This document provides information about inverse functions including:
- The inverse of a function is formed by reversing the coordinates of each ordered pair. A function has an inverse only if it is one-to-one.
- The domain of the inverse function is the range of the original function, and the range of the inverse is the domain of the original.
- To find the inverse of a one-to-one function, replace f(x) with y, interchange x and y, then solve for y in terms of x and replace y with f^-1(x).
- Several examples are provided to demonstrate finding the inverse of different one-to-one functions by following the given steps.
The document introduces differentiation and the concept of the derivative. It discusses how the derivative can be used to find the rate of change of a function and the slope of its tangent line. The main rules covered are:
1) If f(x) = x^n, then the derivative is f'(x) = nx^(n-1).
2) Examples are provided of finding the derivative of functions like f(x) = 6x^3, which is f'(x) = 18x^2.
3) The derivative can be used to find the slope of a tangent line at specific points, like finding the derivative of f(x) = (x + 5)^2 at x
1. The document discusses the concept of the derivative and differentiation using the first principle. It explains how to calculate the slope of a tangent line to a curve at a point using limits, which gives the derivative of the function at that point.
2. Rules for differentiating common functions like polynomials, exponentials, and logarithms are covered. Higher-order derivatives and applications of derivatives to business and economics are also mentioned.
3. The goals of the class are to explain the concept of the derivative, differentiate functions using the first principle (limits), and understand various differentiation rules.
This document discusses how to solve quadratic equations by graphing. It explains that a quadratic equation is of the form y = ax^2 + bx + c, with a being the quadratic term, b the linear term, and c the constant. The number of real solutions is at most two: one solution, two solutions, or no solutions. Solutions are found by setting the equation equal to 0 and finding the x-intercepts of the graph. The graph of a quadratic is a parabola with the roots as the x-intercepts and the vertex as the maximum or minimum point. Quadratic equations can be graphed by making a table of x and y values to plot the parabola.
- The document discusses quadratic functions and their graphs. It explains that the graph of a quadratic function is a parabola, which is a U-shaped curve.
- It describes how to write quadratic functions in standard form and use that form to sketch the graph and find features like the vertex and axis of symmetry.
- Examples are provided to demonstrate how to graph quadratic functions in standard form and how to find the minimum or maximum value of a quadratic function by setting its derivative equal to zero.
This document provides information about inverse functions including:
- The inverse of a function is formed by reversing the coordinates of each ordered pair. A function has an inverse only if it is one-to-one.
- The domain of the inverse function is the range of the original function, and the range of the inverse is the domain of the original.
- To find the inverse of a one-to-one function, replace f(x) with y, interchange x and y, then solve for y in terms of x and replace y with f^-1(x).
- Several examples are provided to demonstrate finding the inverse of different one-to-one functions by following the given steps.
The document introduces differentiation and the concept of the derivative. It discusses how the derivative can be used to find the rate of change of a function and the slope of its tangent line. The main rules covered are:
1) If f(x) = x^n, then the derivative is f'(x) = nx^(n-1).
2) Examples are provided of finding the derivative of functions like f(x) = 6x^3, which is f'(x) = 18x^2.
3) The derivative can be used to find the slope of a tangent line at specific points, like finding the derivative of f(x) = (x + 5)^2 at x
1. The document discusses the concept of the derivative and differentiation using the first principle. It explains how to calculate the slope of a tangent line to a curve at a point using limits, which gives the derivative of the function at that point.
2. Rules for differentiating common functions like polynomials, exponentials, and logarithms are covered. Higher-order derivatives and applications of derivatives to business and economics are also mentioned.
3. The goals of the class are to explain the concept of the derivative, differentiate functions using the first principle (limits), and understand various differentiation rules.
This document discusses how to solve quadratic equations by graphing. It explains that a quadratic equation is of the form y = ax^2 + bx + c, with a being the quadratic term, b the linear term, and c the constant. The number of real solutions is at most two: one solution, two solutions, or no solutions. Solutions are found by setting the equation equal to 0 and finding the x-intercepts of the graph. The graph of a quadratic is a parabola with the roots as the x-intercepts and the vertex as the maximum or minimum point. Quadratic equations can be graphed by making a table of x and y values to plot the parabola.
- The document discusses quadratic functions and their graphs. It explains that the graph of a quadratic function is a parabola, which is a U-shaped curve.
- It describes how to write quadratic functions in standard form and use that form to sketch the graph and find features like the vertex and axis of symmetry.
- Examples are provided to demonstrate how to graph quadratic functions in standard form and how to find the minimum or maximum value of a quadratic function by setting its derivative equal to zero.
1) The document provides an overview of continuity, including defining continuity as a function having a limit equal to its value at a point.
2) It discusses several theorems related to continuity, such as the sum of continuous functions being continuous and various trigonometric, exponential, and logarithmic functions being continuous on their domains.
3) The document also covers inverse trigonometric functions and their domains of continuity.
Increasing and decreasing functions ap calc sec 3.3Ron Eick
The document discusses increasing and decreasing functions and the first derivative test. It defines that a function is increasing if the derivative is positive, decreasing if the derivative is negative, and constant if the derivative is zero. It provides examples of finding the intervals where a function is increasing or decreasing by identifying critical numbers and testing points in each interval. The document also summarizes the first derivative test, stating that a critical point is an extremum if the derivative changes sign there, and whether it is a maximum or minimum depends on if the derivative changes from negative to positive or positive to negative.
