The document describes the method of halving intervals to find approximations of roots. It begins by stating if a function f(x) is continuous on an interval [a,b] where f(a) and f(b) have opposite signs, then there exists a root between a and b. It then works through an example of finding the root of x4 + 2x - 19 = 0 between 1 and 3 by repeatedly halving intervals and evaluating the function at the midpoint until reaching an approximation of 1.96 to two decimal places.
12 X1 T04 07 approximations to roots (2010)Nigel Simmons
The document describes the method of halving intervals to find approximations of roots. It begins by stating if a function f(x) is continuous on an interval [a,b] where f(a) and f(b) have opposite signs, then there exists a root between a and b. It then works through an example of finding the root of x4 + 2x - 19 = 0 between 1 and 3 by repeatedly halving intervals and evaluating the function at the midpoint until reaching an approximation of 1.96 to two decimal places.
The document summarizes the factor theorem and remainder theorem. [1] The factor theorem states that a polynomial P(x) has a factor x - a if and only if P(a) = 0. [2] The remainder theorem states that if p(x) is a polynomial, then p(a) is equal to the remainder when p(x) is divided by x - a. [3] Examples are provided to demonstrate using synthetic division and evaluating polynomials using these theorems to find factors and remainders.
This document provides instructions for solving quadratic equations by completing the square. It gives examples of solving equations by:
1) Dividing both sides by the leading coefficient, if present
2) Subtracting any constant term not involving the variable
3) Completing the square by adding the square of half the coefficient of the second term
4) Taking the square root of both sides to isolate the variable
The document summarizes reflections from Patrick Hyatt and Quincie McCalla on a project where they created complex math problems to review concepts they had learned. Both note that while the project provided some review, it took a significant amount of time to develop the problems and stretched their understanding too far at times. They suggest modifying the format of the project to have students work in pairs and take responsibility for specific concepts, then present problems to each other, for a more focused review that clarifies misunderstandings.
Topic 4 dividing a polynomial by a monomialLori Rapp
The document discusses dividing monomials by monomials by applying the rules of exponents. It explains that when dividing monomials with the same base, the exponent of the denominator is subtracted from the exponent of the numerator. The result will be in the numerator if the larger exponent was in the numerator, and in the denominator if it was in the denominator. Examples are provided to demonstrate dividing monomials and simplifying expressions involving division of monomials.
The document provides examples of factoring sums and differences of powers of polynomials with real and complex coefficients. It demonstrates factoring polynomials using the sums and differences of cubes theorem and sums and differences of odd powers theorem. Examples factor polynomials of the form x^n - y^n, x^n + y^n, t^7 - w^7, x - y^10, and more.
This document discusses solving inequalities and compound inequalities. It explains that inequalities are solved similarly to equations, with the additional rule that the inequality symbol must be reversed if multiplying or dividing by a negative number. Compound inequalities contain two inequalities joined by "and" or "or". For "and" the graph is the intersection of both solutions, while for "or" it is the union of both solutions. Examples are provided to demonstrate solving single and compound inequalities graphically.
This document provides instruction on multiplying, dividing, and simplifying rational expressions. It begins with mini lessons on multiplying fractions and simplifying fractions before and after multiplying. It then works through 4 example problems, showing the step-by-step work of multiplying and simplifying rational expressions. The last section provides a mini lesson on dividing fractions and works through an example of dividing two rational expressions.
12 X1 T04 07 approximations to roots (2010)Nigel Simmons
The document describes the method of halving intervals to find approximations of roots. It begins by stating if a function f(x) is continuous on an interval [a,b] where f(a) and f(b) have opposite signs, then there exists a root between a and b. It then works through an example of finding the root of x4 + 2x - 19 = 0 between 1 and 3 by repeatedly halving intervals and evaluating the function at the midpoint until reaching an approximation of 1.96 to two decimal places.
