Learn To Sketch Curves Using Calculus
Last updated 7/2015
MP4 | Video: h264, 1280x720 | Audio: AAC, 44.1 KHz
Language: English | Size: 27.34 GB | Duration: 13h 28m
Last updated 7/2015
MP4 | Video: h264, 1280x720 | Audio: AAC, 44.1 KHz
Language: English | Size: 27.34 GB | Duration: 13h 28m
Build up a strong toolbox of techniques, including differentiation, to enable you to sketch a range of functions.
What you'll learn
Use Function Notation
Recognise Transformations of Functions
Differentiate Polynomials
Differentiate Trigonometric Functions
Differentiate Exponential and Logarithmic Functions
Differentiate using the Chain Rule
Differentiate using the Product Rule
Differentiate using the Quotient Rule
Use Differentiation to find Stationary Points
Sketch Linear Graphs
Sketch Quadratic Graphs
Sketch Cubics and Higher Polynomials
Sketch Rational Functions
Sketch Trigonometric Functions
Sketch Exponential and Logarithmic Functions
Sketch Modulus Functions
Requirements
You should be able to solve linear equations.
You should be able to expand single and double brackets.
You should be able to factorise into single and (some) double brackets.
You should have met and understand y = mx + c.
It would help if you have met the quadratic formula before (not essential).
It would help if you have met completing the square before (not essential).
Description
Curve Sketching is an incredibly useful tool in mathematical problem solving, as well as an opportunity to improve and test your algebraic understanding.As you study mathematics through school and college to degree level, your algebraic skills will be increasingly tested. In order to become a strong mathematician, you need to understand what the algebra is telling you. Curve Sketching is often examinable and can be a challenging topic to master due to the multitude of techniques that need to be learnt. This course is here to help.Computer programs like Autograph, Desmos, Maple and Matlab can all plot curves, but understanding why a curve behaves the way it does relies on your understanding of algebra and calculus. Using techniques that we will learn on this course, you will be able to successfully sketch complicated functions and learn about the behaviour of different graphs.The course is structured so that you will learn about Graph Transformations and Differentiation and its uses in the initial sections. You will not need to have met these concepts before. I go through Differentiation from its basics, through the derivatives of different functions, and up to the Chain Rule, Product Rule and Quotient Rule.We then start Sketching, and within this we will learn many different techniques along the way.Linear Graphs:Find where the graph crosses the coordinate axes.Learn how to deal with different forms of Linear equations.Quadratic Graphs: Learn methods of Factorising.Learn how to use the Quadratic Formula.Learn about the Discriminant and what it tells us.Learn how to Complete the Square.Cubics and Higher Polynomials:Learn about the Remainder Theorem and the Factor Theorem.Learn how to perform and use Polynomial Division.Rational Functions:Learn about Asymptotes and how to determine how each section of the graph behaves.Learn how to determine how a graph behaves for large positive or negative values of x.Trigonometric Functions:Learn about sin(x), cos(x) and tan(x) from the Unit Circle.Learn how to sketch cosec(x), sec(x) and cot(x).Learn how to sketch transformations of each trigonometric curve.Exponential and Logarithmic Functions:Learn about e^x and be introduced to Logarithms.Learn about the Laws of Logarithms.Learn how to solve equations involving Exponentials and Logarithms.Modulus Functions:Learn about |x| and how to sketch a host of graphs involving the Modulus Function. Learn about the difference between y = |f(x)| and y = f(|x|).Each of these sections is introduced from scratch and involves several worked examples and exercises for you to complete. There are several quizzes to try along the way to test your understanding, and if there are any problems please do not hesitate to start a discussion and ask for help. With over 100 lectures and 13 hours of content, this course is perfect for anybody studying a Calculus course or A-Level Mathematics, or for those wanting to test and improve their mathematical ability before tackling a Mathematics-related undergraduate degree course at university.
Overview
Section 1: Introductions
Lecture 1 Who is this course for?
Lecture 2 What will be covered on this course and what do I need to know?
Lecture 3 How will my progress be assessed?
Lecture 4 Sketching vs Plotting - what is the difference?
