4.3 derivatives of inv erse trig. functions
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Lesson 11: Implicit Differentiation
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
Multiplication of polynomials
The document discusses multiplying polynomials, including multiplying monomials, combining like terms, and special cases such as the sum and difference of binomials, squares of binomials, and cubes of binomials. Examples are provided for multiplying polynomials with 2, 3, or 4 terms. Formulas and step-by-step workings are shown for finding products of binomial expressions.
10 1 Adding Subtracting Polynomials
This document discusses two methods for adding and subtracting polynomials: horizontal and vertical. For addition, terms with the same variables are combined by adding the coefficients. For subtraction, the signs of the terms in the second polynomial are changed to addition signs before combining like terms. Both horizontal and vertical methods can be used, with horizontal grouping like terms and vertical lining them up before combining coefficients.
Module 9 Topic 1 - Adding & Subtracting polynomials
The document provides notes on polynomials, including defining polynomials, describing their terms and degrees, adding and subtracting polynomials, and working through examples of finding degrees, adding, subtracting, and combining like terms of polynomials. The notes include 5 pages on adding and subtracting polynomials and working through examples step-by-step to show the process.
Topic 1 adding & subtracting polynomials
The document describes a lesson on adding and subtracting polynomials. It includes examples of finding the degree of polynomials, adding polynomials by combining like terms, and subtracting polynomials by distributing the negative terms. The examples start simply and increase in complexity, covering the key concepts of adding and subtracting polynomials of various variables and degrees. Student practice problems are provided throughout for students to work through the skills.
7 2 adding and subtracting polynomials
The document discusses several methods for adding and subtracting polynomials: using algebra tiles, the horizontal method, and the vertical method. It provides examples of adding and subtracting polynomials with one and two variables. Terms with the same variables are combined by adding the coefficients. For subtraction, the signs of the terms in the subtracted polynomial are changed before adding.
Notes 12.1 identifying, adding & subtracting polynomials
This document provides examples and explanations of identifying and simplifying polynomials. It begins with defining polynomials as expressions with variables that have non-negative integer exponents. Examples are provided of identifying polynomials and non-polynomials. Later pages discuss adding and subtracting polynomials by combining like terms. Worked examples are shown of simplifying polynomial expressions by removing parentheses and combining like terms. Degrees of polynomials are identified.
Practice power point
1) The document provides step-by-step solutions to 3 problems involving complex roots and factoring polynomials.
2) For problem 1, the technique of sum of cubes is used to solve the equation 27x^3 + 1 = 0, yielding solutions of x = -1/3, x = [1 + i sqrt(3)] / 6, and x = [1 - i sqrt(3)] / 6.
3) Problems 2 and 3 similarly use techniques like sum of cubes, quadratic formula, and substitution to solve the equations x^3 + 8 = 0 and x^4 – 2x^2 – 8, providing the techniques and solutions.
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Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
Multiplication of polynomials
The document discusses multiplying polynomials, including multiplying monomials, combining like terms, and special cases such as the sum and difference of binomials, squares of binomials, and cubes of binomials. Examples are provided for multiplying polynomials with 2, 3, or 4 terms. Formulas and step-by-step workings are shown for finding products of binomial expressions.
10 1 Adding Subtracting Polynomials
This document discusses two methods for adding and subtracting polynomials: horizontal and vertical. For addition, terms with the same variables are combined by adding the coefficients. For subtraction, the signs of the terms in the second polynomial are changed to addition signs before combining like terms. Both horizontal and vertical methods can be used, with horizontal grouping like terms and vertical lining them up before combining coefficients.
Module 9 Topic 1 - Adding & Subtracting polynomials
The document provides notes on polynomials, including defining polynomials, describing their terms and degrees, adding and subtracting polynomials, and working through examples of finding degrees, adding, subtracting, and combining like terms of polynomials. The notes include 5 pages on adding and subtracting polynomials and working through examples step-by-step to show the process.
