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Num Integration

This document discusses numerical integration methods for calculating ship geometrical properties. It introduces the Trapezoidal rule, Simpson's 1st rule, and Simpson's 2nd rule for numerical integration when the ship's shape cannot be represented by a mathematical equation. It then provides examples of applying Simpson's 1st rule to calculate properties like waterplane area, sectional area, submerged volume, and the longitudinal center of floatation (LCF). The document explains the calculation steps and provides generalized Simpson's equations for these examples.

Nilai tugas statistika

This document summarizes the results of chi-square tests performed on assignment, midterm, and final exam score data from 30 students. The chi-square tests for assignment scores had a value of 19.531, df of 15, and p-value of 0.191. The tests for midterm scores had values of 60.625, df of 60, and p-value of 0.453. For final exam scores, the values were 59.375, df of 60, and p-value of 0.499. All p-values were greater than 0.05, so the null hypothesis was accepted in each case.

Introduction to Regression line

This topic is based on research in Computer Science | Pattern Recognition | Probability and Statistics.
Here, we discuss Regression Line with a simple example. Basics of Line equation is demonstrated with a real-world example.
Eg: A railway track is paved in a line connecting various cities. It can be associated with the google maps with cities connected by the flight route / railway route.
Eg: Politicians demanding alternative route for railway constructions
This video is contributed by
https://sites.google.com/view/amarnath-r
Sample Program:
https://sites.google.com/view/amarnath-r/introduction-to-regression-analysis

Cmo sample-papers-for-class-8

CREST Maths Olympiad exam sample paper for class 8th. Sample papers are useful in understanding the question pattern of CMO Olympiad exam.

Obj. 41 Geometric Probability

1. The document defines geometric probability as probability based on ratios of geometric measures like length and area, where outcomes are represented by points or regions.
2. Examples are provided to demonstrate calculating geometric probabilities for situations like choosing a random point on a line segment or in a plane figure, the probability of light cycles, and spinners.
3. Additional examples find the probabilities of points chosen in a rectangle landing in specific shapes like a circle, trapezoid, or one of two squares.

Mason loosleandresantos honors geometry project

The document explains the Pythagorean theorem and how to use it to determine if a triangle is a right triangle. It provides the formula, A2 + B2 = C2, where C is the hypotenuse and A and B are the legs. It then works through examples of applying the formula to find the length of an unknown side of different right triangles. Finally, it explains the converse of the theorem, which is that if the sum of the squares of two sides equals the square of the third side, then the triangle must be a right triangle.

0580 s10 qp_23

This document is an exam for the International General Certificate of Secondary Education in mathematics. It consists of 12 printed pages and contains 23 multiple choice and written response questions testing a variety of math skills. Candidates are instructed to show their work, use calculators and mathematical tables, and provide answers with an appropriate degree of accuracy.

Mathematics Class 6 Paper

The document is a math assessment for 6th class students at Nawal Public School in Chirawa. It contains 11 questions testing various math concepts like properties of operations, the distributive law, number lines, successor/predecessor of numbers, and operations with whole numbers. The assessment has a maximum mark of 20 and is to be completed within 30 minutes. It includes multiple choice, matching, and short answer questions.

Num Integration

This document discusses numerical integration methods for calculating ship geometrical properties. It introduces the Trapezoidal rule, Simpson's 1st rule, and Simpson's 2nd rule for numerical integration when the ship's shape cannot be represented by a mathematical equation. It then provides examples of applying Simpson's 1st rule to calculate properties like waterplane area, sectional area, submerged volume, and the longitudinal center of floatation (LCF). The document explains the calculation steps and provides generalized Simpson's equations for these examples.

Nilai tugas statistika

This document summarizes the results of chi-square tests performed on assignment, midterm, and final exam score data from 30 students. The chi-square tests for assignment scores had a value of 19.531, df of 15, and p-value of 0.191. The tests for midterm scores had values of 60.625, df of 60, and p-value of 0.453. For final exam scores, the values were 59.375, df of 60, and p-value of 0.499. All p-values were greater than 0.05, so the null hypothesis was accepted in each case.

