IB Maths. Turning points. First derivative test
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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.
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- 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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- 3. • • • • A B P Q At A ⇒ f '(x) > 0 ⇒ f is increasing At P ⇒ f '(x) < 0 ⇒ f is decreasing At B and Q ⇒ f '(x) = 0 ⇒ B and Q are stationary points. f ' = 0 f ' < 0 f ' > 0 f ' > 0 f ' = 0
- 4. If the derivative is positive then the function is increasing. If the derivative is negative then the function is decreasing.
- 5. f '( a ) = 0 ⇒ (a, f(a)) is a stationary point • • • • A B P Q A point on a curve at which the gradient is zero is called a stationary point. At a stationary point, the tangent to the curve is horizontal.
- 6. • • • Local Maximum point f ' > 0 f ' = 0 f ' < 0 P To the left of P At point P To the right of P f ' > 0 f ' < 0 f ' = 0 P is a local maximum point
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- 8. Point of inflexion • • • f ' =0 f ' > 0 f ' > 0P • • • f ' = 0 f ' < 0 f ' < 0 P f ' ( a) = 0 but f ' has the same sign to the right and left of a, a is called a horizontal point of inflexion.(because the tangent is horizontal at P) Point of inflection.ggb
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- 11. y= x3 +3x2 +1
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- 13. y= x4 4 x3