Topic 1 adding & subtracting polynomials
The document describes a lesson on adding and subtracting polynomials. It includes examples of finding the degree of polynomials, adding polynomials by combining like terms, and subtracting polynomials by distributing the negative terms. The examples start simply and increase in complexity, covering the key concepts of adding and subtracting polynomials of various variables and degrees. Student practice problems are provided throughout for students to work through the skills.
7 2 adding and subtracting polynomials
The document discusses several methods for adding and subtracting polynomials: using algebra tiles, the horizontal method, and the vertical method. It provides examples of adding and subtracting polynomials with one and two variables. Terms with the same variables are combined by adding the coefficients. For subtraction, the signs of the terms in the subtracted polynomial are changed before adding.
Notes 12.1 identifying, adding & subtracting polynomials
This document provides examples and explanations of identifying and simplifying polynomials. It begins with defining polynomials as expressions with variables that have non-negative integer exponents. Examples are provided of identifying polynomials and non-polynomials. Later pages discuss adding and subtracting polynomials by combining like terms. Worked examples are shown of simplifying polynomial expressions by removing parentheses and combining like terms. Degrees of polynomials are identified.
Practice power point
1) The document provides step-by-step solutions to 3 problems involving complex roots and factoring polynomials.
2) For problem 1, the technique of sum of cubes is used to solve the equation 27x^3 + 1 = 0, yielding solutions of x = -1/3, x = [1 + i sqrt(3)] / 6, and x = [1 - i sqrt(3)] / 6.
3) Problems 2 and 3 similarly use techniques like sum of cubes, quadratic formula, and substitution to solve the equations x^3 + 8 = 0 and x^4 – 2x^2 – 8, providing the techniques and solutions.
Tugas matematika menemukan konsep persamaan kuadrat
1. The document contains examples of solving quadratic equations by factorizing, completing the square, and using the quadratic formula.
2. The examples include finding the area of land by setting up and solving a quadratic equation, finding the volume of objects using quadratic equations, and determining the roots of various quadratic equations.
3. The key steps shown are factorizing, completing the square, setting the discriminant equal to zero, and using the quadratic formula to solve for the roots of the equations.
Adding and subtracting polynomials
This document provides an overview of adding and subtracting polynomials. It defines monomials, binomials, and trinomials as polynomials with 1, 2, or 3 terms respectively. Polynomials can have any number of terms. Terms must have like variables to be combined using addition or subtraction. Examples are provided for determining the type of polynomial, simplifying polynomials by combining like terms, and adding and subtracting polynomials. Practice problems are assigned from the workbook.
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1. The document provides examples of adding and subtracting polynomials by grouping like terms and using column form.
2. Students are asked to perform operations like adding (9y - 7x + 15a) + (-3y + 8x - 8a) and subtracting (4x^2 - 2xy + 3y^2) - (-3x^2 - xy + 2y^2).
3. The objectives are for students to learn how to add and subtract polynomials.
1 4 homework
This document contains examples of multiplying, expanding, and simplifying rational expressions. Some examples involve breaking rational expressions into sums and differences of fractions. Other examples use long division to write rational expressions in the form of a quotient plus a remainder over the divisor. The rational expressions involve variables and operations.
Factorización aplicando Ruffini o Método de Evaluación
1. The document discusses factorizing polynomials using Ruffini's method.
2. It provides two examples of factorizing polynomials into their prime factors: (1) x3 + 5x2 - 2x - 24 = (x-2)(x+3)(x+4) and (2) 2x3 - 3x2 - 11x + 6 = (2x - 1)(x + 2)(x - 3).
3. For each polynomial, it lists the possible factors of the constant term and coefficients, then uses those factors to find the roots that allow factoring the polynomial.
Understanding the remainder theorem
The document discusses using the remainder theorem to find the remainder of a polynomial function divided by a binomial function without fully performing the division. It provides two examples:
1) When f(x) = x^3 - 3x^2 + 7 is divided by g(x) = x + 2, the remainder is -13.
