Ib maths sl product and quotient rules
•
1 like•2,125 views
- The document discusses product and quotient rules for finding derivatives of functions. - The product rule states that the derivative of two functions multiplied together (f(x) = u(x)v(x)) is the first function multiplied by the derivative of the second plus the second function multiplied by the derivative of the first. - The quotient rule is provided but not explained for finding the derivative of a function that is a quotient of two other functions.
Report
Share
Report
Share
Download to read offline
Recommended
AQA Biology B10 [ Answers ]
The document provides information about the human nervous system, including:
- Homeostasis and the roles of receptors, coordination centers, and effectors in maintaining internal conditions.
- The structure and functions of the central and peripheral nervous systems, including different types of neurons.
- Reflex actions and how stimuli are passed through neurons to elicit fast, involuntary responses.
- The structure and functions of the brain regions including the cerebral cortex, cerebellum, and medulla.
- The structure of the eye, including components like the iris, retina, and optic nerve.
AQA Biology B4 [ Answers ]
This document contains answers and guidance for questions about organ systems in animals and plants. It discusses the structure and function of the blood, blood vessels, heart, breathing and gas exchange, plant tissues and organs, and transport systems in plants. Key points covered include the components and roles of blood, the double circulation system, heart structure and valves, respiratory gas exchange in lungs, leaf anatomy and transpiration.
AQA Biology B9 [ Answers ]
- Aerobic respiration takes place in mitochondria and involves glucose and oxygen breaking down into carbon dioxide and water, transferring energy to the environment. Muscle cells need many mitochondria for their high energy needs.
- The liver plays an important role in metabolism by converting lactic acid back to glucose after exercise, storing glucose as glycogen, and providing energy through aerobic respiration for its various metabolic functions.
P4 Electric Circuits [ Answers AQA GCSE Physics ]
This document provides answers to questions about electric circuits from a student book.
1) It defines key terms like current, potential difference, resistance, and charge.
2) It explains how to calculate values in series and parallel circuits such as total resistance, current, and potential difference across components.
3) Sample calculations are provided for circuits involving combinations of cells, resistors, ammeters, and voltmeters.
AQA Physics P5 [ Summary answers ]
This document contains answers and guidance for questions related to electricity and circuits. Key points:
1) As amplitude of waves increases, waves get taller. As time for each cycle decreases, waves get closer together.
2) If a live wire touches a metal case, anyone touching the case could be electrocuted as current passes through their body to the ground.
3) Appliances plugged into different wall sockets can be turned on or off independently, as current on one circuit does not affect another. If too much current draws, a fuse will melt and cut off the circuit.
AQA Chemistry C5 Answers Key
This document contains answers to questions about chemical changes from a student book. It includes answers about:
- Reactions of metals like lithium and zinc with acids producing salts and hydrogen gas.
- Displacement reactions where more reactive metals displace less reactive ones from solutions.
- Extracting metals like zinc through chemical reduction of metal oxides.
- Producing salts by reacting metals with acids or insoluble bases with acids.
- Measuring pH and differences between strong and weak acids.
- Calculations involving moles of gases.
3.4 pascal principle answer
This document discusses Pascal's principle and its applications in hydraulic systems. It provides examples of Pascal's principle in everyday devices like toothpaste tubes and car jacks. It also explains how hydraulic systems work as force multipliers by transferring pressure throughout an incompressible fluid. The document contains sample problems and questions involving calculations with hydraulic systems, as well as explanations of how systems like car brakes and hydraulic lifts operate based on Pascal's principle.
#18 Key
The document contains 6 chemistry problems involving gas phase chemical equilibria calculations:
1) Calculating the equilibrium partial pressure of N2O4 and the equilibrium constant Kp for the reaction N2O4 ⇌ 2NO2.
2) Calculating the equilibrium concentrations of H2, Br2, and HBr and the equilibrium constant Kc for the reaction H2 + Br2 ⇌ 2HBr.
3) Calculating the equilibrium constant Kc for the reaction SO2 + Cl2 ⇌ SO2Cl2 given the value of Kp.
4) Calculating the value of Kp for the reverse reaction of 2NO2 ⇌
Recommended
AQA Biology B10 [ Answers ]
The document provides information about the human nervous system, including:
- Homeostasis and the roles of receptors, coordination centers, and effectors in maintaining internal conditions.
- The structure and functions of the central and peripheral nervous systems, including different types of neurons.
- Reflex actions and how stimuli are passed through neurons to elicit fast, involuntary responses.
- The structure and functions of the brain regions including the cerebral cortex, cerebellum, and medulla.
- The structure of the eye, including components like the iris, retina, and optic nerve.