The document discusses function transformations including shifts, reflections, and stretches/compressions. It defines these transformations and provides examples of how they affect the graph of a function. Specifically, it explains that a shift moves a graph up/down or left/right along an axis, a reflection flips the graph across an axis, and a stretch or compression changes the scale of the graph along an axis. Examples are given of reflecting across the x-axis or y-axis and horizontally or vertically stretching/compressing a function. In the end, students are asked to write equations for specific transformations of a quadratic function.
Lesson 7-8: Derivatives and Rates of Change, The Derivative as a functionMatthew Leingang
The derivative is one of the fundamental quantities in calculus, partly because it is ubiquitous in nature. We give examples of it coming about, a few calculations, and ways information about the function an imply information about the derivative
The Rational Root Theorem provides a method to find all possible rational roots of a polynomial with integer coefficients. It states that every rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. Examples are provided to demonstrate finding all possible rational roots using this theorem and then checking them to determine the actual rational roots. Once a rational root is found, synthetic division can be used to find the depressed polynomial which can then be fully factored to obtain all factors of the original polynomial.
The document provides an overview of calculus and analytical geometry presented by Group D. It discusses various topics related to integration including the history and development of integration, the definition and purpose of integration, common integration rules and formulas like integration by parts and integration by partial fractions, and real-life applications of integration in fields like engineering, medicine, and physics. Examples are provided to demonstrate how to use integration techniques like integration by parts and integration by partial fractions to evaluate definite integrals.
The document discusses differentiation and tangents. It explains that differentiation finds the gradient of a curve at a point and is needed because curves have changing gradients. It provides examples of differentiating simple functions like y=3x^5. Tangents are lines that touch a curve at a single point, with the same gradient as the curve at that point. To find the equation of a tangent, take the derivative and plug in the x-value to get the slope, then use the point to find the y-intercept. Normals are perpendicular to tangents, with a gradient equal to the negative reciprocal of the tangent's gradient.
The document discusses function transformations including translations, reflections, dilations, and compressions. It defines these transformations and provides examples of how they affect the graph of a function. Translations slide the graph left or right without changing its shape or orientation. Reflections create a mirror image of the graph across an axis, flipping it. Compressions squeeze the graph towards or away from an axis. Dilations stretch or shrink the graph away from an axis. The document explains how to interpret various function notation and applies the transformations to example graphs.
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
The document discusses the factor theorem and how to determine if a polynomial is a factor of another polynomial. It provides examples of using the factor theorem to show that (x + 1) is a factor of 2x^3 + 5x^2 - 3 and that (x - 2) is a factor of x^4 + x^3 - x^2 - x - 18. It also gives an example of finding a polynomial function given its zeros as -2, 1, -1. The document provides exercises for using the factor theorem to determine unknown values in polynomials.
Lesson 10 derivative of exponential functionsRnold Wilson
1. The document discusses differentiating exponential functions by applying properties of exponents and logarithms. It provides formulas for differentiating exponentials and natural logarithms.
2. Examples are given of differentiating various exponential functions using the formulas and properties provided. Logarithmic differentiation is also described as a method to differentiate complicated algebraic functions.
3. Steps in applying logarithmic differentiation are outlined, including taking the logarithm of both sides and applying logarithm properties before differentiating.
This document introduces function notation and how to evaluate functions. It explains that f(x) represents the output of the function f for a given input x. Examples show different functions defined using this notation, such as f(x) = 2x + 1. The document also demonstrates how to evaluate functions for given x-values, find x-values for which a function is a certain amount, and graph functions using this notation. It compares translating and shifting graphs of functions.
1) A composite function is formed by combining two functions, where one function is substituted into the other.
2) Notations like fg(x) indicate that function g(x) is substituted into function f(x).
3) Composite functions are non-commutative, meaning the order of the functions matters - fg(x) may not equal gf(x).
This document provides information about derivatives and their applications:
1. It defines the derivative as the limit of the difference quotient, and explains how to calculate derivatives using first principles. It also covers rules for finding derivatives of sums, products, quotients, exponentials, and logarithmic functions.
2. Higher order derivatives are introduced, with examples of how to take second and third derivatives.
3. Applications of derivatives like finding velocity and acceleration from a position-time function are demonstrated. Maximum/minimum values and how to find local and absolute extrema are also discussed with an example.
The document is a lesson on implicit differentiation and related concepts:
1) Implicit differentiation allows one to take the derivative of an implicitly defined relation between x and y, even if y is not explicitly defined as a function of x.
2) Examples are provided to demonstrate implicit differentiation, such as finding the slope of a tangent line to a curve.
3) The van der Waals equation is introduced to describe non-ideal gas properties, and implicit differentiation is used to find the isothermal compressibility of a van der Waals gas.
The document provides an introduction to the precise definition of a limit in calculus. It begins with a heuristic definition of a limit using an error-tolerance game between two players. It then presents Cauchy's precise definition, where the limit is defined using epsilon-delta relationships such that for any epsilon tolerance around the proposed limit L, there exists a corresponding delta tolerance around the point a such that the function values are within epsilon of L when the input values are within delta of a. Examples are provided to illustrate the definition. Pathologies where limits may not exist are also discussed.
The document discusses coordinate geometry and the Cartesian plane. It defines the key terms like the x-axis, y-axis, and origin (0,0). Any point in the plane can be located using its x and y coordinates. The gradient or slope of a line is defined as the vertical distance over the horizontal distance between two points on the line. Examples are given to demonstrate how to calculate the gradient using the gradient formula and by finding the ratio of the vertical to horizontal distances.
Infinity is a dangerous place where the rules of arithmetic break down. But it is a useful concept and study both infinite limits and limits at infinity.