2) When f(x) = x^3 - 3x^2 + 5x - 1 is divided by g(x) = x - 1, the remainder is 2.
The key steps are to set the binomial factor equal to 0 to find the value to substitute into the polynomial, then evaluate the polynomial at that value to determine the remainder.
Adding and subtracting polynomials
This document discusses adding and subtracting polynomials. It explains that to add polynomials, terms with the same variables and exponents are combined. To subtract polynomials, the opposite of the second polynomial is taken and then the polynomials are added like normal. Examples are provided of rearranging terms in decreasing exponential order, stacking like terms, and combining them to solve polynomial addition and subtraction problems.
Sudėtingesnės trupmeninės lygtys
This document contains solutions to three equations involving fractions equal to values.
The first equation, x^2 + x - 2 = 0, is solved to find the values of x that satisfy the equation, which are x = 1 and x = -2.
The second equation, (x-1)/(x+1) = 3, is solved to find the value of x that makes the fraction equal to 3, which is x = -2.
The third equation, x/(x-1) - 1/(x+2) - 3/(x-1)(x+2) = 0, is solved but does not have any real number solutions for x.
Add/Subtracting Polynomials
The student will learn to:
1. Add and subtract polynomials by grouping like terms.
2. Use column form to add and subtract polynomials with three terms.
3. Simplify polynomial expressions involving addition and subtraction.
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This document discusses integration by substitution. It explains that instead of calling the function of x "STUFF", it will be called u, and its derivative will be du. The reader is instructed to identify part of the integrand as u and part as du. Examples are provided of using this technique to integrate functions of the form u^n, including (x^2 + 1)^3 as an example. The reader is prompted to try integrating (3x - 9)/(9x^2) and other functions using this substitution method.
Addition and subtraction of polynomials
The document provides information about adding and subtracting algebraic expressions:
- Like terms are algebraic expressions with the same variables and exponents.
- Unlike terms cannot be combined.
- To add algebraic expressions, combine like terms by adding the coefficients.
- To subtract expressions, change the sign of the second expression and then add as if adding.
Unit 3 polynomials
This document provides an overview of polynomials, including adding, subtracting, and multiplying polynomials. It begins by reviewing integer rules and exponent rules. It then covers combining like terms, adding polynomials by combining like terms and using integer addition/subtraction rules, subtracting polynomials using distribution and combining like terms, and multiplying polynomials using distribution and the box method. Examples are provided for each topic.
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The document summarizes how to subtract polynomials. It explains that to subtract polynomials, one changes the sign of all terms in the subtrahend, changes the operation to addition, and then proceeds to add the polynomials by aligning like terms and combining coefficients. It provides examples of subtracting various polynomials, as well as exercises for students to practice subtracting polynomials.
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Quadratic equations can be solved in several ways:
1) Factorizing, by finding two numbers whose product is the constant term and sum is the coefficient of the x term.
2) Using the quadratic formula.
3) Substitution, by letting an expression like x^2 + 2x equal a variable k, and solving the simplified equation for k and back substituting.
4) Squaring both sides, but this can introduce extraneous solutions so one must check solutions.
Adding & Subtracting Polynomials
This document discusses two methods for adding and subtracting polynomials:
1) The horizontal method involves grouping like terms together and keeping the signs with each term.
2) The vertical method lines up like terms and keeps the signs with each term.
To subtract polynomials, change all the signs in the second set and then add the polynomials as if it were an addition problem. Either the horizontal or vertical method can be used, depending on how the problem is laid out.
Operations on Polynomials
This is a short Powerpoint presentation that addresses some of the steps necessary to add, subtract, and multiply polynomials.
Adding Polynomials
This document defines polynomials and describes how to perform operations on them such as addition and subtraction. It provides examples of adding and subtracting monomials and polynomials. Monomials are terms with variables and coefficients, and polynomials are the sum of monomials. Like terms refer to monomials with the same variables and exponents that can be combined. To add polynomials, like terms are lined up and their coefficients are summed. To subtract polynomials, the operation is changed to addition by using the keep-change-change method and then like terms are combined.