AQA Biology B4 [ Answers ]
This document contains answers and guidance for questions about organ systems in animals and plants. It discusses the structure and function of the blood, blood vessels, heart, breathing and gas exchange, plant tissues and organs, and transport systems in plants. Key points covered include the components and roles of blood, the double circulation system, heart structure and valves, respiratory gas exchange in lungs, leaf anatomy and transpiration.
AQA Biology B9 [ Answers ]
- Aerobic respiration takes place in mitochondria and involves glucose and oxygen breaking down into carbon dioxide and water, transferring energy to the environment. Muscle cells need many mitochondria for their high energy needs.
- The liver plays an important role in metabolism by converting lactic acid back to glucose after exercise, storing glucose as glycogen, and providing energy through aerobic respiration for its various metabolic functions.
P4 Electric Circuits [ Answers AQA GCSE Physics ]
This document provides answers to questions about electric circuits from a student book.
1) It defines key terms like current, potential difference, resistance, and charge.
2) It explains how to calculate values in series and parallel circuits such as total resistance, current, and potential difference across components.
3) Sample calculations are provided for circuits involving combinations of cells, resistors, ammeters, and voltmeters.
AQA Physics P5 [ Summary answers ]
This document contains answers and guidance for questions related to electricity and circuits. Key points:
1) As amplitude of waves increases, waves get taller. As time for each cycle decreases, waves get closer together.
2) If a live wire touches a metal case, anyone touching the case could be electrocuted as current passes through their body to the ground.
3) Appliances plugged into different wall sockets can be turned on or off independently, as current on one circuit does not affect another. If too much current draws, a fuse will melt and cut off the circuit.
AQA Chemistry C5 Answers Key
This document contains answers to questions about chemical changes from a student book. It includes answers about:
- Reactions of metals like lithium and zinc with acids producing salts and hydrogen gas.
- Displacement reactions where more reactive metals displace less reactive ones from solutions.
- Extracting metals like zinc through chemical reduction of metal oxides.
- Producing salts by reacting metals with acids or insoluble bases with acids.
- Measuring pH and differences between strong and weak acids.
- Calculations involving moles of gases.
3.4 pascal principle answer
This document discusses Pascal's principle and its applications in hydraulic systems. It provides examples of Pascal's principle in everyday devices like toothpaste tubes and car jacks. It also explains how hydraulic systems work as force multipliers by transferring pressure throughout an incompressible fluid. The document contains sample problems and questions involving calculations with hydraulic systems, as well as explanations of how systems like car brakes and hydraulic lifts operate based on Pascal's principle.
#18 Key
The document contains 6 chemistry problems involving gas phase chemical equilibria calculations:
1) Calculating the equilibrium partial pressure of N2O4 and the equilibrium constant Kp for the reaction N2O4 ⇌ 2NO2.
2) Calculating the equilibrium concentrations of H2, Br2, and HBr and the equilibrium constant Kc for the reaction H2 + Br2 ⇌ 2HBr.
3) Calculating the equilibrium constant Kc for the reaction SO2 + Cl2 ⇌ SO2Cl2 given the value of Kp.
4) Calculating the value of Kp for the reverse reaction of 2NO2 ⇌
Food tests exam questions and mark scheme
This document describes biochemical tests that can be used to identify different biological molecules, including lipids, proteins, starches, and sugars. It provides tables to complete with the appropriate tests, reagents, and expected results. The key tests described are testing for reducing sugars using Benedict's reagent and testing for proteins using the biuret test. Instructions are also given for hydrolyzing non-reducing sugars before testing for reducing sugars.
04_0862_01_7RP_AFP_tcm143-686152 (1).pdf
This document is a 16-page mathematics assessment for students taking the Cambridge Lower Secondary Checkpoint. It contains 26 multiple choice and written response questions testing a variety of math skills, including operations with fractions, decimals, percentages, geometry, algebra, statistics, and probability. The questions provide context, diagrams, tables, and graphs to aid students in solving word problems and showing their work.
Heat and Thermodynamics cheat sheet
The document summarizes key concepts in thermodynamics and heat transfer. It defines important terms like temperature, heat, work, and entropy. It also outlines the first and second laws of thermodynamics. The first law states that energy is conserved in thermal processes, such that the change in internal energy of a system equals heat added minus work done. The second law states that the entropy of the universe increases over time as energy spreads out. The document provides important equations related to heat, work, and efficiency and lists common units used in thermodynamics.
Force & Motion
This document is a collection of 12 photo credits from various photographers. It concludes by inviting the reader to create their own presentation using Haiku Deck on SlideShare.
AQA Chemistry C4 Answers Key
This document provides answers to chemical calculation problems from a student workbook. It includes:
1) Answers to questions about relative atomic masses, chemical equations, mass to mole conversions, reaction yields, atom economy, concentration expressions, titration procedures and calculations, gas volumes, and molar volume.