Introduction to Regression line

This topic is based on research in Computer Science | Pattern Recognition | Probability and Statistics.
Here, we discuss Regression Line with a simple example. Basics of Line equation is demonstrated with a real-world example.
Eg: A railway track is paved in a line connecting various cities. It can be associated with the google maps with cities connected by the flight route / railway route.
Eg: Politicians demanding alternative route for railway constructions
This video is contributed by
https://sites.google.com/view/amarnath-r
Sample Program:
https://sites.google.com/view/amarnath-r/introduction-to-regression-analysis

Cmo sample-papers-for-class-8

CREST Maths Olympiad exam sample paper for class 8th. Sample papers are useful in understanding the question pattern of CMO Olympiad exam.

Obj. 41 Geometric Probability

1. The document defines geometric probability as probability based on ratios of geometric measures like length and area, where outcomes are represented by points or regions.
2. Examples are provided to demonstrate calculating geometric probabilities for situations like choosing a random point on a line segment or in a plane figure, the probability of light cycles, and spinners.
3. Additional examples find the probabilities of points chosen in a rectangle landing in specific shapes like a circle, trapezoid, or one of two squares.

Mason loosleandresantos honors geometry project

The document explains the Pythagorean theorem and how to use it to determine if a triangle is a right triangle. It provides the formula, A2 + B2 = C2, where C is the hypotenuse and A and B are the legs. It then works through examples of applying the formula to find the length of an unknown side of different right triangles. Finally, it explains the converse of the theorem, which is that if the sum of the squares of two sides equals the square of the third side, then the triangle must be a right triangle.

0580 s10 qp_23

This document is an exam for the International General Certificate of Secondary Education in mathematics. It consists of 12 printed pages and contains 23 multiple choice and written response questions testing a variety of math skills. Candidates are instructed to show their work, use calculators and mathematical tables, and provide answers with an appropriate degree of accuracy.

Mathematics Class 6 Paper

The document is a math assessment for 6th class students at Nawal Public School in Chirawa. It contains 11 questions testing various math concepts like properties of operations, the distributive law, number lines, successor/predecessor of numbers, and operations with whole numbers. The assessment has a maximum mark of 20 and is to be completed within 30 minutes. It includes multiple choice, matching, and short answer questions.

9 maths sample papers 2

This document contains a mock test for the CBSE Class 9 math final exam. It has 34 multiple choice and constructed response questions across 4 sections (A, B, C, D) testing topics like lines and angles, probability, data representation, surface area, and geometric proofs. The test has a total of 90 marks and is to be completed within 3 hours.

Eliminators into dependent types

This document describes eliminators (also called induction principles) for dependent types in dependent type theory, including:
- Empty, unit, sum, product, and function types
- Dependent pairs and dependent functions
- Booleans, natural numbers, lists, and vectors
- Identity types
For each type, it provides the eliminator signature and definition, allowing values of that type to be analyzed in a dependent context.

19 - Scala. Eliminators into dependent types (induction)

This document describes eliminators (also called induction principles) for dependent types in dependent type theory, including:
- Empty, unit, sum, product, and function types
- Dependent pairs and dependent functions
- Boolean, natural numbers, lists, and vectors
- Identity types
For each type, it provides the eliminator signature and definition, describing how to eliminate values of that type into a dependent type.

Traffic congestion2

The document discusses a math problem about modeling traffic congestion using a function. It provides the function f(t) and asks students to analyze properties of the traffic jam like its length at different times, when it is longest, and how quickly it increases and decreases. It also includes artwork created by students depicting traffic jams and the solutions to the math questions about the modeled congestion based on the given function.

Quadratic Functions

This document provides information about graphing parabolas on a graphing calculator. It discusses the key parts of a parabola graph like the vertex, axis of symmetry, zeros, and y-intercept. It explains how to input equations into the graphing calculator and use functions to find the vertex, zeros, and maximum or minimum points. The document also covers how changing variables in the standard parabola equation affects the shape, direction, and location of the graphed parabola. Some examples of word problems involving parabolas are presented as well.

How to test

To solve a quadratic equation by completing the square, rewrite the left side in the form of x^2 + bx, add (b/2)^2 to both sides to create a perfect square on the left, then take the square root of both sides. To solve by the quadratic formula, identify the coefficients a, b, and c and plug them into the formula x = (-b ± √(b^2 - 4ac)) / 2a to find the solutions.