Finding the area under a curve using integration
This document discusses using integration to find the area under a curve. It defines integration as the reverse of differentiation and shows how to find antiderivatives. It provides examples of indefinite and definite integrals, and explains how definite integrals between limits can be used to calculate the area under a curve over an interval. The document demonstrates this by finding the area under various curves bounded between given x-values. It also discusses how to handle areas below or partly above and below the x-axis.
Polinomials division
This document demonstrates how to divide two polynomials. It shows the step-by-step process of dividing the polynomial x3 + 2x2 - 4x + 3 by the polynomial x2 - 1. The quotient is x + 2 and the remainder is -3x + 5.
5.2 first and second derivative test
This document provides guidance on sketching graphs of functions by considering key features such as symmetries, intercepts, extrema, asymptotes, concavity, and inflection points. It then works through an example of sketching the graph of the function f(x) = (2x^2 - 8)/(x^2 - 16). Key steps include finding vertical and horizontal asymptotes, critical points and inflection points, intervals of increase/decrease, and finally sketching the graph.
7.1 area between curves
This document discusses calculating the volume of solids of revolution formed by rotating an area about an axis. It provides the formula for calculating the volume of a cylindrical shell as well as the process for finding the total volume of a solid of revolution by summing the volumes of infinitely thin cylindrical shells. It includes two example problems demonstrating how to set up and solve the integrals to find the volume of solids formed by rotating specific regions about the y-axis. It concludes by providing another example problem for the reader to practice solving.
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Jewish Family Service is a nonprofit organization that has served the community since 1915. It provides various counseling, education, financial assistance, and social services to individuals and families in need. These services include counseling, consulting, emergency financial aid, dental care, guardianship services for older adults, assistance for Holocaust survivors, educational programs, and support for older adults and families through services like meals on wheels and a kosher food pantry. The organization aims to enhance quality of life through professional social work and a mission of caring for those in need.
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1. The document contains examples of solving quadratic equations by factorizing, completing the square, and using the quadratic formula.
2. The examples include finding the area of land by setting up and solving a quadratic equation, finding the volume of objects using quadratic equations, and determining the roots of various quadratic equations.
3. The key steps shown are factorizing, completing the square, setting the discriminant equal to zero, and using the quadratic formula to solve for the roots of the equations.
Adding and subtracting polynomials
This document provides an overview of adding and subtracting polynomials. It defines monomials, binomials, and trinomials as polynomials with 1, 2, or 3 terms respectively. Polynomials can have any number of terms. Terms must have like variables to be combined using addition or subtraction. Examples are provided for determining the type of polynomial, simplifying polynomials by combining like terms, and adding and subtracting polynomials. Practice problems are assigned from the workbook.
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1. The document provides examples of adding and subtracting polynomials by grouping like terms and using column form.
2. Students are asked to perform operations like adding (9y - 7x + 15a) + (-3y + 8x - 8a) and subtracting (4x^2 - 2xy + 3y^2) - (-3x^2 - xy + 2y^2).
3. The objectives are for students to learn how to add and subtract polynomials.
1 4 homework
This document contains examples of multiplying, expanding, and simplifying rational expressions. Some examples involve breaking rational expressions into sums and differences of fractions. Other examples use long division to write rational expressions in the form of a quotient plus a remainder over the divisor. The rational expressions involve variables and operations.
Factorización aplicando Ruffini o Método de Evaluación
1. The document discusses factorizing polynomials using Ruffini's method.
2. It provides two examples of factorizing polynomials into their prime factors: (1) x3 + 5x2 - 2x - 24 = (x-2)(x+3)(x+4) and (2) 2x3 - 3x2 - 11x + 6 = (2x - 1)(x + 2)(x - 3).
3. For each polynomial, it lists the possible factors of the constant term and coefficients, then uses those factors to find the roots that allow factoring the polynomial.