2) The answers are accompanied by guidance on the number of marks awarded for each part of the answers.
3) The resource may have been modified from the original but provides a comprehensive summary of answers to practice problems covering key topics in chemical calculations.
2018 Cambridge Lower Second Progression Test Science Stage 9 QP Paper 1_tcm14...
1. The document is a science exam paper containing multiple choice and short answer questions testing students' knowledge of various science topics.
2. The questions cover topics such as photosynthesis, chemical reactions, food chains, circuits, the periodic table, genetics, density, reactivity of metals, evaporation, and pressure.
3. Students are required to demonstrate their understanding by completing equations, explaining concepts, analyzing diagrams, calculating values, and describing experiments.
IGCSE Biology Revision Quiz - Plant Physiology
This document contains a 63 question quiz about plant structure and function, including questions about photosynthesis, plant tissues, transport processes, reproduction, and growth. It tests knowledge of topics like plant responses to stimuli, the roles of hormones and mineral nutrients, methods of asexual reproduction, pollination, seed and fruit development, seed dispersal, and germination requirements. The quiz appears to be designed to comprehensively assess understanding of key concepts in plant biology.
0620_s14_qp_32
This 12-page document consists of chemistry exam questions covering topics such as electron configurations, fossil fuels, plant growth and fertilizers, carboxylic acids, carbon monoxide production, electrolysis, scandium properties, and salt preparation and identification. The questions include multiple choice, short answer, drawing structures, explaining observations, and calculations.
Rules of derivative
This document discusses differentiation and derivatives. It defines differentiation as finding the average rate of change of one variable with respect to another. It then discusses various methods of finding derivatives, including the direct method using derivative rules, as well as discussing specific rules like the power rule, product rule, quotient rule, chain rule, and rules for derivatives of trigonometric, exponential, and logarithmic functions.
On Application of Power Series Solution of Bessel Problems to the Problems of...
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.03_AJMS_209_19_RA.pdf
This document discusses power series solutions to differential equations, specifically Bessel's equations. It provides background on power series expansions and their properties. It explains that solutions to differential equations can be written as power series when the coefficients of the equation are analytic at a point. As an example, it finds the general solution to a second order differential equation using the power series method. In summary, it outlines techniques for solving differential equations using power series expansions at ordinary points.
03_AJMS_209_19_RA.pdf
This document discusses applying power series solutions of Bessel equations to solve problems involving struts with variable moments of inertia. It begins by reviewing the basics of power series solutions to differential equations and Bessel equations. It then develops a model of the variable strut problem as a Bessel-form differential equation. Finally, it solves this equation using the power series method for Bessel equations to obtain a result for problems involving struts with non-constant moments of inertia.
M3 16marks
This document contains 10 units of questions related to Fourier series, Fourier transforms, partial differential equations, applications of partial differential equations, and Z-transforms. The questions range from deriving equations to solving problems to proving theorems. Overall, the document covers a wide range of topics in applied mathematics and asks the reader to apply mathematical techniques to solve problems in these areas.
Derivatives
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.
Grph quad fncts
The document discusses graphing quadratic functions. It defines a quadratic function as f(x) = ax^2 + bx + c where a, b, and c are real numbers and a is not equal to 0. The graph of a quadratic function is a parabola that is symmetrical about an axis. When the leading coefficient a is positive, the parabola opens upward and the vertex is a minimum. When a is negative, the parabola opens downward and the vertex is a maximum. Standard forms for quadratic functions and methods for finding characteristics like the vertex, axis of symmetry, and x-intercepts from the equation are also presented.
AJMS_389_22.pdf
This document discusses linear multistep methods for solving initial value differential problems. It makes three key points:
1) Linear multistep methods can be viewed as fixed point iterative methods with a differential operator instead of an integral operator. They converge to a fixed point if consistent and stable.
2) The document proves that any linear multistep method defines a contraction mapping, ensuring convergence to a unique fixed point when the problem is Lipschitz continuous.
3) Examples of explicit and implicit linear multistep methods are given, along with their fixed point iterative formulations. These include Euler's method, trapezoidal method, and Adams-Moulton method.
Quadratic Equations Graphing
The document discusses how to graph quadratic functions of the form y = ax^2 + bx + c. It explains that the graph is a parabola that may open up or down. It describes how to find the line of symmetry, which is -b/2a, and how this line passes through the vertex. It provides steps to find the vertex by plugging the x-value from the line of symmetry into the original equation. Finally, it demonstrates graphing a parabola by finding two additional points and reflecting them across the line of symmetry.