Depreciation

The document provides information about depreciation methods for computer equipment and a building. Computer equipment is depreciated using the reducing balance method at 30% annually, resulting in $18,960 of depreciation. The building is depreciated using the straight line method with a $1,000 scrap value over 40 years of useful life, resulting in $1,200 of annual depreciation.

Eco human resources university

This document discusses economic concepts related to demand and elasticity. It provides calculations of price elasticity of demand for various pairs of goods using the midpoint formula. Specifically, it calculates:
1) The price elasticity of demand between goods A and B, between B and C, between D and E, and between A and D.
2) It also finds the price elasticity of demand between goods B and D.
3) For each pair of goods, it calculates the elasticity using the midpoint formula and provides the specific values.

Obj. 39 Composite Figures

The document discusses finding the area of composite figures, which are shapes made up of simple shapes like triangles, rectangles, trapezoids, and circles. To find the area of a composite figure, you divide it into its simple shapes, find the area of each shape, and add or subtract the areas together. It provides examples of calculating the areas of different composite figures by breaking them into triangles, squares, circles and other basic shapes and combining those areas.

maths Individual assignment on differentiation

The document contains an individual assignment with 14 math problems. The assignment includes problems on calculus topics such as derivatives, limits, implicit differentiation, and optimization. The solutions show the steps and work to arrive at the answers for each problem.

Ejercicio 6

The document provides steps to solve the integral of Sen 3X CosX dX. It rewrites the integral in terms of u-substitution where u = SenX and du = CosX dX. It then evaluates the integral as (SenX) + c = Sen X + c.

2.1 graphing quadratic functions

The document discusses quadratic functions and their graphs known as parabolas. It provides examples of graphing quadratic functions and finding key features such as the vertex, axis of symmetry, and x-intercepts. Specifically, it explains that the graph of a quadratic function f(x) = ax^2 + bx + c is a parabola. The leading coefficient a determines whether the parabola opens upward or downward, and the vertex is located at (-b/2a, f(-b/2a)). Examples are given to demonstrate how to graph quadratic functions and find the vertex and intercepts.

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.

Actividad 19 mrc

This document appears to be a handwritten note listing the names Austin Mahone and Mariana Rivera Cabrera along with the date 5/12/13 and designation 1F. It also includes a list of song titles by Austin Mahone including "11:11", "Say Somethin", "What About Love", "Heard It In A Love Song", and "Say You're Just A Friend".

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.

5.1 analysis of function i

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.

JFS NEPA 2015 Presentation

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.

Defining an audience

This document discusses how media producers select content and define their target audiences. It provides examples of how different elements like images, words, colors, and fonts are used tailored to specific audiences. Images on the front covers of magazines are carefully chosen to represent the magazine's content and attract their target readers. Captions, headlines, and other text elements are also used to shape how audiences interpret and understand images. The layout, color schemes, and other conventions established "codes and rules" that publishers follow to engage their target demographics. Feedback from focus groups, panels, and complaints help publishers evaluate audience reactions and make improvements.

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.

9 maths sample papers 2

This document contains a mock test for the CBSE Class 9 math final exam. It has 34 multiple choice and constructed response questions across 4 sections (A, B, C, D) testing topics like lines and angles, probability, data representation, surface area, and geometric proofs. The test has a total of 90 marks and is to be completed within 3 hours.

Eliminators into dependent types

This document describes eliminators (also called induction principles) for dependent types in dependent type theory, including:
- Empty, unit, sum, product, and function types
- Dependent pairs and dependent functions
- Booleans, natural numbers, lists, and vectors
- Identity types
For each type, it provides the eliminator signature and definition, allowing values of that type to be analyzed in a dependent context.

19 - Scala. Eliminators into dependent types (induction)

This document describes eliminators (also called induction principles) for dependent types in dependent type theory, including:
- Empty, unit, sum, product, and function types
- Dependent pairs and dependent functions
- Boolean, natural numbers, lists, and vectors
- Identity types
For each type, it provides the eliminator signature and definition, describing how to eliminate values of that type into a dependent type.

Traffic congestion2

The document discusses a math problem about modeling traffic congestion using a function. It provides the function f(t) and asks students to analyze properties of the traffic jam like its length at different times, when it is longest, and how quickly it increases and decreases. It also includes artwork created by students depicting traffic jams and the solutions to the math questions about the modeled congestion based on the given function.