The document summarizes the factor theorem and remainder theorem. [1] The factor theorem states that a polynomial P(x) has a factor x - a if and only if P(a) = 0. [2] The remainder theorem states that if p(x) is a polynomial, then p(a) is equal to the remainder when p(x) is divided by x - a. [3] Examples are provided to demonstrate using synthetic division and evaluating polynomials using these theorems to find factors and remainders.
This document provides instructions for solving quadratic equations by completing the square. It gives examples of solving equations by:
1) Dividing both sides by the leading coefficient, if present
2) Subtracting any constant term not involving the variable
3) Completing the square by adding the square of half the coefficient of the second term
4) Taking the square root of both sides to isolate the variable
The document summarizes reflections from Patrick Hyatt and Quincie McCalla on a project where they created complex math problems to review concepts they had learned. Both note that while the project provided some review, it took a significant amount of time to develop the problems and stretched their understanding too far at times. They suggest modifying the format of the project to have students work in pairs and take responsibility for specific concepts, then present problems to each other, for a more focused review that clarifies misunderstandings.
Topic 4 dividing a polynomial by a monomialLori Rapp
The document discusses dividing monomials by monomials by applying the rules of exponents. It explains that when dividing monomials with the same base, the exponent of the denominator is subtracted from the exponent of the numerator. The result will be in the numerator if the larger exponent was in the numerator, and in the denominator if it was in the denominator. Examples are provided to demonstrate dividing monomials and simplifying expressions involving division of monomials.
The document provides examples of factoring sums and differences of powers of polynomials with real and complex coefficients. It demonstrates factoring polynomials using the sums and differences of cubes theorem and sums and differences of odd powers theorem. Examples factor polynomials of the form x^n - y^n, x^n + y^n, t^7 - w^7, x - y^10, and more.
This document discusses solving inequalities and compound inequalities. It explains that inequalities are solved similarly to equations, with the additional rule that the inequality symbol must be reversed if multiplying or dividing by a negative number. Compound inequalities contain two inequalities joined by "and" or "or". For "and" the graph is the intersection of both solutions, while for "or" it is the union of both solutions. Examples are provided to demonstrate solving single and compound inequalities graphically.
This document provides instruction on multiplying, dividing, and simplifying rational expressions. It begins with mini lessons on multiplying fractions and simplifying fractions before and after multiplying. It then works through 4 example problems, showing the step-by-step work of multiplying and simplifying rational expressions. The last section provides a mini lesson on dividing fractions and works through an example of dividing two rational expressions.
The document discusses travel graphs showing the height of a bouncing ball over time. It provides the equation to model the ball's height and uses it to:
1) Find the ball's height after 1 second (35m) and the time it reaches this height again (7 seconds)
2) Calculate the average velocity during the 1st second (35m/s) and 5th second (-5m/s)
3) Determine that the average velocity over the ball's 8 seconds in the air is 0.
12 x1 t04 03 further growth & decay (2012)Nigel Simmons
The document discusses equations to model growth and decay over time by accounting for limiting conditions. It presents an equation for population change over time and its solution. It then gives an example using Newton's law of cooling, showing that an equation in the form T = A + Ce^kt satisfies the law, where T is temperature, A is the constant outside temperature, and C and k are constants. It then solves this equation to find that given information about a room's temperature dropping over half an hour, the temperature will reach 10 degrees Celsius after approximately 2 hours.
12 x1 t04 06 integrating functions of time (2012)Nigel Simmons
The document discusses integrating functions of time to determine changes in displacement, distance, velocity, and speed. It explains that the integral of position over time equals displacement, while subtracting integrals of position over different time intervals equals distance. Similarly, the integral of velocity over time equals speed, while the integral of acceleration over time equals velocity. Graphs of functions and their derivatives are also presented, showing the relationships between integration and differentiation.
The document discusses calculating rates of change for variables in terms of other variables. It provides two examples:
1) A spherical balloon deflating at a constant radius decrease rate of 10 mm/s. When the radius is 100 mm, the volume decrease rate is calculated to be 400000π mm3/s.