Ms2 gradient of a curve and equation
Math Studies Year 2 - Intro to Calculus
Find gradient of a curve and equation of a tangent to a curve
Limits and derivatives
The document discusses functions and their derivatives. It defines functions, different types of functions, and notation used for functions. It then covers the concept of limits, theorems on limits, and limits at infinity. The document defines the slope of a tangent line to a curve and increments. It provides definitions and rules for derivatives, including differentiation from first principles and various differentiation rules. It includes examples of finding derivatives using these rules and taking multiple derivatives.
Taller 2. 4264. (2)
This document discusses linear algebra concepts applied to telecommunications engineering. It establishes objectives like exploring applications of vector spaces and subspaces in telecommunications, generating interest in the topic through a workshop, and identifying linear dependence using systems of equations. The theoretical foundations section defines vector spaces and subspaces. It discusses concepts like linearly independent sets, generators, bases, and using the Wronskian theorem to determine linear independence. Examples are provided to identify if sets of functions are linearly independent or dependent. The document aims to strengthen student understanding of linear algebra concepts relevant to their field of study.
Supporting Vector Machine
Support Vector Machine (SVM) is a supervised machine learning algorithm that can be used for both classification and regression tasks. It works by finding a hyperplane in an N-dimensional space that distinctly classifies the data points. SVM finds this optimal separation by maximizing the margin between the two classes. The algorithm transforms the data into a higher dimension using kernels to handle non-linear classification. Popular kernels include polynomial kernels and Gaussian radial basis function kernels. SVMs are effective because they use a convex optimization which guarantees a global optimal solution and avoids local optima issues.
3 graphs of second degree functions x
The document discusses quadratic functions and parabolas. It begins by defining quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It then provides an example of graphing the quadratic function y = x2 - 4x - 12. To do this, it finds the vertex by setting x = -b/2a, and uses the vertex and other points like the y-intercept to sketch the parabolic shape. It also discusses general properties of parabolas, such as being symmetric around a center line and having a highest/lowest point called the vertex that sits on this line.
More Related Content
What's hot
Food tests exam questions and mark scheme
This document describes biochemical tests that can be used to identify different biological molecules, including lipids, proteins, starches, and sugars. It provides tables to complete with the appropriate tests, reagents, and expected results. The key tests described are testing for reducing sugars using Benedict's reagent and testing for proteins using the biuret test. Instructions are also given for hydrolyzing non-reducing sugars before testing for reducing sugars.
04_0862_01_7RP_AFP_tcm143-686152 (1).pdf
This document is a 16-page mathematics assessment for students taking the Cambridge Lower Secondary Checkpoint. It contains 26 multiple choice and written response questions testing a variety of math skills, including operations with fractions, decimals, percentages, geometry, algebra, statistics, and probability. The questions provide context, diagrams, tables, and graphs to aid students in solving word problems and showing their work.
Heat and Thermodynamics cheat sheet
The document summarizes key concepts in thermodynamics and heat transfer. It defines important terms like temperature, heat, work, and entropy. It also outlines the first and second laws of thermodynamics. The first law states that energy is conserved in thermal processes, such that the change in internal energy of a system equals heat added minus work done. The second law states that the entropy of the universe increases over time as energy spreads out. The document provides important equations related to heat, work, and efficiency and lists common units used in thermodynamics.
Force & Motion
This document is a collection of 12 photo credits from various photographers. It concludes by inviting the reader to create their own presentation using Haiku Deck on SlideShare.
AQA Chemistry C4 Answers Key
This document provides answers to chemical calculation problems from a student workbook. It includes:
1) Answers to questions about relative atomic masses, chemical equations, mass to mole conversions, reaction yields, atom economy, concentration expressions, titration procedures and calculations, gas volumes, and molar volume.
2) The answers are accompanied by guidance on the number of marks awarded for each part of the answers.
3) The resource may have been modified from the original but provides a comprehensive summary of answers to practice problems covering key topics in chemical calculations.
2018 Cambridge Lower Second Progression Test Science Stage 9 QP Paper 1_tcm14...
1. The document is a science exam paper containing multiple choice and short answer questions testing students' knowledge of various science topics.
2. The questions cover topics such as photosynthesis, chemical reactions, food chains, circuits, the periodic table, genetics, density, reactivity of metals, evaporation, and pressure.
3. Students are required to demonstrate their understanding by completing equations, explaining concepts, analyzing diagrams, calculating values, and describing experiments.
IGCSE Biology Revision Quiz - Plant Physiology
This document contains a 63 question quiz about plant structure and function, including questions about photosynthesis, plant tissues, transport processes, reproduction, and growth. It tests knowledge of topics like plant responses to stimuli, the roles of hormones and mineral nutrients, methods of asexual reproduction, pollination, seed and fruit development, seed dispersal, and germination requirements. The quiz appears to be designed to comprehensively assess understanding of key concepts in plant biology.