This document discusses stationary points (SPs) and how to identify and classify them. It explains that SPs occur when the derivative of a function is equal to 0. It provides examples of finding SPs by taking the derivative, setting it equal to 0, and solving for x. It also introduces the concept of using a nature table to determine whether a SP is a maximum or minimum turning point by examining the sign of the derivative just before and after the SP. The document demonstrates this process on several examples, finding the SPs and using the nature table to classify them.
IB Maths. Turning points. First derivative testestelav
By the end of the lesson, students will be able to use derivatives to find maximum and minimum points of a function, and use second derivatives to determine the nature of stationary points and points of inflection. Specifically, they will learn that: (1) if the first derivative is zero at a point, it is a stationary point; (2) the second derivative test can determine if it is a local max, min or point of inflection; and (3) points of inflection occur when the curve changes concavity. Students will apply these concepts to find the stationary points of sample functions and classify their nature.
1) The document provides an overview of continuity, including defining continuity as a function having a limit equal to its value at a point.
2) It discusses several theorems related to continuity, such as the sum of continuous functions being continuous and various trigonometric, exponential, and logarithmic functions being continuous on their domains.
3) The document also covers inverse trigonometric functions and their domains of continuity.
Increasing and decreasing functions ap calc sec 3.3Ron Eick
The document discusses increasing and decreasing functions and the first derivative test. It defines that a function is increasing if the derivative is positive, decreasing if the derivative is negative, and constant if the derivative is zero. It provides examples of finding the intervals where a function is increasing or decreasing by identifying critical numbers and testing points in each interval. The document also summarizes the first derivative test, stating that a critical point is an extremum if the derivative changes sign there, and whether it is a maximum or minimum depends on if the derivative changes from negative to positive or positive to negative.
The document discusses function transformations including shifts, reflections, and stretches/compressions. It defines these transformations and provides examples of how they affect the graph of a function. Specifically, it explains that a shift moves a graph up/down or left/right along an axis, a reflection flips the graph across an axis, and a stretch or compression changes the scale of the graph along an axis. Examples are given of reflecting across the x-axis or y-axis and horizontally or vertically stretching/compressing a function. In the end, students are asked to write equations for specific transformations of a quadratic function.
Lesson 7-8: Derivatives and Rates of Change, The Derivative as a functionMatthew Leingang
The derivative is one of the fundamental quantities in calculus, partly because it is ubiquitous in nature. We give examples of it coming about, a few calculations, and ways information about the function an imply information about the derivative
The Rational Root Theorem provides a method to find all possible rational roots of a polynomial with integer coefficients. It states that every rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. Examples are provided to demonstrate finding all possible rational roots using this theorem and then checking them to determine the actual rational roots. Once a rational root is found, synthetic division can be used to find the depressed polynomial which can then be fully factored to obtain all factors of the original polynomial.
The document provides an overview of calculus and analytical geometry presented by Group D. It discusses various topics related to integration including the history and development of integration, the definition and purpose of integration, common integration rules and formulas like integration by parts and integration by partial fractions, and real-life applications of integration in fields like engineering, medicine, and physics. Examples are provided to demonstrate how to use integration techniques like integration by parts and integration by partial fractions to evaluate definite integrals.
The document discusses differentiation and tangents. It explains that differentiation finds the gradient of a curve at a point and is needed because curves have changing gradients. It provides examples of differentiating simple functions like y=3x^5. Tangents are lines that touch a curve at a single point, with the same gradient as the curve at that point. To find the equation of a tangent, take the derivative and plug in the x-value to get the slope, then use the point to find the y-intercept. Normals are perpendicular to tangents, with a gradient equal to the negative reciprocal of the tangent's gradient.
The document discusses function transformations including translations, reflections, dilations, and compressions. It defines these transformations and provides examples of how they affect the graph of a function. Translations slide the graph left or right without changing its shape or orientation. Reflections create a mirror image of the graph across an axis, flipping it. Compressions squeeze the graph towards or away from an axis. Dilations stretch or shrink the graph away from an axis. The document explains how to interpret various function notation and applies the transformations to example graphs.
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
The document discusses the factor theorem and how to determine if a polynomial is a factor of another polynomial. It provides examples of using the factor theorem to show that (x + 1) is a factor of 2x^3 + 5x^2 - 3 and that (x - 2) is a factor of x^4 + x^3 - x^2 - x - 18. It also gives an example of finding a polynomial function given its zeros as -2, 1, -1. The document provides exercises for using the factor theorem to determine unknown values in polynomials.
Lesson 10 derivative of exponential functionsRnold Wilson
1. The document discusses differentiating exponential functions by applying properties of exponents and logarithms. It provides formulas for differentiating exponentials and natural logarithms.
2. Examples are given of differentiating various exponential functions using the formulas and properties provided. Logarithmic differentiation is also described as a method to differentiate complicated algebraic functions.
3. Steps in applying logarithmic differentiation are outlined, including taking the logarithm of both sides and applying logarithm properties before differentiating.
This document introduces function notation and how to evaluate functions. It explains that f(x) represents the output of the function f for a given input x. Examples show different functions defined using this notation, such as f(x) = 2x + 1. The document also demonstrates how to evaluate functions for given x-values, find x-values for which a function is a certain amount, and graph functions using this notation. It compares translating and shifting graphs of functions.
1) A composite function is formed by combining two functions, where one function is substituted into the other.
2) Notations like fg(x) indicate that function g(x) is substituted into function f(x).
3) Composite functions are non-commutative, meaning the order of the functions matters - fg(x) may not equal gf(x).
This document provides information about derivatives and their applications:
1. It defines the derivative as the limit of the difference quotient, and explains how to calculate derivatives using first principles. It also covers rules for finding derivatives of sums, products, quotients, exponentials, and logarithmic functions.