2) A spherical bubble expanding with a constant volume increase rate of 70 mm3/s. When the radius is 10 mm, the surface area increase rate is calculated to be 140 mm2/s.
The document discusses exponential growth and decay models. It shows that the rate of change of population (dP/dt) is proportional to the existing population (P), with dP/dt = kP. This leads to the differential equation P = Ae^kt, where A is the initial population and k is the growth or decay constant. Examples are given to show how to use the model to determine population sizes at different times, and to calculate growth or decay rates from population data.
The document discusses graphing sine curve functions. It explains that a basic sine curve has the equation y = sin(x) with a period of 2π units. More generally, a sine curve has the equation y = a sin(bx + c) where the period is 2π/b units, the amplitude is a units, and c shifts the curve left or right. It provides an example of y = 5 sin(9x - π/2) with a period of 2π/9 units, amplitude of 5 units, and a right shift of π/18.
The document discusses key concepts in calculus that are applied to physical systems, including:
- Displacement (distance from a point with direction)
- Velocity (rate of change of displacement over time, including direction)
- Acceleration (rate of change of velocity over time)
It provides examples of how to calculate acceleration, velocity, and displacement from equations describing physical phenomena. The relationships between these concepts are illustrated, such as how displacement can be determined by integrating velocity or differentiating acceleration. Calculus allows describing physical systems quantitatively and determining values like speed, acceleration, and position at different points in time.
The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(
12X1 T04 05 approximations to roots (2011)Nigel Simmons
The document describes the method of halving intervals to find approximations of roots. It begins by stating if a function f(x) is continuous over an interval [a,b] where f(a) and f(b) have opposite signs, then there exists a root between a and b. It then works through an example of finding the root of x4 + 2x - 19 = 0 between 1 and 3 by repeatedly halving intervals and evaluating the function at the midpoint until reaching an approximation of 1.96 to two decimal places.
12 x1 t04 07 approximations to roots (2012)Nigel Simmons
The document describes the method of halving intervals to find approximations of roots. It begins by stating if a function f(x) is continuous on an interval [a,b] where f(a) and f(b) have opposite signs, then there exists a root between a and b. It then works through an example of finding the root of x4 + 2x - 19 = 0 between 1 and 3 by repeatedly halving intervals and evaluating the function at the midpoint until reaching an approximation of 1.96 to two decimal places.
The document provides information on determining limits of algebraic functions. It discusses different methods for calculating limits, including dividing the numerator and denominator by the highest power term, and multiplying by the conjugate of the numerator and denominator. Examples are provided to illustrate each method and determine limits as the variable approaches a value.
This test contains 19 multiple choice questions about quantitative methods. It will be administered on Friday April 25, 2008 for 60 minutes and is worth a total of 35 marks. The test covers topics such as derivatives, optimization, and linear programming.
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.
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.
- The document discusses factoring expressions using the greatest common factor (GCF) method and grouping method.
- With the GCF method, the greatest common factor is determined for each term and factored out.
- The grouping method is used when there are 4 or more terms. The first two terms are factored and the second two terms are factored, then the common factor is extracted.
- Examples of applying both methods to factor polynomials are provided and worked through step-by-step.
The document discusses using Taylor series to estimate the value of functions and introduces the Lagrange form of the remainder to determine the accuracy of estimates based on partial sums. It defines the Lagrange form of the remainder as the next term in the Taylor series with "a" replaced by a value "c" between the point "a" and the point "x" being estimated. The document also introduces the Remainder Estimation Theorem which uses the maximum value of the derivative in the interval to bound the size of the remainder term.
The document discusses using Taylor series to estimate the value of functions and introduces the Lagrange form of the remainder to determine the accuracy of estimates based on partial sums. It defines the Lagrange form of the remainder as the next term in the Taylor series with "a" replaced by a value "c" between the point "a" and the point "x" being estimated. The document also introduces the Remainder Estimation Theorem, which uses the maximum value of the derivative in the interval to bound the size of the remainder term.