Lecture 5 Introducing the "Algebra Skills Practice" Quiz
Section 2: Transformations
Lecture 6 Introducing Function Notation
Lecture 7 EXERCISE: Using Function Notation
Lecture 8 Introducing the "Using Function Notation for Substitution" Quiz
Lecture 9 Introducing Transformations
Lecture 10 Introducing Translations
Lecture 11 Introducing Stretches in the x and y-direction
Lecture 12 Introducing Reflections in the x and y-axes
Lecture 13 EXERCISE: Describing Transformations
Lecture 14 Introducing the "Transformations" Quiz
Section 3: Introducing Differentiation
Lecture 15 Differentiation: before we begin…
Lecture 16 Introducing Differentiation
Lecture 17 Differentiating Linear and Constant Terms
Lecture 18 Differentiating ax^n
Lecture 19 Differentiating Polynomials
Lecture 20 EXERCISE: Differentiating Polynomials
Lecture 21 Introducing the "Differentiating Polynomials" Quiz
Lecture 22 Differentiating sin(x) and cos(x)
Lecture 23 EXERCISE: Differentiating sin(x) and cos(x)
Lecture 24 Differentiating exp(x) and ln(x)
Lecture 25 EXERCISE: Differentiating exp(x) and ln(x)
Lecture 26 Introducing the "Differentiation so far" Quiz
Lecture 27 Introducing the Chain Rule
Lecture 28 Basic Examples of using The Chain Rule
Lecture 29 More Examples of using the Chain Rule
Lecture 30 EXERCISE: The Chain Rule
Lecture 31 Introducing The Product Rule
Lecture 32 Examples of using the Product Rule
Lecture 33 Examples of using the Product Rule with the Chain Rule
Lecture 34 EXERCISE: The Product Rule
Lecture 35 Introducing The Quotient Rule
Lecture 36 Examples of using the Quotient Rule
Lecture 37 Examples of using the Quotient Rule with the Chain Rule
Lecture 38 EXERCISE: The Quotient Rule
Lecture 39 Introducing the "Identifying which method to use" Quiz
Section 4: Using Differentiation
Lecture 40 Introducing Stationary Points
Lecture 41 An Example of finding Stationary Points for a Polynomial
Lecture 42 EXERCISE 1: Stationary Points
Lecture 43 Examples of finding Stationary Points using the Chain Rule
Lecture 44 EXERCISE 2: Stationary Points
Lecture 45 An Example of finding Stationary Points using the Product Rule
Lecture 46 EXERCISE 3: Stationary Points
Lecture 47 An Example of finding Stationary Points using the Quotient Rule
Lecture 48 EXERCISE 4: Stationary Points
Lecture 49 Introducing the Second Derivative
Lecture 50 Examples of finding the Second Derivative
Lecture 51 EXERCISE: Finding the Second Derivative
Lecture 52 Local Minimums and Local Maximums
Lecture 53 Example of determining the Type of Stationary Point
Lecture 54 EXERCISE 1: Finding and Determining Types of Stationary Points
Lecture 55 EXERCISE 2: Finding and Determining Types of Stationary Points
Lecture 56 EXERCISE 3: Finding and Determining Types of Stationary Points
Lecture 57 EXERCISE 4: Finding and Determining Types of Stationary Points
Section 5: Sketching Linear Graphs
Lecture 58 Introducing Sketching Linear Graphs
Lecture 59 Some important straight lines we need to know
Lecture 60 Transformations and the line y = x
Lecture 61 Finding where a Linear Graph crosses the coordinate axes
Lecture 62 Examples of Sketching Linear Graphs
Lecture 63 EXERCISE: Sketching Linear Graphs
Lecture 64 Introducing the "Linear Graphs" Quiz
Section 6: Sketching Quadratic Graphs
Lecture 65 Introducing Sketching Quadratic Graphs
Lecture 66 Methods for Factorising Quadratics
Lecture 67 Using the Quadratic Formula
Lecture 68 Using the Discriminant
Lecture 69 Transformations of y = x^2
Lecture 70 Completing the Square
Lecture 71 An Alternative Method for Finding the Vertex of a Parabola
Lecture 72 Examples of Sketching Quadratic Graphs
Lecture 73 EXERCISE: Sketching Quadratic Graphs
Lecture 74 Introducing the "Quadratic Graphs" Quiz
Section 7: Sketching Cubics and Higher Polynomials
Lecture 75 Introducing Sketching Cubics and Higher Polynomials
Lecture 76 Transformations of y = x^3
Lecture 77 Shapes of Cubics and Higher Polynomials
Lecture 78 Introducing the Remainder Theorem and the Factor Theorem