2. Higher order derivatives are introduced, with examples of how to take second and third derivatives.
3. Applications of derivatives like finding velocity and acceleration from a position-time function are demonstrated. Maximum/minimum values and how to find local and absolute extrema are also discussed with an example.
The document is a lesson on implicit differentiation and related concepts:
1) Implicit differentiation allows one to take the derivative of an implicitly defined relation between x and y, even if y is not explicitly defined as a function of x.
2) Examples are provided to demonstrate implicit differentiation, such as finding the slope of a tangent line to a curve.
3) The van der Waals equation is introduced to describe non-ideal gas properties, and implicit differentiation is used to find the isothermal compressibility of a van der Waals gas.
The document provides an introduction to the precise definition of a limit in calculus. It begins with a heuristic definition of a limit using an error-tolerance game between two players. It then presents Cauchy's precise definition, where the limit is defined using epsilon-delta relationships such that for any epsilon tolerance around the proposed limit L, there exists a corresponding delta tolerance around the point a such that the function values are within epsilon of L when the input values are within delta of a. Examples are provided to illustrate the definition. Pathologies where limits may not exist are also discussed.
The document discusses coordinate geometry and the Cartesian plane. It defines the key terms like the x-axis, y-axis, and origin (0,0). Any point in the plane can be located using its x and y coordinates. The gradient or slope of a line is defined as the vertical distance over the horizontal distance between two points on the line. Examples are given to demonstrate how to calculate the gradient using the gradient formula and by finding the ratio of the vertical to horizontal distances.
Infinity is a dangerous place where the rules of arithmetic break down. But it is a useful concept and study both infinite limits and limits at infinity.
This document discusses stationary points (SPs) and how to identify and classify them. It explains that SPs occur when the derivative of a function is equal to 0. It provides examples of finding SPs by taking the derivative, setting it equal to 0, and solving for x. It also introduces the concept of using a nature table to determine whether a SP is a maximum or minimum turning point by examining the sign of the derivative just before and after the SP. The document demonstrates this process on several examples, finding the SPs and using the nature table to classify them.
IB Maths. Turning points. First derivative testestelav
By the end of the lesson, students will be able to use derivatives to find maximum and minimum points of a function, and use second derivatives to determine the nature of stationary points and points of inflection. Specifically, they will learn that: (1) if the first derivative is zero at a point, it is a stationary point; (2) the second derivative test can determine if it is a local max, min or point of inflection; and (3) points of inflection occur when the curve changes concavity. Students will apply these concepts to find the stationary points of sample functions and classify their nature.
There are three types of stationary points: maximum points, minimum points, and points of inflection. A maximum point occurs when the gradient is positive on one side and negative on the other. A minimum point occurs when the gradient is negative on one side and positive on the other. A point of inflection occurs when the gradient is zero but the second derivative is also zero. To determine the type of stationary point, the gradient on each side is considered and the second derivative at that point is examined. Sketching the curve involves finding any stationary points, where the curve meets the axes, and using the information to plot the general shape.
IB Maths.Turning points. Second derivative testestelav
The document discusses methods for finding maximum, minimum, and points of inflection on a curve:
1. Use the first derivative test to find stationary points where f'(a) = 0, then examine the sign of f' left and right of a to determine if it is a maximum or minimum.
2. Use the second derivative test where f'(a) = 0, and if f''(a) < 0 it is a maximum, f''(a) > 0 it is a minimum, and if f'' changes sign at a it is a point of inflection.
Several examples are provided to demonstrate finding stationary points and determining their nature using these two methods, as well as sketching
This document contains an exam paper for Core Mathematics C4. It includes 5 questions testing calculus skills. The paper provides instructions for candidates, advising them to show working, write answers in the spaces provided, and use an appropriate degree of accuracy when using a calculator. It also lists the materials candidates may use and information about the duration, marks and structure of the exam.
This document provides instructions and information for a mathematics exam. It includes:
1) Details about the exam such as the date, time allotted, and materials allowed.
2) Instructions for candidates on how to identify their work and provide their information.
3) Information for candidates about the structure of the exam including the total number and types of questions, and the total marks available.
4) Advice to candidates about showing their working and obtaining full credit.
Este documento contiene 10 problemas de sistemas de ecuaciones que deben resolverse mediante sustitución. Se proporcionan las ecuaciones para cada sistema, así como las respuestas una vez resueltos.
The document provides information about Arnold Schoenberg's piece "Peripetie" including definitions of musical terms used in the piece such as hexachord, diminution, and glissando. It gives one example of Schoenberg's use of canon and explains that atonal music has no set key or tonality. The document also identifies that symbols "H" and "N" represent the most and second most important melodies. It cites five features showing the piece was composed in the 20th century including its atonal angular style, use of extended techniques, focus on timbre, and lack of clear form.
This document provides information about an exam for the Edexcel GCE Core Mathematics C3 Bronze Level B1 qualification. It lists the paper reference, time allowed, materials required and permitted calculators. It provides instructions for candidates on writing details on the front page and information about the structure of the paper. It also lists the 9 questions that make up the exam, covering topics like functions, graphs, derivatives, iterations and logarithms. The final section suggests grade boundaries for the exam.
The document discusses the benefits of meditation for reducing stress and anxiety. Regular meditation practice can help calm the mind and body by lowering heart rate and blood pressure. Studies have shown that meditating for just 10-20 minutes per day can have significant positive impacts on both mental and physical health.
This document appears to be an exam paper for a mechanics course. It contains 6 multiple part questions testing concepts in mechanics such as forces, kinematics, and dynamics. The questions provide contextual word problems and diagrams requiring students to set up and solve equations to find requested values. The exam paper provides space for work and answers and includes instructions for candidates on providing responses. It is signed and includes information for examiners.