The document discusses travel graphs showing the height of a bouncing ball over time. It provides the equation to model the ball's height and uses it to:
1) Find the ball's height after 1 second (35m) and the time it reaches this height again (7 seconds)
2) Calculate the average velocity during the 1st second (35m/s) and 5th second (-5m/s)
3) Determine that the average velocity over the ball's 8 seconds in the air is 0.
12 x1 t04 03 further growth & decay (2012)Nigel Simmons
The document discusses equations to model growth and decay over time by accounting for limiting conditions. It presents an equation for population change over time and its solution. It then gives an example using Newton's law of cooling, showing that an equation in the form T = A + Ce^kt satisfies the law, where T is temperature, A is the constant outside temperature, and C and k are constants. It then solves this equation to find that given information about a room's temperature dropping over half an hour, the temperature will reach 10 degrees Celsius after approximately 2 hours.
12 x1 t04 06 integrating functions of time (2012)Nigel Simmons
The document discusses integrating functions of time to determine changes in displacement, distance, velocity, and speed. It explains that the integral of position over time equals displacement, while subtracting integrals of position over different time intervals equals distance. Similarly, the integral of velocity over time equals speed, while the integral of acceleration over time equals velocity. Graphs of functions and their derivatives are also presented, showing the relationships between integration and differentiation.
The document discusses calculating rates of change for variables in terms of other variables. It provides two examples:
1) A spherical balloon deflating at a constant radius decrease rate of 10 mm/s. When the radius is 100 mm, the volume decrease rate is calculated to be 400000π mm3/s.
2) A spherical bubble expanding with a constant volume increase rate of 70 mm3/s. When the radius is 10 mm, the surface area increase rate is calculated to be 140 mm2/s.
The document discusses exponential growth and decay models. It shows that the rate of change of population (dP/dt) is proportional to the existing population (P), with dP/dt = kP. This leads to the differential equation P = Ae^kt, where A is the initial population and k is the growth or decay constant. Examples are given to show how to use the model to determine population sizes at different times, and to calculate growth or decay rates from population data.
The document discusses graphing sine curve functions. It explains that a basic sine curve has the equation y = sin(x) with a period of 2π units. More generally, a sine curve has the equation y = a sin(bx + c) where the period is 2π/b units, the amplitude is a units, and c shifts the curve left or right. It provides an example of y = 5 sin(9x - π/2) with a period of 2π/9 units, amplitude of 5 units, and a right shift of π/18.
The document discusses key concepts in calculus that are applied to physical systems, including:
- Displacement (distance from a point with direction)
- Velocity (rate of change of displacement over time, including direction)
- Acceleration (rate of change of velocity over time)
It provides examples of how to calculate acceleration, velocity, and displacement from equations describing physical phenomena. The relationships between these concepts are illustrated, such as how displacement can be determined by integrating velocity or differentiating acceleration. Calculus allows describing physical systems quantitatively and determining values like speed, acceleration, and position at different points in time.
The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(
12X1 T04 05 approximations to roots (2011)Nigel Simmons
The document describes the method of halving intervals to find approximations of roots. It begins by stating if a function f(x) is continuous over an interval [a,b] where f(a) and f(b) have opposite signs, then there exists a root between a and b. It then works through an example of finding the root of x4 + 2x - 19 = 0 between 1 and 3 by repeatedly halving intervals and evaluating the function at the midpoint until reaching an approximation of 1.96 to two decimal places.
12 x1 t04 07 approximations to roots (2012)Nigel Simmons
The document describes the method of halving intervals to find approximations of roots. It begins by stating if a function f(x) is continuous on an interval [a,b] where f(a) and f(b) have opposite signs, then there exists a root between a and b. It then works through an example of finding the root of x4 + 2x - 19 = 0 between 1 and 3 by repeatedly halving intervals and evaluating the function at the midpoint until reaching an approximation of 1.96 to two decimal places.