Quadratic Functions

This document provides information about graphing parabolas on a graphing calculator. It discusses the key parts of a parabola graph like the vertex, axis of symmetry, zeros, and y-intercept. It explains how to input equations into the graphing calculator and use functions to find the vertex, zeros, and maximum or minimum points. The document also covers how changing variables in the standard parabola equation affects the shape, direction, and location of the graphed parabola. Some examples of word problems involving parabolas are presented as well.

How to test

To solve a quadratic equation by completing the square, rewrite the left side in the form of x^2 + bx, add (b/2)^2 to both sides to create a perfect square on the left, then take the square root of both sides. To solve by the quadratic formula, identify the coefficients a, b, and c and plug them into the formula x = (-b ± √(b^2 - 4ac)) / 2a to find the solutions.

Depreciation

The document provides information about depreciation methods for computer equipment and a building. Computer equipment is depreciated using the reducing balance method at 30% annually, resulting in $18,960 of depreciation. The building is depreciated using the straight line method with a $1,000 scrap value over 40 years of useful life, resulting in $1,200 of annual depreciation.

Eco human resources university

This document discusses economic concepts related to demand and elasticity. It provides calculations of price elasticity of demand for various pairs of goods using the midpoint formula. Specifically, it calculates:
1) The price elasticity of demand between goods A and B, between B and C, between D and E, and between A and D.
2) It also finds the price elasticity of demand between goods B and D.
3) For each pair of goods, it calculates the elasticity using the midpoint formula and provides the specific values.

Obj. 39 Composite Figures

The document discusses finding the area of composite figures, which are shapes made up of simple shapes like triangles, rectangles, trapezoids, and circles. To find the area of a composite figure, you divide it into its simple shapes, find the area of each shape, and add or subtract the areas together. It provides examples of calculating the areas of different composite figures by breaking them into triangles, squares, circles and other basic shapes and combining those areas.

maths Individual assignment on differentiation

The document contains an individual assignment with 14 math problems. The assignment includes problems on calculus topics such as derivatives, limits, implicit differentiation, and optimization. The solutions show the steps and work to arrive at the answers for each problem.

Ejercicio 6

The document provides steps to solve the integral of Sen 3X CosX dX. It rewrites the integral in terms of u-substitution where u = SenX and du = CosX dX. It then evaluates the integral as (SenX) + c = Sen X + c.

2.1 graphing quadratic functions

The document discusses quadratic functions and their graphs known as parabolas. It provides examples of graphing quadratic functions and finding key features such as the vertex, axis of symmetry, and x-intercepts. Specifically, it explains that the graph of a quadratic function f(x) = ax^2 + bx + c is a parabola. The leading coefficient a determines whether the parabola opens upward or downward, and the vertex is located at (-b/2a, f(-b/2a)). Examples are given to demonstrate how to graph quadratic functions and find the vertex and intercepts.

9 maths sample papers 2

9 maths sample papers 2

Eliminators into dependent types

Eliminators into dependent types

19 - Scala. Eliminators into dependent types (induction)

19 - Scala. Eliminators into dependent types (induction)

Traffic congestion2

Traffic congestion2

Quadratic Functions

Quadratic Functions

How to test

How to test

Depreciation

Depreciation

Eco human resources university

Eco human resources university

Obj. 39 Composite Figures

Obj. 39 Composite Figures

maths Individual assignment on differentiation

maths Individual assignment on differentiation

Ejercicio 6

Ejercicio 6

2.1 graphing quadratic functions

2.1 graphing quadratic functions

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.

Actividad 19 mrc

This document appears to be a handwritten note listing the names Austin Mahone and Mariana Rivera Cabrera along with the date 5/12/13 and designation 1F. It also includes a list of song titles by Austin Mahone including "11:11", "Say Somethin", "What About Love", "Heard It In A Love Song", and "Say You're Just A Friend".

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.

5.1 analysis of function i

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.

JFS NEPA 2015 Presentation

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.