This document provides information about a Core Mathematics C3 exam taken by Edexcel students. It includes instructions for students taking the exam, information about materials allowed and provided, and 8 questions testing various calculus, geometry, and trigonometry concepts. The exam is 1 hour and 30 minutes long and contains a total of 75 marks across the 8 questions. Students are advised to show their working clearly and label answers to parts of questions.
This document contains an exam paper for a Core Mathematics C3 Advanced exam. It provides instructions for candidates on how to fill out their details, contains 9 questions to answer, and specifies the time allotted and materials allowed. Candidates are to show their working and answers must be written in the spaces provided after each question.
This document provides an introduction to dynamics and forces acting on particles moving in a straight line. It introduces Newton's second law of motion, which states that force is equal to mass times acceleration (F=ma). It defines key concepts like weight, normal reaction force, friction, tension, and thrust. Examples are provided on using F=ma to calculate acceleration given force and mass, or force given mass and acceleration. The document also discusses resolving forces into perpendicular components and using this to solve problems involving multiple forces acting on an object.
The document provides information on kinematics equations for particles moving with constant acceleration in a straight line (SUVAT equations). It introduces the variables used in the equations (s, u, v, a, t) and provides examples of using the equations to solve kinematics problems involving displacement, velocity, acceleration, and time. It also presents three additional SUVAT equations and works through examples of solving problems using these equations.
This document discusses kinematics of a particle moving in a straight line. It explains that motion can be represented using speed-time graphs, distance-time graphs, or acceleration-time graphs. The gradient of a speed-time graph represents acceleration, while the area under the graph represents distance traveled. Several examples are provided of constructing and interpreting these graphs to analyze different scenarios of linear motion.
The document provides revision notes on various mathematics topics including:
1) Binomial expansions, partial fractions, trigonometry formulas, and techniques for integration like volumes of revolution.
2) Parametric equations, vectors, vector equations, planes, and differential equations.
3) Details are given for solving problems involving these topics, such as using compound angle formulas, rewriting algebraic fractions as partial fractions, and separating variables to solve first-order differential equations.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help boost feelings of calmness, happiness and focus.
The document discusses increasing and decreasing functions. It defines an increasing function as one where the gradient (dy/dx) is always positive, and a decreasing function as one where the gradient is always negative. To determine if a function is increasing or decreasing, you find where the derivative (dy/dx) is equal to 0 to find critical points, and use a nature table to analyze the sign of the derivative on each interval. The derivative also needs to be analyzed to determine if a function is always increasing or decreasing.
Numerical analysis stationary variablesSHAMJITH KM
This document discusses optimization methods for functions of single and two variables. It defines stationary points as points where the derivative of a function is equal to zero. For a function of a single variable, a necessary condition for a relative maximum is that the derivative is equal to zero at that point. The sufficient condition depends on higher order derivatives. For functions of two variables, the gradient vector must be equal to zero at stationary points. The Hessian matrix formed from second order derivatives is used to classify stationary points as maxima, minima or neither. Several examples are provided to demonstrate finding stationary points and classifying them.
This document provides an overview of the key topics covered in Lecture 4, including:
1. How to sketch quadratic and cubic curves by finding intercepts and stationary points.
2. How to use the second derivative to determine if a stationary point is a maximum, minimum, or point of inflection.
3. Rules for simplifying expressions using indices and how to convert numbers to and from standard form.
4.2 derivatives of logarithmic functionsdicosmo178
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).
This document provides an overview of topics covered in intermediate algebra revision including: collecting like terms, multiplying terms, indices, expanding single and double brackets, substitution, solving equations, finding nth terms of sequences, simultaneous equations, inequalities, factorizing common factors and quadratics, solving quadratic equations, rearranging formulas, and graphing curves and lines. The document contains examples and practice problems for each topic.
This document contains solutions to three equations involving fractions equal to values.
The first equation, x^2 + x - 2 = 0, is solved to find the values of x that satisfy the equation, which are x = 1 and x = -2.
The second equation, (x-1)/(x+1) = 3, is solved to find the value of x that makes the fraction equal to 3, which is x = -2.
The third equation, x/(x-1) - 1/(x+2) - 3/(x-1)(x+2) = 0, is solved but does not have any real number solutions for x.
Quadratic equations can be solved in several ways:
1) Factorizing, by finding two numbers whose product is the constant term and sum is the coefficient of the x term.
2) Using the quadratic formula.
3) Substitution, by letting an expression like x^2 + 2x equal a variable k, and solving the simplified equation for k and back substituting.
4) Squaring both sides, but this can introduce extraneous solutions so one must check solutions.
This document contains the solutions to 5 questions related to calculus concepts like integration, derivatives, series approximation, and geometry of curves and surfaces. Some of the key steps include:
- Using integration to find volumes, masses, and centroids
- Finding critical points and classifying extrema
- Approximating a series to evaluate an integral
- Solving a geometric series problem to find an initial height
- Analyzing motion problems using kinematic equations
- Finding equations of planes and tangent lines to surfaces
A quadratic equation is a second-order polynomial with terms up to x^2. It has two roots, or solutions, which may be real or complex. The quadratic formula can be used to find the exact numeric values of the roots. A root is a value that makes the quadratic equation equal to zero. A double root occurs when the two roots are equal, making the quadratic a perfect square trinomial.
1) The document explains various methods for dividing and factoring polynomials, including: dividing polynomials using long division; using Ruffini's rule to divide polynomials; applying the remainder theorem and factor theorem; and factoring polynomials through finding common factors, using identities, solving quadratic equations, and finding polynomial roots.