The document provides information on determining limits of algebraic functions. It discusses different methods for calculating limits, including dividing the numerator and denominator by the highest power term, and multiplying by the conjugate of the numerator and denominator. Examples are provided to illustrate each method and determine limits as the variable approaches a value.
This test contains 19 multiple choice questions about quantitative methods. It will be administered on Friday April 25, 2008 for 60 minutes and is worth a total of 35 marks. The test covers topics such as derivatives, optimization, and linear programming.
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.
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.
- The document discusses factoring expressions using the greatest common factor (GCF) method and grouping method.
- With the GCF method, the greatest common factor is determined for each term and factored out.
- The grouping method is used when there are 4 or more terms. The first two terms are factored and the second two terms are factored, then the common factor is extracted.
- Examples of applying both methods to factor polynomials are provided and worked through step-by-step.
The document discusses using Taylor series to estimate the value of functions and introduces the Lagrange form of the remainder to determine the accuracy of estimates based on partial sums. It defines the Lagrange form of the remainder as the next term in the Taylor series with "a" replaced by a value "c" between the point "a" and the point "x" being estimated. The document also introduces the Remainder Estimation Theorem which uses the maximum value of the derivative in the interval to bound the size of the remainder term.
The document discusses using Taylor series to estimate the value of functions and introduces the Lagrange form of the remainder to determine the accuracy of estimates based on partial sums. It defines the Lagrange form of the remainder as the next term in the Taylor series with "a" replaced by a value "c" between the point "a" and the point "x" being estimated. The document also introduces the Remainder Estimation Theorem, which uses the maximum value of the derivative in the interval to bound the size of the remainder term.
This document provides questions about finding the equations of tangents and normals to various polynomial functions at given points. It contains 8 parts with multiple questions each about finding the equations of tangents to polynomial curves and normals to polynomial curves at specified points using differentiation.
The document discusses Fourier series and periodic functions. Some key points:
1. A periodic function f(x) satisfies f(x+T)=f(x) for some period T. The Fourier series of a periodic function f(x) represents f(x) as the sum of sines and cosines with frequencies that are integer multiples of 1/T.
2. The Fourier coefficients are calculated by taking integrals of f(x) multiplied by sine and cosine functions over one period.
3. The Fourier series converges to the average of the left and right limits of f(x) at points of discontinuity, and converges to f(x) itself at points of continuity.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and describes special types of discontinuities. The IVT states that if a function is continuous on a closed interval and takes on intermediate values, there exists a number in the interval where the function value is equal to the intermediate value. Examples are provided to illustrate the IVT, including proving the existence of the square root of two.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and establishes theorems about continuity. The IVT states that if a function is continuous on a closed interval and takes on intermediate values within its range, there exists at least one value in the domain where the function value is intermediate. An example proves the existence of the square root of two using the IVT and bisection method.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and establishes theorems about continuity. The IVT states that if a function is continuous on a closed interval and takes on intermediate values within its range, there exists at least one value in the domain where the function value is intermediate. An example proves the existence of the square root of two using the IVT and bisection method.
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.
Continuity is the property that the limit of a function near a point is the value of the function near that point. An important consequence of continuity is the intermediate value theorem, which tells us we once weighed as much as our height.