Defining an audience

This document discusses how media producers select content and define their target audiences. It provides examples of how different elements like images, words, colors, and fonts are used tailored to specific audiences. Images on the front covers of magazines are carefully chosen to represent the magazine's content and attract their target readers. Captions, headlines, and other text elements are also used to shape how audiences interpret and understand images. The layout, color schemes, and other conventions established "codes and rules" that publishers follow to engage their target demographics. Feedback from focus groups, panels, and complaints help publishers evaluate audience reactions and make improvements.

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.

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.

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.

4.3 derivative of exponential functions

L'Hopital's rule provides a method for evaluating indeterminate limits of the form 0/0 and ∞/∞ by taking the derivative of the numerator and denominator. It can be applied when the limit of f(x)/g(x) is an indeterminate form, by finding the limit of f'(x)/g'(x) instead. Examples are provided of using L'Hopital's rule to evaluate limits that are indeterminate forms such as 0/0, ∞/∞, 0×∞, and ∞-∞. Other indeterminate forms like 0^∞ can sometimes be evaluated by introducing a new variable and taking the limit of its logarithm

4.2 derivatives of logarithmic functions

This document discusses implicit and explicit differentiation.
It provides examples of taking the derivative of equations in both implicit and explicit form. It also shows how to find the derivative at a point, such as finding the slope of an implicitly defined equation at the point (1,1).

7.2 volumes by slicing disks and washers

This document discusses different methods for calculating the volumes of solids of revolution: the disk method and washer method. It provides step-by-step explanations of how to set up and evaluate the definite integrals needed to calculate these volumes, whether the region is revolved about an axis that forms a border or not. Examples are given to illustrate each method. The key steps are to divide the solid into slices, approximate the volume of each slice, add the slice volumes using a limit of a Riemann sum, and evaluate the resulting definite integral.

Logistic Regression/Markov Chain presentation

Presentation on Logistic Regression/Markov Chain algorithm for forecasting NCAA Basketball tournament outcomes. For Regression and ANOVA course

8.2 integration by parts

8.2 integration by parts

5.3 curve sketching

5.3 curve sketching

Actividad 19 mrc

Actividad 19 mrc

5.7 rolle's thrm & mv theorem

5.7 rolle's thrm & mv theorem

8.7 numerical integration

8.7 numerical integration

5.1 analysis of function i

5.1 analysis of function i

6.3 integration by substitution

6.3 integration by substitution

JFS NEPA 2015 Presentation

JFS NEPA 2015 Presentation

Defining an audience

Defining an audience

7.1 area between curves

7.1 area between curves

5.2 first and second derivative test

5.2 first and second derivative test

6.2 the indefinite integral

6.2 the indefinite integral

4.3 derivatives of inv erse trig. functions

4.3 derivatives of inv erse trig. functions

6.1 & 6.4 an overview of the area problem area

6.1 & 6.4 an overview of the area problem area

4.3 derivative of exponential functions

4.3 derivative of exponential functions

4.2 derivatives of logarithmic functions

4.2 derivatives of logarithmic functions

7.2 volumes by slicing disks and washers

7.2 volumes by slicing disks and washers

7.3 volumes by cylindrical shells

7.3 volumes by cylindrical shells

4.1 implicit differentiation

4.1 implicit differentiation

Logistic Regression/Markov Chain presentation

Logistic Regression/Markov Chain presentation

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

This document discusses different methods for calculating the volumes of solids of revolution: the disk method and washer method. It provides step-by-step explanations of how to set up and evaluate the definite integrals needed to calculate these volumes, whether the region is revolved about an axis that forms a border or not. Examples are given to illustrate each method. The key steps are to divide the solid into slices, approximate the volume of each slice, add the slice volumes using a limit of a Riemann sum, and evaluate the resulting definite integral.

7.1 area between curves

The document discusses calculating the area between two curves. It explains that this area is defined as the limit of sums of the areas of rectangles between the curves as the number of rectangles approaches infinity, which is represented by a definite integral. It provides examples of finding the area between curves defined by various functions through setting up and evaluating the appropriate definite integrals.

6.3 integration by substitution

This document discusses integration by substitution. It provides an example of recognizing a composite function and rewriting the integral in terms of the inside and outside functions. Specifically, it shows rewriting the integral of (x2 +1)2x dx as the integral of the outside function (x2 + 1) with the inside function (x) plugged in, plus a constant. It then provides additional practice problems applying the technique of substitution to rewrite integrals in terms of u-substitutions.