2) Specific factorization methods covered are removing common factors, using identities like a^2 - b^2, factoring quadratic trinomials, using the remainder theorem and Ruffini's rule to find factors for polynomials of degree greater than two, and identifying irreducible polynomials.
3) Additional algebraic identities explained are for cubing binomials like (a ± b)^3 and taking the square of trinomial
This document contains 5 math problems involving factorizing expressions, solving equations, evaluating expressions for given values, expanding expressions, and finding the highest common factor. It also provides context on working with straight line graphs, including finding the gradient and y-intercept of a line from its equation, finding the gradient between two points, finding the midpoint and a point that divides a line segment in a given ratio, and finding the x- and y-intercepts of a line.
This document provides a summary of core mathematics concepts including:
1) Linear graphs and equations such as y=mx+c and finding the equation of a line.
2) Quadratic equations and graphs including using the quadratic formula, completing the square, and finding the vertex and axis of symmetry.
3) Simultaneous equations and interpreting their solutions geometrically as the intersection of graphs.
4) Other topics covered include surds, polynomials, differentiation, integration, and areas under graphs.
(1) The student solved several integrals and derivatives.
(2) They sketched regions bounded by curves and found the areas.
(3) Properties of functions like extremes and concavity were examined.
This document provides information about integration in higher mathematics. It begins with an overview of integration as the opposite of differentiation. It then discusses using antidifferentiation to find integrals by reversing the power rule for differentiation. Several examples are provided to illustrate integrating polynomials. The document also discusses using integrals to find the area under a curve or between two curves. It provides examples of calculating areas bounded by graphs and the x-axis. Finally, it presents some exam-style integration questions for practice.
1) The document contains an exam with 4 problems related to ordinary differential equations.
2) The first problem involves solving a homogeneous Cauchy-Euler differential equation.
3) The second problem involves using an initial value problem to find the general solution of a non-homogeneous differential equation, then applying initial conditions to determine constants.
4) The third problem uses the method of variation of parameters to solve a non-homogeneous differential equation.
5) The fourth problem applies an equation of motion to a physical system of a mass on a spring to determine the instantaneous velocity when the mass passes through equilibrium.
1) An antiderivative of a function f(x) is any function F(x) whose derivative is equal to f(x).
2) The general antiderivative of a function f(x) is written as F(x) + C, where F(x) is a particular antiderivative and C is an arbitrary constant.
3) The indefinite integral notation ∫f(x)dx represents the entire family of antiderivatives for a function f(x), since each value of C defines a different antiderivative.
1) The student solved several integral evaluation problems and derivative problems.
2) They sketched the region bounded by two curves and found its area.
3) Several functions were analyzed, including finding their derivatives, extrema, concavity, asymptotes and sketching their graphs.
4) Some proofs and word problems involving applications of calculus like radioactive decay were also addressed.
maths Individual assignment on differentiationtenwoalex
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.
This document contains the work of a student on a calculus test. It includes:
1) Solving limits, finding derivatives, and applying L'Hopital's rule.
2) Using induction to prove an identity.
3) Providing epsilon-delta proofs of limits.
4) Finding where a tangent line is parallel to a secant line.
5) Proving statements about limits of functions.
The student provides detailed solutions showing their work for each problem on the test.
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P.D. Wimalasiri is a lecturer in computer science at Rajarata University of Sri Lanka who has a B.Sc. in Maths and M.Sc. in Computer Science. The document defines differentiation and derivatives of simple functions such as x^2 and x^3. It also covers the product rule, quotient rule, and derivatives of exponential functions.
Factoring polynomials involves finding common factors that can be divided out of terms, similar to factoring numbers but with variables; this is done by looking for a single variable or number that is a common factor of all terms that can be pulled out in front of parentheses. The document provides examples of different types of factoring polynomials including using the greatest common factor, difference of squares, grouping, and perfect squares and cubes.
This document introduces vectors and how they can be used to describe displacements and solve problems involving displacement. It discusses that vectors have both direction and magnitude, and provides examples. Vectors can be added and represented using line segments. The triangle law of addition allows vectors to be added using a triangle. Vectors can also be described using i and j notation, where i and j are unit vectors along the x and y axes, and any two-dimensional vector can be written as ai + bj. Problems can then be solved by adding or subtracting the i and j terms. The magnitude of a vector can be found using Pythagoras' theorem, and the angle between a vector and an axis can be found using trig
- A particle starts from the point with position vector (3i + 7j) m and then moves with constant velocity (2i – j) ms-1. The question asks to find the position vector of the particle 4 seconds later.
- Substituting the values into the displacement equation gives the final position vector as (12i + 3j) m.
- A second particle is given a position vector of (2i + 4j) m at time t = 0 and a position vector of (12i + 16j) m five seconds later. Using the displacement equation gives the velocity of the particle as (2i + 4j) ms-1.
- For a third particle
(1) The document discusses moments, which are turning forces that cause an object to rotate rather than push it linearly. Moments depend on the magnitude of the applied force and its distance from the pivot point.
(2) To calculate the moment of a single force, the formula is Moment = Force x Perpendicular Distance from the pivot. When multiple forces are present, their individual moments are summed.
(3) Systems in equilibrium have their clockwise and counter-clockwise moments equal, allowing one to solve for unknown values like reactions at supports. Diagrams are drawn and moments taken to set up and solve equations of equilibrium.