Understanding the remainder theorem
The document discusses using the remainder theorem to find the remainder of a polynomial function divided by a binomial function without fully performing the division. It provides two examples:
1) When f(x) = x^3 - 3x^2 + 7 is divided by g(x) = x + 2, the remainder is -13.
2) When f(x) = x^3 - 3x^2 + 5x - 1 is divided by g(x) = x - 1, the remainder is 2.
The key steps are to set the binomial factor equal to 0 to find the value to substitute into the polynomial, then evaluate the polynomial at that value to determine the remainder.
Adding and subtracting polynomials
This document discusses adding and subtracting polynomials. It explains that to add polynomials, terms with the same variables and exponents are combined. To subtract polynomials, the opposite of the second polynomial is taken and then the polynomials are added like normal. Examples are provided of rearranging terms in decreasing exponential order, stacking like terms, and combining them to solve polynomial addition and subtraction problems.
Sudėtingesnės trupmeninės lygtys
This document contains solutions to three equations involving fractions equal to values.
The first equation, x^2 + x - 2 = 0, is solved to find the values of x that satisfy the equation, which are x = 1 and x = -2.
The second equation, (x-1)/(x+1) = 3, is solved to find the value of x that makes the fraction equal to 3, which is x = -2.
The third equation, x/(x-1) - 1/(x+2) - 3/(x-1)(x+2) = 0, is solved but does not have any real number solutions for x.
Add/Subtracting Polynomials
The student will learn to:
1. Add and subtract polynomials by grouping like terms.
2. Use column form to add and subtract polynomials with three terms.
3. Simplify polynomial expressions involving addition and subtraction.
125 5.2
This document discusses integration by substitution. It explains that instead of calling the function of x "STUFF", it will be called u, and its derivative will be du. The reader is instructed to identify part of the integrand as u and part as du. Examples are provided of using this technique to integrate functions of the form u^n, including (x^2 + 1)^3 as an example. The reader is prompted to try integrating (3x - 9)/(9x^2) and other functions using this substitution method.
Addition and subtraction of polynomials
The document provides information about adding and subtracting algebraic expressions:
- Like terms are algebraic expressions with the same variables and exponents.
- Unlike terms cannot be combined.
- To add algebraic expressions, combine like terms by adding the coefficients.
- To subtract expressions, change the sign of the second expression and then add as if adding.
Unit 3 polynomials
This document provides an overview of polynomials, including adding, subtracting, and multiplying polynomials. It begins by reviewing integer rules and exponent rules. It then covers combining like terms, adding polynomials by combining like terms and using integer addition/subtraction rules, subtracting polynomials using distribution and combining like terms, and multiplying polynomials using distribution and the box method. Examples are provided for each topic.
Lesson 6 subtraction of polynomials
The document summarizes how to subtract polynomials. It explains that to subtract polynomials, one changes the sign of all terms in the subtrahend, changes the operation to addition, and then proceeds to add the polynomials by aligning like terms and combining coefficients. It provides examples of subtracting various polynomials, as well as exercises for students to practice subtracting polynomials.
Gr 11 equations
Quadratic equations can be solved in several ways:
1) Factorizing, by finding two numbers whose product is the constant term and sum is the coefficient of the x term.
2) Using the quadratic formula.
3) Substitution, by letting an expression like x^2 + 2x equal a variable k, and solving the simplified equation for k and back substituting.
4) Squaring both sides, but this can introduce extraneous solutions so one must check solutions.
Adding & Subtracting Polynomials
This document discusses two methods for adding and subtracting polynomials:
1) The horizontal method involves grouping like terms together and keeping the signs with each term.
2) The vertical method lines up like terms and keeps the signs with each term.
To subtract polynomials, change all the signs in the second set and then add the polynomials as if it were an addition problem. Either the horizontal or vertical method can be used, depending on how the problem is laid out.
Operations on Polynomials
This is a short Powerpoint presentation that addresses some of the steps necessary to add, subtract, and multiply polynomials.