0620_s14_qp_32
This 12-page document consists of chemistry exam questions covering topics such as electron configurations, fossil fuels, plant growth and fertilizers, carboxylic acids, carbon monoxide production, electrolysis, scandium properties, and salt preparation and identification. The questions include multiple choice, short answer, drawing structures, explaining observations, and calculations.
What's hot (8)
2018 Cambridge Lower Second Progression Test Science Stage 9 QP Paper 1_tcm14...
2018 Cambridge Lower Second Progression Test Science Stage 9 QP Paper 1_tcm14...
Similar to Ib maths sl product and quotient rules
Rules of derivative
This document discusses differentiation and derivatives. It defines differentiation as finding the average rate of change of one variable with respect to another. It then discusses various methods of finding derivatives, including the direct method using derivative rules, as well as discussing specific rules like the power rule, product rule, quotient rule, chain rule, and rules for derivatives of trigonometric, exponential, and logarithmic functions.
On Application of Power Series Solution of Bessel Problems to the Problems of...
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.03_AJMS_209_19_RA.pdf
This document discusses power series solutions to differential equations, specifically Bessel's equations. It provides background on power series expansions and their properties. It explains that solutions to differential equations can be written as power series when the coefficients of the equation are analytic at a point. As an example, it finds the general solution to a second order differential equation using the power series method. In summary, it outlines techniques for solving differential equations using power series expansions at ordinary points.
03_AJMS_209_19_RA.pdf
This document discusses applying power series solutions of Bessel equations to solve problems involving struts with variable moments of inertia. It begins by reviewing the basics of power series solutions to differential equations and Bessel equations. It then develops a model of the variable strut problem as a Bessel-form differential equation. Finally, it solves this equation using the power series method for Bessel equations to obtain a result for problems involving struts with non-constant moments of inertia.
M3 16marks
This document contains 10 units of questions related to Fourier series, Fourier transforms, partial differential equations, applications of partial differential equations, and Z-transforms. The questions range from deriving equations to solving problems to proving theorems. Overall, the document covers a wide range of topics in applied mathematics and asks the reader to apply mathematical techniques to solve problems in these areas.
Derivatives
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.
Grph quad fncts
The document discusses graphing quadratic functions. It defines a quadratic function as f(x) = ax^2 + bx + c where a, b, and c are real numbers and a is not equal to 0. The graph of a quadratic function is a parabola that is symmetrical about an axis. When the leading coefficient a is positive, the parabola opens upward and the vertex is a minimum. When a is negative, the parabola opens downward and the vertex is a maximum. Standard forms for quadratic functions and methods for finding characteristics like the vertex, axis of symmetry, and x-intercepts from the equation are also presented.
AJMS_389_22.pdf
This document discusses linear multistep methods for solving initial value differential problems. It makes three key points:
1) Linear multistep methods can be viewed as fixed point iterative methods with a differential operator instead of an integral operator. They converge to a fixed point if consistent and stable.
2) The document proves that any linear multistep method defines a contraction mapping, ensuring convergence to a unique fixed point when the problem is Lipschitz continuous.
3) Examples of explicit and implicit linear multistep methods are given, along with their fixed point iterative formulations. These include Euler's method, trapezoidal method, and Adams-Moulton method.
Quadratic Equations Graphing
The document discusses how to graph quadratic functions of the form y = ax^2 + bx + c. It explains that the graph is a parabola that may open up or down. It describes how to find the line of symmetry, which is -b/2a, and how this line passes through the vertex. It provides steps to find the vertex by plugging the x-value from the line of symmetry into the original equation. Finally, it demonstrates graphing a parabola by finding two additional points and reflecting them across the line of symmetry.
Ms2 gradient of a curve and equation
Math Studies Year 2 - Intro to Calculus
Find gradient of a curve and equation of a tangent to a curve
Limits and derivatives
The document discusses functions and their derivatives. It defines functions, different types of functions, and notation used for functions. It then covers the concept of limits, theorems on limits, and limits at infinity. The document defines the slope of a tangent line to a curve and increments. It provides definitions and rules for derivatives, including differentiation from first principles and various differentiation rules. It includes examples of finding derivatives using these rules and taking multiple derivatives.
Taller 2. 4264. (2)
This document discusses linear algebra concepts applied to telecommunications engineering. It establishes objectives like exploring applications of vector spaces and subspaces in telecommunications, generating interest in the topic through a workshop, and identifying linear dependence using systems of equations. The theoretical foundations section defines vector spaces and subspaces. It discusses concepts like linearly independent sets, generators, bases, and using the Wronskian theorem to determine linear independence. Examples are provided to identify if sets of functions are linearly independent or dependent. The document aims to strengthen student understanding of linear algebra concepts relevant to their field of study.