This chapter focuses on objects in static equilibrium, where the net force and net torque on the object are both zero. Solving static equilibrium problems involves drawing free body diagrams showing all external forces acting on the object, then resolving forces into components and setting the sums of forces in each direction equal to zero. Three examples are given of solving static equilibrium problems involving particles under the influence of multiple forces. The problems are solved by resolving forces into horizontal and vertical or parallel and perpendicular components, setting the component force equations equal to zero, and solving the equations to determine the magnitudes of unknown forces. Key steps include drawing diagrams, resolving forces, setting force sums to zero, and solving the resulting equations.
This document provides an introduction to dynamics and forces acting on particles moving in a straight line. It introduces Newton's second law of motion, F=ma, and defines key concepts like weight, tension, thrust, friction, and normal reaction. It explains how to resolve forces into horizontal and vertical components when multiple forces are acting. Examples show how to set up force diagrams and use Newton's second law to solve for acceleration, distance, and missing forces. Trigonometry is used to resolve forces acting at angles into their x- and y-direction components.
The document discusses increasing and decreasing functions. An increasing function has a positive gradient, while a decreasing function has a negative gradient. Some functions can be increasing over one interval and decreasing over another. You need to be able to determine the intervals where a function is increasing or decreasing by examining its gradient. The document provides examples of finding where a function is decreasing by taking the derivative, setting it equal to 0, and solving for the range of x-values that make the gradient negative. It also discusses using derivatives to find the coordinates of stationary points like maxima and minima, and using the second derivative to determine if a stationary point is a maximum or minimum.
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it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
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আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
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A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
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Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
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unwillingness to rectify this violation through action requires accountability.
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The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
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2. The stationary points of a curve are the points where
the gradient is zero
e.g.
y = x3 − 3x2 − 9x
A local maximum
x
dy
=0
dx
x
A local minimum
The word local is usually omitted and the points called
maximum and minimum points.
3. e.g.1 Find the coordinates of the stationary points
on the curve y = x 3 − 3 x 2 − 9 x
y = x3 − 3x2 − 9x
Solution:
dy
⇒
= 3x2 − 6x − 9
dx
dy
⇒ 3 x 2 − 6 x − 9 = 0 ⇒ 3( x 2 − 2 x − 3) = 0
=0
dx
3( x − out + 1 = 0 ⇒ x = 3
Tip: Watch 3)( xfor )common factors or x = −1
x = 3 when finding )stationary points.
⇒ y = ( 3 3 − 3( 3) 2 − 9( 3)
= 27 − 27 − 27 = − 27
x = −1 ⇒ y = ( −1) 3 − 3( −1) 2 − 9( −1)
= −1 − 3 + 9 = 5
The stationary points are (3, -27) and ( -1, 5)
4. Exercises
Find the coordinates of the stationary points of the
following functions
2
1. y = x − 4 x + 5
2.
y = 2 x 3 + 3 x 2 − 12 x + 1
Solutions:
dy
1.
= 2x − 4
dx
dy
= 0 ⇒ 2x − 4 = 0
dx
⇒ x=2
x = 2 ⇒ y = ( 2) 2 − 4( 2) + 5 = 1
Ans: St. pt. is ( 2, 1)
5. y = 2 x 3 + 3 x 2 − 12 x + 1
2.
Solution:
dy
= 6 x 2 + 6 x − 12
dx
dy
= 0 ⇒ 6( x 2 + x − 2) = 0 ⇒ 6( x − 1)( x + 2) = 0
dx
⇒ x = 1 or x = −2
x = 1 ⇒ y = −6
x = −2 ⇒ y = 2( −2) 3 + 3( −2) 2 − 12( −2) + 1 = 21
Ans: St. pts. are ( 1, −6) and ( −2, 21 )
6. We need to be able to determine the nature of a
stationary point ( whether it is a max or a min ).
There are several ways of doing this. e.g.
On the left of
a maximum,
the gradient is
positive
+
On the right of
a maximum,
the gradient is
negative
−
7. So, for a max the gradients are
0 At the max
On the left of
On the right of
the max
the max
−
+
The opposite is true for a minimum
−
0
+
Calculating the gradients on the left and right of a
stationary point tells us whether the point is a max or a
min.
8. e.g.2 Find the coordinates of the stationary point of the
2
curve y = x − 4 x + 1 . Is the point a max or min?
− − − − − − (1)
y = x2 − 4x + 1
Solution:
dy
⇒
= 2x − 4
dx
dy
=0
⇒
2x − 4 = 0 ⇒ x = 2
dx
y = ( 2) 2 − 4( 2) + 1
⇒ y = −3
Substitute in (1):
dy
= 2(1) − 4 = − 2 < 0
On the left of x = 2 e.g. at x = 1,
dx
dy
On the right of x = 2 e.g. at x = 3,
= 2( 3) − 4 = 2 > 0
dx
+
−
⇒ ( 2, − 3) is a min
We have
0
9. Another method for determining the nature of a
stationary point.
e.g.3 Consider
y = x 3 + 3 x 2 − 9 x + 10
The gradient function
is given by
dy
= 3x2 + 6x − 9
dx
dy
dx
3
2
At the max of y = x + 3 x − 9 x + 10 the gradient is 0
but the gradient of the gradient is negative.
10. Another method for determining the nature of a
stationary point.
e.g.3 Consider
y = x 3 + 3 x 2 − 9 x + 10
The gradient function
is given by
dy
= 3x2 + 6x − 9
dx
dy
dx
At the min of
y = x 3 + 3 x 2 − 9 x + 10
the gradient of the
gradient is positive.
d2y
The notation for the gradient of the gradient is
dx 2
“d 2 y by d x squared”
11. e.g.3 ( continued ) Find the stationary points on the
curve y = x 3 + 3 x 2 − 9 x + 10 and distinguish between
the max and the min.
y = x 3 + 3 x 2 − 9 x + 10
Solution:
dy
d2y
2
⇒
= 3x + 6x − 9 ⇒
= 6x + 6
2
dx
dx
2
dy
2 d y
Stationary points:
= 0 ⇒ 3 x + 6 x −is called the
9=0
dx
dx 2 nd
2 derivative
⇒ 3( x 2 + 2 x − 3) = 0
⇒ 3( x + 3)( x − 1) = 0
⇒
x = −3 or x = 1
We now need to find the y-coordinates of the st. pts.
