IB Maths . Gradient at a point. Tangent and normal lines
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This document provides instruction on calculating the gradient of a curve at a point and finding the equation of the tangent line to a curve at a point. It includes examples of finding the gradient at specific points on curves defined by equations and finding the coordinates of points where a curve has a horizontal tangent. It also discusses using calculators or software to find gradients and draw tangents and normals to curves at given points.
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One of the central tasks in computational mathematics and statistics is to accurately approximate unknown target functions. This is typically done with the help of data — samples of the unknown functions. The emergence of Big Data presents both opportunities and challenges. On one hand, big data introduces more information about the unknowns and, in principle, allows us to create more accurate models. On the other hand, data storage and processing become highly challenging. In this talk, we present a set of sequential algorithms for function approximation in high dimensions with large data sets. The algorithms are of iterative nature and involve only vector operations. They use one data sample at each step and can handle dynamic/stream data. We present both the numerical algorithms, which are easy to implement, as well as rigorous analysis for their theoretical foundation.Class xii assignment chapters (1 to 9)
1. Find the equation of tangent to the curve x = sin 3t, y = cos 2t at t = π/4.
2. Solve the differential equation: cos2x dy/dx + y = tan x.
3. Using integration, find the area of the region bounded by the parabola y2 = 4x and the circle 4x2 + 4y2 = 9.
4. Evaluate several integrals using properties of integrals, derivatives, and determinants.
Equations of tangents
This document discusses how to find the equation of a tangent line to a curve at a given point. It explains that you need to find the derivative to get the slope (m) of the tangent line, use the original equation to find the y-coordinate (b) of the point, and then plug those values into the equation for a line, y - b = m(x - a), to get the equation of the tangent line. It provides examples finding the derivatives, slopes, points and tangent equations for curves like y = x^2, y = x^3 - 2x^2 + 5, and y = x^3 - 7x - 3 at given x-values.
PRESENT.pptx this paper will help the next
This document discusses the fixed point iteration method for finding approximate solutions to nonlinear equations. It begins with an introduction to roots of nonlinear equations and converting them to a fixed point problem. It then presents the fixed point iteration method theorem and algorithm. Examples are provided to demonstrate applying the method to find the roots of equations up to four decimal places. The document concludes by noting the method can be implemented using loops in code and is useful for finding real roots expressed as infinite series.
Further pure mathmatics 3 vectors
This document provides an overview of key concepts in vectors and vector algebra covered in an Edexcel FP3 module. It begins with an introduction to vectors and quaternions. Section 1 defines vectors and the vector product. Section 2 defines the triple scalar product and its use in calculating the volume of a tetrahedron. It discusses properties of the vector product, including that it yields a vector perpendicular to the two original vectors and has magnitude equal to the product of the lengths of the original vectors and the sine of the angle between them. It provides examples of using the vector and triple scalar products to calculate the area of a parallelogram and the volume of a tetrahedron.
Applications of Differential Calculus in real life
Applications of Differentiation in economics, optimization, kinematics, tangent and normal lines, related rates, linear approximation or small changes
Mathematics
1. The document provides a sample paper for mathematics class 12 with questions covering various topics like vectors, matrices, integrals, probability, linear programming, etc. divided into three sections - short 1 mark questions, 4 mark questions and 6 mark questions.
2. It also includes the marking scheme with answers for all the questions.
3. The questions aim to test different cognitive levels from remembering to applications and higher order thinking skills.
Mathematics
1. The document provides a sample paper for mathematics class 12 with questions covering various topics like vectors, matrices, integrals, probability, linear programming, etc. divided into three sections - short 1 mark questions, 4 mark questions and 6 mark questions.
2. It also includes the marking scheme with answers for all the questions.
3. The questions assess different cognitive levels like remembering, understanding, application and higher order thinking skills.
Cbse 12 Class Maths Sample Paper
In pursuit of excellence, the CBSE board conducts a thorough research on emerging educational requirements. While designing the syllabus, the board ensures that every topic meets the learning needs of students in the best possible manner. CBSE Class 12 Maths - http://cbse.edurite.com/cbse-maths/cbse-class-12-maths.html
Rate of change and tangent lines
The document discusses key concepts related to rates of change:
- Average rate of change is defined as the change in y over the change in x. Instantaneous rate of change is the limit of this ratio as the change approaches zero.
- Average velocity is the total distance traveled divided by the total time. Instantaneous velocity is the limit of the change in position over the change in time as the time interval approaches zero.
- The slope of a curve at a point equals the slope of the tangent line at that point, which can be determined by taking the limit of the difference quotient as the change approaches zero.
CLASS X MATHS Coordinate geometry
This document provides an overview of coordinate geometry. It defines key concepts like the Cartesian coordinate system, quadrants, and using an ordered pair (x,y) to locate points on a plane. It then explains how to calculate the distance between two points using the distance formula. Other topics covered include finding the area of a triangle using the coordinates of its vertices, using the section formula to divide a line segment internally, and finding the midpoint of two points.
Solutions of AHSEC Mathematics Paper 2015
1) The document provides solutions to 10 math questions involving relations, trigonometric functions, matrices, integrals, and derivatives.
2) It proves that if a function f(x) is differentiable at a point c, then it is continuous at c. The proof uses the definition of the derivative to show the limit definition of continuity is satisfied.
3) One integral evaluated is the integral from 0 to 1/3 of (x-x3)/(1/3)x4 dx, which is solved by a u-substitution of u = 1/x2 - 1.
Parabola complete
This summary combines slides from Melanie Tomlinson and Morrobea on the topic of parabolas. The key points covered include:
- The geometric definition of a parabola as the set of all points equidistant from a fixed point (the focus) and fixed line (the directrix).
- Parabolas can be represented using various equation forms including vertex form, standard form, and general form.
- Methods for graphing parabolas by identifying features like the vertex, axis of symmetry, x-intercepts, focus, and directrix.
- Applications of parabolas to model real-world situations like searchlights and radio telescopes.
Solving quadratic equations[1]
The document discusses various methods for solving quadratic equations: factoring, using square roots, and completing the square. It provides examples of using each method to solve equations arising from applications like finding the dimensions of a rectangle given its area. Sample problems are worked through step-by-step to find the solutions or roots of quadratic equations.
Solving quadratic equations
The document discusses various methods for solving quadratic equations: factoring, using square roots, and completing the square. It provides examples of using each method to solve equations as well as word problems involving quadratic equations. Factoring involves writing the quadratic as a product of two binomials and setting each factor equal to 0. Using square roots involves taking the square root of both sides of the equation to isolate the variable. Completing the square rewrites the equation by adding or subtracting terms to put it in the form (x+a)2=b and then taking the square root of both sides.
D1_YT_English_VEctor_Calculus_Gradient_&_Directional_Derivative.pptx
This document provides an overview of vector calculus concepts including gradient, directional derivative, dot product, and cross product. It includes examples of calculating directional derivatives of multivariate functions at given points in specified directions. It also provides practice problems testing the concepts covered, such as finding the gradient of a function, or the directional derivative of a function at a point along a given direction.
dist- midpt - pythag PP.ppt
This document discusses formulas for distance, midpoint, and the Pythagorean theorem. It provides the formulas to find the distance between two points, the midpoint of a line segment, and the length of the hypotenuse of a right triangle. It includes examples of using the formulas to find distances, midpoints, and missing lengths in right triangles. It assigns homework problems involving using these formulas to solve geometric problems.
Similar to IB Maths . Gradient at a point. Tangent and normal lines (20)
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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).
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Y9 algebra 1 Simultaneous Equations byElimination Method
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