Lecture 79 Using the Remainder Theorem and the Factor Theorem
Lecture 80 Polynomial Division Method 1
Lecture 81 Polynomial Division Method 2
Lecture 82 Examples of Sketching Cubic Graphs
Lecture 83 EXERCISE: Sketching Cubic Graphs
Lecture 84 An Example of Sketching a Higher Polynomial
Lecture 85 EXERCISE: Sketching Higher Polynomials Part 1
Lecture 86 EXERCISE: Sketching Higher Polynomials Part 2
Lecture 87 EXERCISE: Sketching Higher Polynomials Part 3
Lecture 88 Introducing the "Cubics and Higher Polynomials" Quiz
Section 8: Sketching Rational Functions
Lecture 89 Introducing Asymptotes
Lecture 90 Translating y = 1/x
Lecture 91 Examples of a Higher-order Polynomial in the Denominator
Lecture 92 EXERCISE 1: Sketching Rational Functions
Lecture 93 Examples of Same-ordered Polynomial in both the Numerator and Denominator
Lecture 94 EXERCISE 2: Sketching Rational Functions
Lecture 95 Further Polynomial Division
Lecture 96 Examples of a Higher-ordered Polynomial in the Numerator
Lecture 97 EXERCISE 3: Sketching Rational Functions Part 1
Lecture 98 EXERCISE 3: Sketching Rational Functions Part 2
Lecture 99 EXERCISE 3: Sketching Rational Functions Part 3
Lecture 100 An Example of a Rational Function with no Vertical Asymptotes
Lecture 101 Introducing the "Rational Functions" Quiz
Section 9: Sketching Trigonometric Functions
Lecture 102 The Unit Circle
Lecture 103 Sketching y = sin(x)
Lecture 104 Sketching y = cosec(x)
Lecture 105 Sketching y = cos(x)
Lecture 106 Sketching y= sec(x)
Lecture 107 Sketching y = tan(x)
Lecture 108 Sketching y = cot(x)
Lecture 109 Examples of Sketching Transformations of Trigonometric Functions
Lecture 110 EXERCISE: Sketching Transformations of Trigonometric Functions
Lecture 111 Introducing the "Trigonometric Functions" Quiz
Section 10: Sketching Exponential and Logarithmic Functions
Lecture 112 Introducing Exponentials and Logarithms
Lecture 113 Introducing the Exponential and Logarithmic Functions
Lecture 114 The Laws of Logarithms
Lecture 115 Solving equations involving the Exponential Function
Lecture 116 Solving equations involving the Logarithmic Function
Lecture 117 Examples of Sketching Exponential Functions
Lecture 118 EXERCISE: Sketching Exponential Functions
Lecture 119 Examples of Sketching Logarithmic Functions
Lecture 120 EXERCISE: Sketching Logarithmic Functions
Lecture 121 Introducing the "Exponential and Logarithmic Functions" Quiz
Section 11: Sketching Modulus Functions
Lecture 122 Introducing the Modulus Function
Lecture 123 Sketching the Modulus of Linear Functions
Lecture 124 Sketching the Modulus of Quadratic Functions
Lecture 125 Sketching the Modulus of Cubics and Higher Polynomials
Lecture 126 Sketching the Modulus of Rational Functions
Lecture 127 Sketching the Modulus of Trigonometric Functions
Lecture 128 Sketching the Modulus of Exponential and Logarithmic Functions
Lecture 129 The Difference between |f(x)| and f(|x|)
Lecture 130 EXERCISE: Sketching Modulus Functions
Lecture 131 Introducing the "Modulus Functions" Quiz
Section 12: Sketching More Complicated Functions
Lecture 132 Sketching Linear Combinations of Functions
Lecture 133 Sketching Functions of Functions
Lecture 134 Sketching Products of Functions
Lecture 135 Sketching Quotients of Functions
Section 13: Summary and Conclusions
Lecture 136 Summary and Conclusions
This course in Curve Sketching is designed for students currently studying A-Level Maths or A-Level Further Maths (or at an equivalent level, roughly post-16 education), or for those going on to study a first year degree course with a mathematics element.,It is perfect as a refresher course, and can also be used as a self-study course for those having studied Higher GCSE Maths and gained at least a grade A.,This course is not suitable to those without a relatively strong algebraic background - techniques like using the quadratic formula and completing the square will be covered, but it is expected that you will have met a lot of the basic processes at GCSE (or equivalent).