12. y = x 3 + 3 x 2 − 9 x + 10
x = −3 ⇒
y = ( −3) 3 + 3( −3) 2 − 9( −3) + 10 = 37
x =1
y = 1 + 3 − 9 + 10 = 5
⇒
To distinguish between max and min we use the 2nd
derivative, at the stationary points.
d2y
2
= 6x + 6
dx
d y
= 6( −3) + 6 = −12 < 0 ⇒ max at (−3, 37 )
At x = −3 ,
2
dx
2
At x = 1 ,
d2y
dx
2
= 6 + 6 = 12 > 0 ⇒ min at (1, 5)
13. SUMMARY
To find stationary points, solve the equation
dy
=0
dx
Determine the nature of the stationary points
•
either by finding the gradients on the left
and right of the stationary points
+
−
•
⇒ minimum
0
0
+
−
⇒
maximum
or by finding the value of the 2nd derivative
at the stationary points
d2y
dx
2
< 0 ⇒ max
d2y
dx
2
> 0 ⇒ min
14. Exercises
Find the coordinates of the stationary points of the
following functions, determine the nature of each
and sketch the functions.
3
2
3
2
y = x + 3x − 2
Ans. (0, − 2) is a min.
1.
(−2 , 2)
2.
y = x + 3x − 2
is a max.
y = 2 + 3x − x3
Ans. (−1, 0)
(1 , 4)
is a min.
is a max.
y = 2 + 3x − x3
15.
16. The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
17. The stationary points of a curve are the points where
the gradient is zero
e.g.
y = x3 − 3x2 − 9x
A local maximum
x
dy
=0
dx
x
A local minimum
The word local is usually omitted and the points called
maximum and minimum points.
18. e.g.1 Find the coordinates of the stationary points
y = x3 − 3x2 − 9x
on the curve
Solution:
⇒
dy
=0
dx
⇒
y = x3 − 3x2 − 9x
dy
= 3x2 − 6x − 9
dx
3x2 − 6x − 9 = 0 ⇒
3( x 2 − 2 x − 3) = 0
3( x − 3)( x + 1) = 0 ⇒ x = 3 or x = −1
x = 3 ⇒ y = ( 3) 3 − 3( 3) 2 − 9( 3)
= 27 − 27 − 27 = − 27
x = −1 ⇒ y = ( −1) 3 − 3( −1) 2 − 9( −1)
= −1 − 3 + 9 = 5
The stationary points are (3, -27) and ( -1, 5)
19. Determining the nature of a Stationary Point
For a max we have
On the left of
the max
+
0
At the max
−
On the right of
the max
The opposite is true for a minimum
−
0
+
Calculating the gradients on the left and right
of a stationary point tells us whether the point
is a max or a min.
20. Another method for determining the nature of a
stationary point.
e.g. Consider
y
y = x 3 + 3 x 2 − 9 x + 10
The gradient function
is given by
dy
= 3x2 + 6x − 9
dx
dy
dx
3
2
At the max of y = x + 3 x − 9 x + 10 the gradient is
0, but the gradient of the gradient is negative.
21. y = x 3 + 3 x 2 − 9 x + 10
The gradient function
is given by
dy
= 3x2 + 6x − 9
dx
dy
dx
At the min of
y = x 3 + 3 x 2 − 9 x + 10
the gradient of the
gradient is positive.
d2y
The notation for the gradient of the gradient is
dx 2
“d 2 y by d x squared”
22. The gradient of the gradient is called the 2nd
derivative and is written as
d2y
dx 2
23. e.g. Find the stationary points on the curve
3
y = xand 3distinguish between the max
+ x 2 − 9 x + 10
and the=min.+ 3 x 2 − 9 x + 10
y x3
Solution:
dy
d2y
2
⇒
= 3x + 6x − 9 ⇒
= 6x + 6
2
dx
dx
dy
Stationary points:
= 0 ⇒ 3x2 + 6x − 9 = 0
dx
⇒ 3( x 2 + 2 x − 3) = 0
⇒ 3( x + 3)( x − 1) = 0
⇒
x = −3 or x = 1
We now need to find the y-coordinates of the st. pts.
24. y = x 3 + 3 x 2 − 9 x + 10
x = −3 ⇒
y = ( −3) 3 + 3( −3) 2 − 9( −3) + 10 = 37
x =1
y = 1 + 3 − 9 + 10 = 5
⇒
To distinguish between max and min we use the 2nd
derivative,
d2y
2
= 6x + 6
dx
d2y
At x = −3 , 2 = 6( −3) + 6 = −12 < 0 ⇒ max at (−3, 37 )
dx
At
x =1 ,
d2y
dx
2
= 6 + 6 = 12 > 0 ⇒ min at (1, 5)
25. SUMMARY
To find stationary points, solve the equation
dy
=0
dx
Determine the nature of the stationary points
•
either by finding the gradients on the left
and right of the stationary points
0
−
+
−
⇒ maximum
⇒ minimum +
0
• or by finding the value of the 2nd derivative
at the stationary points
d2y
dx
2
< 0 ⇒ max
d2y
dx
2
> 0 ⇒ min