sec 2 xdx
0
1
u n du when x 0, u 0
0
u n 1 1
x ,u 1
4
n 1 0
39. (iii) (2004 Question 8b)
Let I n 4 tan n xdx and let J n 1 I 2 n for n 0,1, 2,
n
0
1
a ) Show that I n I n2
n 1
I n I n2 4 tan n dx 4 tan n2 dx
0 0
4 tan n x 1 tan 2 x dx
0
u tan x
4 tan n x sec 2 xdx du sec 2 xdx
0
1
u n du when x 0, u 0
0
u n 1 1
x ,u 1
4
n 1 0
1 1
0
n 1 n 1
40. 1
n
b) Deduce that J n J n1 for n 1
2n 1
41. 1
n
b) Deduce that J n J n1 for n 1
2n 1
J n J n1 1 I 2 n 1 I 2 n2
n n 1
42. 1
n
b) Deduce that J n J n1 for n 1
2n 1
J n J n1 1 I 2 n 1 I 2 n2
n n 1
1 I 2 n 1 I 2 n2
n n
43. 1
n
b) Deduce that J n J n1 for n 1
2n 1
J n J n1 1 I 2 n 1 I 2 n2
n n 1
1 I 2 n 1 I 2 n2
n n
1 I 2 n I 2 n2
n
44. 1
n
b) Deduce that J n J n1 for n 1
2n 1
J n J n1 1 I 2 n 1 I 2 n2
n n 1
1 I 2 n 1 I 2 n2
n n
1 I 2 n I 2 n2
n
1
n
2n 1
45. 1
n
m
c) Show that J m
4 n 1 2n 1
46. 1
n
m
c) Show that J m
4 n 1 2n 1
1
m
Jm J m1
2m 1
47. 1
n
m
c) Show that J m
4 n 1 2n 1
1
m
Jm J m1
2m 1
1 1
m m 1
J m2
2m 1 2m 3
48. 1
n
m
c) Show that J m
4 n 1 2n 1
1
m
Jm J m1
2m 1
1 1
m m 1
J m2
2m 1 2m 3
1 1 1
m m 1 1
J0
2m 1 2m 3 1
49. 1
n
m
c) Show that J m
4 n 1 2n 1
1
m
Jm J m1
2m 1
1 1
m m 1
J m2
2m 1 2m 3
1 1 1
m m 1 1
J0
2m 1 2m 3 1
1
n
m
4 dx
n 1 2n 1 0
50. 1
n
m
c) Show that J m
4 n 1 2n 1
1
m
Jm J m1
2m 1
1 1
m m 1
J m2
2m 1 2m 3
1 1 1
m m 1 1
J0
2m 1 2m 3 1
1
n
m
4 dx
n 1 2n 1 0
1
n
m
x 04
n 1 2n 1
1
n
m
n 1 2n 1 4
51. 1 un
d ) Use the substitution u tan x to show that I n du
0 1 u 2
52. 1 un
d ) Use the substitution u tan x to show that I n du
0 1 u 2
I n 4 tan n xdx
0
53. 1 un
d ) Use the substitution u tan x to show that I n du
0 1 u 2
I n 4 tan n xdx
0 u tan x x tan 1 u
du
dx
1 u2
54. 1 un
d ) Use the substitution u tan x to show that I n du
0 1 u 2
I n 4 tan n xdx
0 u tan x x tan 1 u
du
dx
1 u2
when x 0, u 0
x ,u 1
4
55. 1 un
d ) Use the substitution u tan x to show that I n du
0 1 u 2
I n 4 tan n xdx
0 u tan x x tan 1 u
1 du du
In un dx
0 1 u2 1 u2
1 u n du when x 0, u 0
In
0 1 u2
x ,u 1
4
56. 1 un
d ) Use the substitution u tan x to show that I n du
0 1 u 2
I n 4 tan n xdx
0 u tan x x tan 1 u
1 du du
In un dx
0 1 u2 1 u2
1 u n du when x 0, u 0
In
0 1 u2
x ,u 1
4
1
e) Deduce that 0 I n and conclude that J n 0 as n
n 1
57. 1 un
d ) Use the substitution u tan x to show that I n du
0 1 u 2
I n 4 tan n xdx
0 u tan x x tan 1 u
1 du du
In un dx
0 1 u2 1 u2
1 u n du when x 0, u 0
In
0 1 u2
x ,u 1
4
1
e) Deduce that 0 I n and conclude that J n 0 as n
n 1
un
0, for all u 0
1 u 2
58. 1 un
d ) Use the substitution u tan x to show that I n du
0 1 u 2
I n 4 tan n xdx
0 u tan x x tan 1 u
1 du du
In un dx
0 1 u2 1 u2
1 u n du when x 0, u 0
In
0 1 u2
x ,u 1
4
1
e) Deduce that 0 I n and conclude that J n 0 as n
n 1
un
0, for all u 0
1 u 2
n
1 u
In du 0, for all u 0
0 1 u2
61. 1
I n I n 2
n 1
1
In I n 2
n 1
1
In , as I n2 0
n 1
62. 1
I n I n 2
n 1
1
In I n 2
n 1
1
In , as I n2 0
n 1
1
0 In
n 1
63. 1
I n I n 2
n 1
1
In I n 2
n 1
1
In , as I n2 0
n 1
1
0 In
n 1
1
as n , 0
n 1
64. 1
I n I n 2
n 1
1
In I n 2
n 1
1
In , as I n2 0
n 1
1
0 In
n 1
1
as n , 0
n 1
In 0
65. 1
I n I n 2
n 1
1
In I n 2
n 1
1
In , as I n2 0
n 1
1
0 In
n 1
1
as n , 0
n 1
In 0
J n 1 I 2 n 0
n
66. 1
I n I n 2
n 1
1
In I n 2
n 1
1
In , as I n2 0
n 1
1
0 In Exercise 2D; 1, 2, 3, 6, 8,
n 1 9, 10, 12, 14
1
as n , 0
n 1
In 0
J n 1 I 2 n 0
n