Adding Polynomials
This document defines polynomials and describes how to perform operations on them such as addition and subtraction. It provides examples of adding and subtracting monomials and polynomials. Monomials are terms with variables and coefficients, and polynomials are the sum of monomials. Like terms refer to monomials with the same variables and exponents that can be combined. To add polynomials, like terms are lined up and their coefficients are summed. To subtract polynomials, the operation is changed to addition by using the keep-change-change method and then like terms are combined.
Finding the area under a curve using integration
This document discusses using integration to find the area under a curve. It defines integration as the reverse of differentiation and shows how to find antiderivatives. It provides examples of indefinite and definite integrals, and explains how definite integrals between limits can be used to calculate the area under a curve over an interval. The document demonstrates this by finding the area under various curves bounded between given x-values. It also discusses how to handle areas below or partly above and below the x-axis.
Polinomials division
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7.1 area between curves
This document discusses calculating the volume of solids of revolution formed by rotating an area about an axis. It provides the formula for calculating the volume of a cylindrical shell as well as the process for finding the total volume of a solid of revolution by summing the volumes of infinitely thin cylindrical shells. It includes two example problems demonstrating how to set up and solve the integrals to find the volume of solids formed by rotating specific regions about the y-axis. It concludes by providing another example problem for the reader to practice solving.
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Jewish Family Service is a nonprofit organization that has served the community since 1915. It provides various counseling, education, financial assistance, and social services to individuals and families in need. These services include counseling, consulting, emergency financial aid, dental care, guardianship services for older adults, assistance for Holocaust survivors, educational programs, and support for older adults and families through services like meals on wheels and a kosher food pantry. The organization aims to enhance quality of life through professional social work and a mission of caring for those in need.
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1. Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints.
2. To solve maximum/minimum problems: draw a figure, write the primary equation relating quantities, reduce to one variable if needed, take the derivative(s) to find critical points, and check solutions in the domain.
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8.7 numerical integration
The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval. The area is calculated as the sum of the areas of each trapezoid multiplied by the width of the sub-interval. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals, giving an approximate result of 1.26953125. The document also provides an example of using the trapezoidal rule with n=8 sub-intervals to estimate the area under the curve of the function y=x from 0 to 3.
8.2 integration by parts
Integration by parts is a technique for evaluating integrals of the form ∫udv, where u and v are differentiable functions. It works by expressing the integral as uv - ∫vdu. Some examples of integrals solved using integration by parts include ∫x cos x dx, ∫xe^x dx, and ∫ln x dx. Repeated integration by parts may be necessary when the integral ∫vdu produced is still difficult to evaluate. Integration by parts also applies to definite integrals between limits a and b using the formula ∫_a^b udv = [uv]_a^b - ∫_a^b vdu.
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4.3 derivative of exponential functions
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8.7 numerical integration
The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval multiplied by the sub-interval width. The general formula sums these values over all sub-intervals divided by the number of intervals. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals and gets an approximate value of 1.26953125.
8.2 integration by parts
Integration by parts is a technique for evaluating integrals of the form ∫udv, where u and v are differentiable functions. It works by expressing the integral as uv - ∫vdu. Some examples of integrals solved using integration by parts include ∫xe^xdx, ∫lnxdx, and ∫xe^-xdx. The technique can also be used repeatedly and for definite integrals between limits a and b using the formula ∫abudv = uv|_a^b - ∫avdu.
7.3 volumes by cylindrical shells
This document discusses calculating the volume of solids of revolution formed by rotating an area bounded by graphs around an axis. It provides the formula for finding the volume of a cylindrical shell as well as the formula for finding the total volume of a solid of revolution by summing the volumes of infinitely thin cylindrical shells. It includes two example problems demonstrating how to set up and solve the integrals to find the volume of solids of revolution.