Supporting Vector Machine
Support Vector Machine (SVM) is a supervised machine learning algorithm that can be used for both classification and regression tasks. It works by finding a hyperplane in an N-dimensional space that distinctly classifies the data points. SVM finds this optimal separation by maximizing the margin between the two classes. The algorithm transforms the data into a higher dimension using kernels to handle non-linear classification. Popular kernels include polynomial kernels and Gaussian radial basis function kernels. SVMs are effective because they use a convex optimization which guarantees a global optimal solution and avoids local optima issues.
3 graphs of second degree functions x
The document discusses quadratic functions and parabolas. It begins by defining quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It then provides an example of graphing the quadratic function y = x2 - 4x - 12. To do this, it finds the vertex by setting x = -b/2a, and uses the vertex and other points like the y-intercept to sketch the parabolic shape. It also discusses general properties of parabolas, such as being symmetric around a center line and having a highest/lowest point called the vertex that sits on this line.
12 graphs of second degree functions x
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It states that the graphs of quadratic equations are called parabolas. Parabolas are symmetric around a central line, with the vertex (highest/lowest point) located on this line. The vertex formula is given as x = -b/2a. Steps for graphing a parabola are outlined, including finding the vertex, another point, and reflections across the central line. An example graphs the parabola y = x2 - 4x - 12, finding the vertex as (2, -16) and x-intercepts as -
2.4 grapgs of second degree functions
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. Parabolas are the graphs of quadratic functions and have certain properties: they are symmetric about a center line, with the highest/lowest point (called the vertex) sitting on the center line. The vertex position can be found using the formula x = -b/2a. Examples are given of finding the vertex and graphing parabolas.
4.11.2 Special Right Triangles
This document discusses special right triangles and their properties. It defines 45-45-90 and 30-60-90 triangles, and provides the key relationships between their sides: for 45-45-90 triangles, the hypotenuse is √2 times the leg; for 30-60-90 triangles, the hypotenuse is 2 times the shorter leg and √3 times the longer leg. It provides examples of using these relationships to solve for missing side lengths.
Chapters 1 8 ( 6 marks)
The document contains a list of 31 multi-part math problems involving matrices, integrals, functions, geometry, and equations. The problems cover topics like finding the composition and inverse of functions, evaluating definite integrals, solving systems of equations using matrices, finding maximums and minimums, and proving geometric properties.
Chapters 1 8 ( 6 marks)
The document contains a list of 31 multi-part math problems involving matrices, integrals, functions, geometry, and equations. The problems cover topics like finding the product and inverse of matrices, evaluating definite integrals, finding maximums and minimums of functions, solving systems of equations using matrices, and calculating areas of geometric shapes.
Chapters 1 8 ( 6 marks)
The document contains a list of 31 multi-part math problems involving matrices, integrals, functions, geometry, and equations. The problems cover topics like finding the product and inverse of matrices, evaluating definite integrals, finding maximums and minimums of functions, solving systems of equations using matrices, and calculating areas of geometric shapes.
Similar to Ib maths sl product and quotient rules (20)
On Application of Power Series Solution of Bessel Problems to the Problems of...
On Application of Power Series Solution of Bessel Problems to the Problems of...
More from estelav
6) equations with exponentials & logarithms
This document provides worked examples for solving various logarithmic and exponential equations. It begins with an introduction stating the learner will be able to solve exponential equations of different bases, logarithmic equations, and exponential equations of the same base. It then shows 7 worked examples of solving equations involving logarithms and exponents, with the conclusion being about expressing exponential powers in the same base when possible.
7) change of base
This document discusses logarithmic equations and calculations. It provides instructions on how to rewrite logarithmic equations without using logarithms, solve simultaneous logarithmic equations, calculate logarithms to specific bases and numbers of significant figures, change logarithmic bases, and solve logarithmic equations. The document aims to teach students how to perform various operations with logarithms including rewriting, solving, calculating, and changing bases.
5) logarithms graphs
This document provides an overview of exponential and logarithmic functions:
1) It explores properties of exponential functions with the forms y = ax, where a > 1 represents exponential growth and 0 < a < 1 represents exponential decay. Key aspects like asymptotes and y-intercepts of these graphs are discussed.
2) Tables are completed and graphs are drawn to represent exponential growth and decay functions.
3) The natural logarithm function ln(x) is introduced and its graph is sketched using a calculator. Common features that logarithmic graphs share with exponential graphs are described.
4) Examples of simplifying logarithmic expressions and solving logarithmic equations are presented.
4) logarithms laws
This document discusses logarithmic laws and their proofs. It contains:
- Exercises to verify logarithmic laws like loga(xy) = loga(x) + loga(y) and nloga(x) = loga(x)n
- Proofs of these laws using the definition of logarithms and index form
- More exercises for students to solve involving simplifying logarithmic expressions using these laws
The goal is for students to understand and be able to apply the key logarithmic laws through worked examples and practice questions.