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:
- Lower estimate (LAM) uses the left endpoints of intervals
- Upper estimate (RAM) uses the right endpoints
- Average estimate (MAM) uses the midpoints
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.

8.7 numerical integration

8.7 numerical integration

8.2 integration by parts

8.2 integration by parts

7.3 volumes by cylindrical shells

7.3 volumes by cylindrical shells

7.2 volumes by slicing disks and washers

7.2 volumes by slicing disks and washers

7.1 area between curves

7.1 area between curves

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...

6.3 integration by substitution

6.3 integration by substitution

6.2 the indefinite integral

6.2 the indefinite integral

6.1 & 6.4 an overview of the area problem area

6.1 & 6.4 an overview of the area problem area

5.8 rectilinear motion

5.8 rectilinear motion

5.7 rolle's thrm & mv theorem

5.7 rolle's thrm & mv theorem

5.5 optimization

5.5 optimization

5.4 absolute maxima and minima

5.4 absolute maxima and minima

5.3 curve sketching

5.3 curve sketching

5.2 first and second derivative test

5.2 first and second derivative test

5.1 analysis of function i

5.1 analysis of function i

4.3 derivatives of inv erse trig. functions

4.3 derivatives of inv erse trig. functions

7.2 volumes by slicing disks and washers

7.2 volumes by slicing disks and washers

8.2 integration by parts

8.2 integration by parts

8.7 numerical integration

8.7 numerical integration

- 1. Applied Maximum and Minimum Problems
- 2. Optimization Problems 1. Draw an appropriate figure and label the quantities relevant to the problem. 2. Write a primary equation that relates the given and unknown quantities. 3. If necessary, reduce the primary equation to 1 variable (use a secondary equation if necessary). 4. Determine the desired max/min using the derivative(s). 5. Check solutions with possible values (domain).
- 3. Example: An open box is to be made from a 16” by 30” piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. What size should the squares be to obtain a box with the largest volume? V = l×w×h l =30-2x w =16-2x h = x V = 30-2x ( )× 16-2x ( )× x V = 30x -2x2 ( )× 16-2x ( )= 480x -60x2 -32x2 + 4x3 V = 4x3 -92x2 +480x V ' =12x2 -184x+480 = 0 V ' = 4 3x2 - 46x +120 ( )= 0 V ' = 4 x -12 ( ) 3x-10 ( )= 0 x x 16 30 0 £ x £8 x = 10 3 and x =12 x = 10 3
- 4. Example: An offshore oil well located at a point W that is 5 km from the closest point A on a straight shoreline. Oil is to be piped from W to a shore point B that is 8 km from A by piping it on a straight line under water from W to some shore point P between A and B and then on to B via pipe along the shoreline. If the cost of laying pipe is $1 million under water and $½ million over land, where should the point P be located to minimize the coast of laying the pipe? 5 km A P B x 8 – x 8 km W WP = x2 +25 PB =8- x C =1× x2 +25 + 1 2 × 8- x ( ) 0 £ x £8 C' = 1 2 x2 +25 ×2x - 1 2 = 0 x x2 +25 = 1 2 x2 +25 = 2x x2 +25= 4x2 3x2 = 25 x = 5 3 x = - 5 3 C 5 3 æ è ç ö ø ÷ » 8.330127 C(8) » 9.433981 abs max
- 5. Example: You have 200 feet of fencing to enclose two adjacent rectangular corrals. What dimensions should be used to maximize the area? A = 2x× 200- 4x 3 = 400x -8x2 3 P = 4x+3y = 200 y = 200- 4x 3 A' = 3 400-16x ( ) 9 = 0 x = 25 0 £ x £ 200 y = 100 3
- 6. Example: Find the radius and height of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 4 inches and height 10 inches. r h V = pr2 h 10-h 10-h r = 10 4 h =10- 5 2 r V = pr2 10- 5 2 r æ è ç ö ø ÷ V =10pr2 - 5p 2 r3 V ' = 20pr - 15p 2 r2 = 0 5pr 4- 3 2 r æ è ç ö ø ÷ = 0 r = 8 3 0 £ r £ 4 r(0) = 0 r 8 3 æ è ç ö ø ÷ » 74.42962 r(4) = 0 abs max h = 10 3