7.2 volumes by slicing disks and washers
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6.3 integration by substitution
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6.2 the indefinite integral
This document discusses the indefinite integral and antiderivatives. It defines an antiderivative as a function whose derivative is the original function, and notes that there are infinitely many antiderivatives that differ by a constant. The process of finding antiderivatives is called indefinite integration or antidifferentiation. Initial conditions can be used to determine a unique particular solution by solving for the constant of integration.
6.1 & 6.4 an overview of the area problem area
The document discusses different methods for approximating the area under a curve:
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Formulas are provided for calculating the area using each method by summing the areas of rectangles. Examples are shown for finding the area under y=x^2 from 0 to 2 using each method. Finally, the document introduces using the antiderivative method to find the exact area under a curve by calculating the antiderivative and evaluating it over the bounds.
5.8 rectilinear motion
This document discusses rectilinear motion and concepts related to position, velocity, speed, and acceleration for objects moving along a straight line. It defines velocity as the rate of change of position with respect to time and speed as the magnitude of velocity. Acceleration is defined as the rate of change of velocity with respect to time. Examples are given to show how to calculate position, velocity, speed, and acceleration functions from a given position function. The document also analyzes position versus time graphs to determine characteristics of the particle's motion at different points in time.
5.7 rolle's thrm & mv theorem
This document discusses Rolle's theorem and the mean value theorem. It provides the definitions and formulas for each theorem. It then gives examples of applying each theorem to find values of c where a derivative is equal to zero or a tangent line is parallel to a secant line. Rolle's theorem examples find values of c where the derivative of a function over an interval is zero. The mean value theorem examples find values of c where the slope of a tangent line equals the slope of a secant line over an interval.
5.5 optimization
1. Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints.
2. To solve maximum/minimum problems: draw a figure, write the primary equation relating quantities, reduce to one variable if needed, take the derivative(s) to find critical points, and check solutions in the domain.
3. Examples show applying this process to find the dimensions that maximize volume of an open box, minimize cost of laying pipe between points, maximize area of two corrals with a fixed fence length, and find the largest volume cylinder that can fit in a cone.
5.4 absolute maxima and minima
The document provides information on finding absolute maximum and minimum values (absolute extrema) of functions on different interval types. It discusses determining absolute extrema on closed, infinite, and open intervals. Examples are provided finding the absolute extrema of specific functions on given intervals, including finding any critical points and limits to determine if absolute extrema exist. Practice problems are also provided at the end to find the absolute extrema of additional functions on specified intervals.
5.3 curve sketching
This document provides guidance on sketching graphs of functions by considering key features such as symmetries, intercepts, extrema, asymptotes, concavity, and inflection points. It then works through an example of sketching the graph of the function f(x) = (2x^2 - 8)/(x^2 - 16). Key steps include finding vertical and horizontal asymptotes, critical points and inflection points, intervals of increase/decrease, and finally sketching the graph.
5.2 first and second derivative test
This document discusses methods for finding relative extrema of functions:
1. The First Derivative Test (FDT) states that a critical point is a relative maximum if the derivative changes from positive to negative, and a relative minimum if the derivative changes from negative to positive.
2. The Second Derivative Test (SDT) states that a critical point is a relative maximum if the second derivative is negative, and a relative minimum if the second derivative is positive.
3. Examples are provided to demonstrate using the FDT and SDT to find relative extrema of functions.
5.1 analysis of function i
This document discusses increasing and decreasing functions, concavity of functions, and finding intervals where functions are increasing, decreasing, concave up, or concave down. It provides examples of finding the intervals for the functions f(x)=x-4x^2+3 and f(x)=x-5x^4+9x showing the steps to determine where the functions are increasing or decreasing and where they are concave up or concave down. It also discusses inflection points and provides an example of finding intervals of increase, decrease, concavity and the inflection point for the function f(x)=x^3-3x^2+1.
