Learn To Sketch Curves Using Calculus

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).