Similar to 12 x1 t04 07 approximations to roots (2013) (17)
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
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.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
2. Approximations To Roots
(1) Halving The Interval
y
y f x
x
If y = f(x) is a continuous function over the interval a x b , and
f(a) and f(b) are opposite in sign,
3. Approximations To Roots
(1) Halving The Interval
y
y f x
f a
a b x
f b
If y = f(x) is a continuous function over the interval a x b , and
f(a) and f(b) are opposite in sign,
4. Approximations To Roots
(1) Halving The Interval
y
y f x
f a
a b x
f b
If y = f(x) is a continuous function over the interval a x b , and
f(a) and f(b) are opposite in sign, then at least one root of the equation
f(x) = 0 lies in the interval a x b
5. e.g Find an approximation to two decimal places for a root of
x 4 2 x 19 0 in the interval 1 x 3
6. e.g Find an approximation to two decimal places for a root of
x 4 2 x 19 0 in the interval 1 x 3
f x x 4 2 x 19 f 1 14 2 19 f 3 34 23 19
16 0 68 0
7. e.g Find an approximation to two decimal places for a root of
x 4 2 x 19 0 in the interval 1 x 3
f x x 4 2 x 19 f 1 14 2 19 f 3 34 23 19
16 0 68 0
1 3
x1 f 2 2 4 22 19
2
1 0 1 2 3
2
8. e.g Find an approximation to two decimal places for a root of
x 4 2 x 19 0 in the interval 1 x 3
f x x 4 2 x 19 f 1 14 2 19 f 3 34 23 19
16 0 68 0
1 3
x1 f 2 2 4 22 19
2
1 0 1 2 3
2
solution lies in interval 1 x 2
9. e.g Find an approximation to two decimal places for a root of
x 4 2 x 19 0 in the interval 1 x 3
f x x 4 2 x 19 f 1 14 2 19 f 3 34 23 19
16 0 68 0
1 3
x1 f 2 2 4 22 19
2
1 0 1 2 3
2
solution lies in interval 1 x 2
1 2
x2 f 1.5 1.54 21.5 19
2
10.9 0 1 1.5 2
1 .5
10. e.g Find an approximation to two decimal places for a root of
x 4 2 x 19 0 in the interval 1 x 3
f x x 4 2 x 19 f 1 14 2 19 f 3 34 23 19
16 0 68 0
1 3
x1 f 2 2 4 22 19
2
1 0 1 2 3
2
solution lies in interval 1 x 2
1 2
x2 f 1.5 1.54 21.5 19
2
10.9 0 1 1.5 2
1 .5
solution lies in interval 1.5 x 2
21. 1.94 2
x6 f 1.97 1.97 4 21.97 19
2
0.001 0 1.94 1.97 2
1.97
solution lies in interval 1.94 x 1.97
1.94 1.97
x7
2 f 1.96 1.96 4 21.96 19
1.94 1.96 1.97
1.96
0.32 0
solution lies in interval 1.96 x 1.97
so is the solution closer to 1.96 or 1.97?
22. 1.94 2
x6 f 1.97 1.97 4 21.97 19
2
0.001 0 1.94 1.97 2
1.97
solution lies in interval 1.94 x 1.97
1.94 1.97
x7
2 f 1.96 1.96 4 21.96 19
1.94 1.96 1.97
1.96
0.32 0
solution lies in interval 1.96 x 1.97
so is the solution closer to 1.96 or 1.97?
1.96 1.965 1.97
23. 1.94 2
x6 f 1.97 1.97 4 21.97 19
2
0.001 0 1.94 1.97 2
1.97
solution lies in interval 1.94 x 1.97
1.94 1.97
x7
2 f 1.96 1.96 4 21.96 19
1.94 1.96 1.97
1.96
0.32 0
solution lies in interval 1.96 x 1.97
so is the solution closer to 1.96 or 1.97?
f 1.965 1.9654 2 1.965 19
0.16 0 1.96 1.965 1.97
24. 1.94 2
x6 f 1.97 1.97 4 21.97 19
2
0.001 0 1.94 1.97 2
1.97
solution lies in interval 1.94 x 1.97
1.94 1.97
x7
2 f 1.96 1.96 4 21.96 19
1.94 1.96 1.97
1.96
0.32 0
solution lies in interval 1.96 x 1.97
so is the solution closer to 1.96 or 1.97?