4.3 derivatives of inv erse trig. functions
This document discusses derivatives of inverse trigonometric functions and differentiability of inverse functions. It provides examples of finding the derivative of inverse trig functions like sin^-1(x^3) and sec^-1(e^x). It also explains that if a function f(x) is differentiable on an interval I, its inverse f^-1(x) will also be differentiable if f'(x) is not equal to 0. It gives the formula for the derivative of the inverse function and an example confirming this formula. It also discusses monotonic functions and how if f'(x) is always greater than 0 or less than 0, f(x) is one-to-one and its inverse will be different
8.2 integration by parts
Integration by parts is a technique for evaluating integrals of the form ∫udv, where u and v are differentiable functions. It works by expressing the integral as uv - ∫vdu. Some examples of integrals solved using integration by parts include ∫xe^xdx, ∫lnxdx, and ∫xe^-xdx. Repeated integration by parts may be necessary when the integral ∫vdu generated cannot be directly evaluated. Integration by parts also applies to definite integrals between limits a and b using the formula ∫_a^budv = uv|_a^b - ∫_a^bvdu.
8.7 numerical integration
The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval. The area is calculated as the sum of the areas of each trapezoid multiplied by the width of the sub-interval. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals, giving an approximate result of 1.26953125. The document also provides an example of using the trapezoidal rule with n=8 sub-intervals to estimate the area under the curve of the function y=x from 0 to 3.
More from dicosmo178 (20)
6.5 & 6.6 & 6.9 the definite integral and the fundemental theorem of calculus...
6.5 & 6.6 & 6.9 the definite integral and the fundemental theorem of calculus...
4.3 derivatives of inv erse trig. functions
- 2. Implicit & Explicit Forms Implicit Form Explicit Form Derivative Explicit: y in terms of x Implicit: y and x together Differentiating: want to be able to use either 1 xy 1 1 x x y 2 2 1 x x dx dy
- 3. Differentiating with respect to x Derivative → Deriving when denominator agrees → use properties Deriving when denominator disagrees → use chain rule & properties dx d 2 4x dx d x 8 Denominator agrees - properties 2 3y dx d dx dy y 6 Denominator disagrees – chain rule
- 4. Implicit Differentiation Trying to find dx dy
- 5. Derive Explicitly Derive Implicitly x2 + y2 = 5 x2 + y2 = 5 y = ± 5- x2 dy dx = 1 ±2 5- x2 × -2x ( ) dy dx = -x ± 5- x2 dy dx = -x y since y = ± 5- x2 2x+2y dy dx =0 2y dy dx =-2x y dy dx =-x dy dx = -x y
- 6. Derive implicitly: y = 3xy4 dy dx = 3× y4 +4y3 dy dx 3x dy dx - 4y3 dy dx ×3x = 3× y4 dy dx 1- 4y3 3x ( )= 3× y4 dy dx 1-12xy3 ( )=3×y4 dy dx = 3×y4 1-12xy3 ( )
- 7. Example: Find the derivative 38 3 2 2 2 3 xy y x x 3x2 - 4xy+ dy dx 2x2 æ è ç ö ø ÷+ 3y2 +2y dy dx 3x æ è ç ö ø ÷= 0 3x2 - 4xy- dy dx 2x2 +3y2 +2y dy dx 3x = 0 -6xy dy dx +2x2 dy dx = 3x2 - 4xy+3y2 dy dx -6xy+2x2 ( )= 3x2 - 4xy+3y2 dy dx = 3x2 - 4xy+3y2 -6xy+2x2 ( )
- 8. Example: Determine the slope at the point (1,1) xy y x 2 3 3 3x2 +3y2 dy dx = 2y+ dy dx 2x 3y2 dy dx -2x dy dx = 2y -3x2 dy dx 3y2 -2x ( )= 2y -3x2 dy dx x=1 y=1 = -1 1 = -1 dy dx = 2y -3x2 3y2 - 2x ( ) dy dx x=1 y=1 = 2×1-3×12 3×12 - 2×1 ( )
- 9. xy y x 2 3 3 dy dx = 2y -3x2 3y2 - 2x ( ) dy dx x=1 y=1 = -1 1 = -1