3) logarithm definition
This document introduces logarithms and how to use them to solve exponential equations. It defines logarithms as the power to which a base number must be raised to equal the value being logged. Examples are provided of writing numbers and powers as logarithms in different bases. The basics that the logarithm of a number to itself is 1 and the logarithm of 1 or 0 is 0 or undefined are covered. Students are directed to an online worksheet and book exercises to practice evaluating and using logarithms.
2) exponential growth and decay
This document discusses exponential functions and their properties. It explores exponential growth and decay through graphs of functions like y=2^x and y=0.5^x. It shows that as x increases, exponential growth functions approach infinity, while decay functions approach zero. The document also introduces the irrational number e as the most important base for modeling continuous growth and decay. It shows how the function A=Ce^rt models continuous compound interest as the compounding period approaches infinity.
1) exponential equations
This document discusses exponential functions and equations. It explores the properties of exponential functions by examining their graphs and understanding exponential growth and decay. It also covers how to solve problems leading to exponential equations by learning about bases, exponents, powers, and index rules. The document teaches how to evaluate, write in different bases, identify exponential equations, and solve exponential equations. It includes exercises to solve from the attached document on exponential equations and functions.
6) simultaneous and problems
This document discusses finding the intersection points between a line and parabola by solving simultaneous equations, finding the value of k that makes a line tangent to a parabola, and finding the maximum area of a rectangle with a perimeter of 28 cm.
5) quadratic equations
This document discusses solving quadratic equations by different methods such as factorisation, using the quadratic formula, and completing the square. It explains how to determine the number of solutions a quadratic equation has based on the sign of its discriminant, namely:
- If the discriminant is positive, there are two real roots
- If the discriminant is zero, there is one repeated root
- If the discriminant is negative, there are no real roots.
Examples are provided to illustrate calculating the discriminant and determining the number and type of roots.
4) quadratic functions factorised form
This document discusses quadratic functions and their properties. It explains how to express quadratics in different forms like factored, general, and vertex form. It provides examples of finding the zeros, line of symmetry, and vertex of various quadratic functions. Students are asked to work through examples of sketching graphs, factorizing expressions, solving equations, and rewriting functions in different forms to investigate the properties of quadratic functions and their relationships between algebraic and graphical representations.
3) quadratic vertex form
The document discusses quadratic functions and their graphs. It aims to explore the properties of quadratic functions and express them in different forms, including vertex form. It provides examples of writing quadratic functions in vertex form given the vertex, as well as writing the function in general form given its vertex. The key points are that the vertex form of a quadratic function y = a(x - h)2 + k shows the vertex is (h, k) and the line of symmetry is x = h. Converting between the vertex and general forms allows finding the maximum point of a quadratic function.
2) quadratics gral form
This document discusses properties of quadratic functions and their graphs. It explores expressing quadratic functions in different forms, sketching graphs of basic quadratic functions, and investigating the effects of changing coefficients on the graph. The key aspects covered are the relationship between the coefficients and the graph properties like the vertex, line of symmetry, and y-intercept. Examples of matching functions to graphs and writing functions in standard form are provided.
1) quadratic functions
The document discusses quadratic functions and their graphs. It explores expressing quadratic functions in different forms, and transforming quadratic functions and their graphs through translations, stretches, and other changes. Key aspects covered include the vertex and line of symmetry of parabolas, and how changing the coefficients affects the graph shape and position. Examples of specific quadratic functions are analyzed and graphed.
Y9 algebra 1 Simultaneous Equations byElimination Method
This document discusses two methods for solving simultaneous equations: the elimination method and the substitution method. It provides examples of using the elimination method to solve several systems of simultaneous equations, showing the steps to eliminate one variable and solve for the other variables. The document concludes by providing two word problems as examples and assigning practice problems from the book.
Y9 algebra 1 simultaneous equations by substitution method
The document provides examples of solving simultaneous equations by substitution. It includes examples of setting up two equations representing relationships between variables, substituting one equation into the other to isolate a single variable, and solving for the value of that variable. It then uses the solved variable to find the value of the other variable.
Y9 algebra 1 Change of Subject
By the end of the lesson, students should be able to change the subject of a formula. The document provides examples of changing the subject of formulas for perimeter of a circle, cost of a taxi, volume of a cone, and other formulas. Students are asked to make different variables the subject of provided formulas, such as making d, a, m, h, or x the subject.
Y9 algebra 1 Inequalities
This document provides an overview of inequalities and how to solve them algebraically. It discusses representing inequalities on number lines, and the rules for adding, subtracting, multiplying, and dividing terms when solving inequalities, such as changing the sign of the inequality when multiplying or dividing by a negative number. Examples are provided of solving inequalities algebraically and representing the solutions on number lines.