f 1.965 1.9654 2 1.965 19
0.16 0 1.96 1.965 1.97
25. 1.94 2
x6 f 1.97 1.97 4 21.97 19
2
0.001 0 1.94 1.97 2
1.97
solution lies in interval 1.94 x 1.97
1.94 1.97
x7
2 f 1.96 1.96 4 21.96 19
1.94 1.96 1.97
1.96
0.32 0
solution lies in interval 1.96 x 1.97
so is the solution closer to 1.96 or 1.97?
f 1.965 1.9654 2 1.965 19
0.16 0 1.96 1.965 1.97
an approximation for the root is x 1.97
26. NOTE:
(2) Newton’s Method of Approximation
y x0 must be a good first
approximation
y f x Newton’s method finds
where the tangent at x0
cuts the x axis
x
27. NOTE:
(2) Newton’s Method of Approximation
y x0 must be a good first
approximation
y f x Newton’s method finds
where the tangent at x0
cuts the x axis
x0 x
28. NOTE:
(2) Newton’s Method of Approximation
y x0 must be a good first
approximation
y f x Newton’s method finds
where the tangent at x0
cuts the x axis
x1 x0 x
29. NOTE:
(2) Newton’s Method of Approximation
y x0 must be a good first
approximation
y f x Newton’s method finds
where the tangent at x0
cuts the x axis
x1 x0 x If f x0 0
i.e. tangent || x axis
the method will fail
30. NOTE:
(2) Newton’s Method of Approximation
y x0 must be a good first
approximation
y f x Newton’s method finds
where the tangent at x0
cuts the x axis
x1 x0 x If f x0 0
i.e. tangent || x axis
the method will fail
Using the tangent at x0 to find x1
31. NOTE:
(2) Newton’s Method of Approximation
y x0 must be a good first
approximation
y f x Newton’s method finds
where the tangent at x0
cuts the x axis
x1 x0 x If f x0 0
i.e. tangent || x axis
the method will fail
Using the tangent at x0 to find x1
f x0 0
slope of tangent
x0 x1
32. NOTE:
(2) Newton’s Method of Approximation
y x0 must be a good first
approximation
y f x Newton’s method finds
where the tangent at x0
cuts the x axis
x1 x0 x If f x0 0
i.e. tangent || x axis
the method will fail
Using the tangent at x0 to find x1
f x0 0
slope of tangent
x0 x1
f x0 0
f x0
x0 x1
33. x0 x1 f x0 f x0
f x0
x0 x1
f x0
34. x0 x1 f x0 f x0
f x0
x0 x1
f x0
If x0 is a good first approximation to a root of the equation f(x) = 0,
then a closer approximation is given by;
f x0
x1 x0
f x0
35. x0 x1 f x0 f x0
f x0
x0 x1
f x0
If x0 is a good first approximation to a root of the equation f(x) = 0,
then a closer approximation is given by;
f x0
x1 x0
f x0
Successive approximations x2 , x3 , , xn , xn 1are given by;
f xn
xn 1 xn
f xn
36. e.g Find an approximation to two decimal places for a root of
x 4 2 x 19 0
37. e.g Find an approximation to two decimal places for a root of
x 4 2 x 19 0
f x x 4 2 x 19
f x 4 x 3 2
38. e.g Find an approximation to two decimal places for a root of
x 4 2 x 19 0
f x x 4 2 x 19
f x 4 x 3 2
x0 1.5 f 1.5 1.54 21.5 19 f 1.5 41.5 2
3
10.9375 15.5
39. e.g Find an approximation to two decimal places for a root of
x 4 2 x 19 0
f x x 4 2 x 19
f x 4 x 3 2
x0 1.5 f 1.5 1.54 21.5 19 f 1.5 41.5 2
3
10.9375 15.5
f x0
x1 x0
f x0
10.9375
1 .5
15.5
2.21
40. e.g Find an approximation to two decimal places for a root of
x 4 2 x 19 0
f x x 4 2 x 19
f x 4 x 3 2
x0 1.5 f 1.5 1.54 21.5 19 f 1.5 41.5 2
3
10.9375 15.5
f x0
x1 x0 f 2.21 2.214 22.21 19
f x0
9.2744
10.9375
1 .5 f 2.21 42.21 2
3
15.5
2.21 45.1754
46. 9.2744 f 2 2 4 22 19
x2 2.21
45.1754 1
2.00
f 2 42 2
3
35
x3 2
1 f 1.97 1.97 4 21.97 19
35 0.001
1.97
f 1.97 41.97 2
3
32.58
0.001
x4 1.97
32.58
1.97
x 1.97 is a better approximation for the root
47. (ii ) Use Newton's Method to obtain an approximation to 23
correct to two decimal places
48. (ii ) Use Newton's Method to obtain an approximation to 23
correct to two decimal places
f x x 2 23
49. (ii ) Use Newton's Method to obtain an approximation to 23
correct to two decimal places
f x x 2 23
f x 2x
50. (ii ) Use Newton's Method to obtain an approximation to 23
correct to two decimal places
xn12 23
f x x 23
2
xn xn1
2 xn1
f x 2x
xn12 23
2 xn1
51. (ii ) Use Newton's Method to obtain an approximation to 23
correct to two decimal places
xn12 23
f x x 23
2
xn xn1
2 xn1
f x 2x
xn12 23
2 xn1
x0 5
52. (ii ) Use Newton's Method to obtain an approximation to 23
correct to two decimal places
xn12 23
f x x 23
2
xn xn1
2 xn1
f x 2x
xn12 23
2 xn1
x0 5
52 23
x1
2 5
x1 4.8
53. (ii ) Use Newton's Method to obtain an approximation to 23
correct to two decimal places
xn12 23
f x x 23
2
xn xn1
2 xn1
f x 2x
xn12 23
2 xn1
x0 5
52 23 4.82 23
x1 x2
2 5 2 4.8
x1 4.8 x2 4.795833333
x2 4.80 (to 2 dp)
54. (ii ) Use Newton's Method to obtain an approximation to 23
correct to two decimal places
xn12 23
f x x 23
2
xn xn1
2 xn1
f x 2x
xn12 23
2 xn1
x0 5
52 23 4.82 23
x1 x2
2 5 2 4.8
x1 4.8 x2 4.795833333
x2 4.80 (to 2 dp)
23 4.80 (to 2 dp)
58. Other Possible Problems with Newton’s Method
y Approximations oscillate
want to find this root
x
59. Other Possible Problems with Newton’s Method
y Approximations oscillate
want to find this root
x1 x2 x
60. Other Possible Problems with Newton’s Method
y Approximations oscillate
want to find this root
x1 x2 x
61. Other Possible Problems with Newton’s Method
y Approximations oscillate
want to find this root
x1 x2 x
wrong side of stationary point
converges to wrong root
62. Other Possible Problems with Newton’s Method
y Approximations oscillate
want to find this root
x1 x2 x
wrong side of stationary point
y converges to wrong root
x
63. Other Possible Problems with Newton’s Method
y Approximations oscillate
want to find this root
x1 x2 x
wrong side of stationary point
y converges to wrong root
x
want to find this root
64. Other Possible Problems with Newton’s Method
y Approximations oscillate
want to find this root
x1 x2 x
wrong side of stationary point
y converges to wrong root
x1 x2 x
want to find this root
65. Other Possible Problems with Newton’s Method
y Approximations oscillate
want to find this root
x1 x2 x
wrong side of stationary point
y converges to wrong root
x1 x2 x
want to find this root
66. Other Possible Problems with Newton’s Method
y Approximations oscillate
want to find this root
Exercise 6E; 1, 3ac,
6adf, 8a, 10, 12
x1 x2 x
wrong side of stationary point
y converges to wrong root
x1 x2 x
want to find this root