Y9 algebra 1 Collecting Terms & Expanding Brackets
The document outlines key concepts and skills to be learned in an algebra lesson, including:
- Simplifying algebraic expressions by collecting like terms
- Expanding brackets
- Multiplying algebraic expressions
It then provides examples of simplifying algebraic expressions, expanding brackets, finding perimeters and areas of shapes, evaluating expressions, and converting between temperatures in Fahrenheit and Celsius.
Y9 algebra 1 Linear Equations
This document provides an overview of solving linear equations, including equations involving brackets, fractions, and changing the subject of a formula. It presents several example equations and step-by-step solutions showing how to isolate the variable. Students are reminded to check their answers by substituting them back into the original equation. The document also introduces solving equations with fractions using the lowest common denominator approach.
Ib maths sl chain rule
The document discusses applying the chain rule to find derivatives of composite functions. It provides examples of identifying composite functions and using the chain rule formula to take derivatives. It lists practice problems for students to find derivatives of various composite functions using the chain rule, including functions with exponents like (x3 + 1)8, as well as trigonometric functions like y = sin(4x).
More from estelav (20)
Y9 algebra 1 Simultaneous Equations byElimination Method
Y9 algebra 1 Simultaneous Equations byElimination Method
Y9 algebra 1 simultaneous equations by substitution method
Y9 algebra 1 simultaneous equations by substitution method
Y9 algebra 1 Collecting Terms & Expanding Brackets
Y9 algebra 1 Collecting Terms & Expanding Brackets
Recently uploaded
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...National Information Standards Organization (NISO)
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar KanvariaBÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...Nguyen Thanh Tu Collection
https://app.box.com/s/y977uz6bpd3af4qsebv7r9b7s21935vdclinical examination of hip joint (1).pdf
described clinical examination all orthopeadic conditions .
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective UpskillingExcellence Foundation for South Sudan
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.Executive Directors Chat Leveraging AI for Diversity, Equity, and Inclusion
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
South African Journal of Science: Writing with integrity workshop (2024)
South African Journal of Science: Writing with integrity workshop (2024)Academy of Science of South Africa
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.Hindi varnamala | hindi alphabet PPT.pdf
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
The History of Stoke Newington Street Names
Presented at the Stoke Newington Literary Festival on 9th June 2024
www.StokeNewingtonHistory.com
How to Manage Your Lost Opportunities in Odoo 17 CRM
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
The Diamonds of 2023-2024 in the IGRA collection
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Azure Interview Questions and Answers PDF By ScholarHat
Azure Interview Questions and Answers PDF By ScholarHat
DRUGS AND ITS classification slide share
Any substance (other than food) that is used to prevent, diagnose, treat, or relieve symptoms of a
disease or abnormal condition
Assessment and Planning in Educational technology.pptx
In an education system, it is understood that assessment is only for the students, but on the other hand, the Assessment of teachers is also an important aspect of the education system that ensures teachers are providing high-quality instruction to students. The assessment process can be used to provide feedback and support for professional development, to inform decisions about teacher retention or promotion, or to evaluate teacher effectiveness for accountability purposes.
How to Fix the Import Error in the Odoo 17
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Main Java[All of the Base Concepts}.docx
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama UniversityAkanksha trivedi rama nursing college kanpur.
Natural birth techniques are various type such as/ water birth , alexender method, hypnosis, bradley method, lamaze method etcHow to Add Chatter in the odoo 17 ERP Module
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Recently uploaded (20)
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Executive Directors Chat Leveraging AI for Diversity, Equity, and Inclusion
Executive Directors Chat Leveraging AI for Diversity, Equity, and Inclusion
South African Journal of Science: Writing with integrity workshop (2024)
South African Journal of Science: Writing with integrity workshop (2024)
How to Manage Your Lost Opportunities in Odoo 17 CRM
How to Manage Your Lost Opportunities in Odoo 17 CRM
Azure Interview Questions and Answers PDF By ScholarHat
Azure Interview Questions and Answers PDF By ScholarHat
Assessment and Planning in Educational technology.pptx
Assessment and Planning in Educational technology.pptx
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Ib maths sl product and quotient rules
- 2. Product and quotient rules By the end of the lesson you will be able to: • Use the product and quotient rules to find the derivative of functions.
- 3. If f (x) = u(x). v(x) then f '(x)= u'(x).v(x)+u(x).v'(x) simpler: (uv)' = u'v+uv' The product rule
- 5. Find if:
- 6. Find if:
- 7. Find if:
- 8. Find if:
- 10. The quotient rule If then: simpler: 2
- 11. Find if:
- 12. Find if:
- 13. Find if:
- 14. Find if:
- 15. Differentiate with respect to x:
- 16. Find if:
- 17. Find if: