Learn to Sketch Curves using Calculus

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Build up a strong toolbox of techniques, including differentiation, to enable you to sketch a range of functions.

122 students enrolled

What Will I 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.

Who is the target audience?

- 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).

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Curriculum For This Course

136 Lectures

13:28:25
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Introductions
5 Lectures
10:32

This quiz will go through several different algebra skills that you will want to be good at before continuing with this course. If you find the quiz too challenging, this course will likely not be at the right level for you at the moment.

Algebra Skills Practice

8 questions

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Transformations
9 Lectures
26:46

Introducing Function Notation

03:35

EXERCISE: Using Function Notation

03:58

Introducing the "Using Function Notation for Substitution" Quiz

00:17

Using Function Notation for Substitution

5 questions

Introducing Transformations

00:36

Introducing Translations

03:41

Introducing Stretches in the x and y-direction

04:28

Introducing Reflections in the x and y-axes

04:10

EXERCISE: Describing Transformations

05:43

Introducing the "Transformations" Quiz

00:18

Transformations

8 questions

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Introducing Differentiation
25 Lectures
01:36:04

Differentiation: before we begin...

01:51

Introducing Differentiation

04:58

Differentiating Linear and Constant Terms

02:01

Differentiating ax^n

03:07

Differentiating Polynomials

01:28

EXERCISE: Differentiating Polynomials

03:26

Introducing the "Differentiating Polynomials" Quiz

00:23

Differentiating Polynomials

8 questions

Differentiating sin(x) and cos(x)

02:15

EXERCISE: Differentiating sin(x) and cos(x)

02:22

Differentiating exp(x) and ln(x)

01:54

EXERCISE: Differentiating exp(x) and ln(x)

02:55

Introducing the "Differentiation so far" Quiz

00:23

Differentiation so far

5 questions

Introducing the Chain Rule

05:10

Basic Examples of using The Chain Rule

10:14

More Examples of using the Chain Rule

06:03

EXERCISE: The Chain Rule

07:05

Introducing The Product Rule

02:19

Examples of using the Product Rule

04:53

Examples of using the Product Rule with the Chain Rule

04:34

EXERCISE: The Product Rule

07:08

Introducing The Quotient Rule

02:43

Examples of using the Quotient Rule

06:11

Examples of using the Quotient Rule with the Chain Rule

03:51

EXERCISE: The Quotient Rule

07:54

Introducing the "Identifying which method to use" Quiz

00:56

Identifying which method to use: Chain, Product or Quotient Rule?

5 questions

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Using Differentiation
18 Lectures
01:59:28

Introducing Stationary Points

03:42

An Example of finding Stationary Points for a Polynomial

04:06

EXERCISE 1: Stationary Points

07:06

Examples of finding Stationary Points using the Chain Rule

10:44

EXERCISE 2: Stationary Points

09:16

An Example of finding Stationary Points using the Product Rule

03:53

EXERCISE 3: Stationary Points

10:47

An Example of finding Stationary Points using the Quotient Rule

03:19

EXERCISE 4: Stationary Points

07:12

Introducing the Second Derivative

06:15

Examples of finding the Second Derivative

08:24

EXERCISE: Finding the Second Derivative

07:10

Local Minimums and Local Maximums

00:27

Example of determining the Type of Stationary Point

08:26

EXERCISE 1: Finding and Determining Types of Stationary Points

08:12

EXERCISE 2: Finding and Determining Types of Stationary Points

05:59

EXERCISE 3: Finding and Determining Types of Stationary Points

08:44

EXERCISE 4: Finding and Determining Types of Stationary Points

05:46

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Sketching Linear Graphs
7 Lectures
21:21

Introducing Sketching Linear Graphs

00:39

Some important straight lines we need to know

02:18

Transformations and the line y = x

03:18

Finding where a Linear Graph crosses the coordinate axes

03:09

Examples of Sketching Linear Graphs

04:53

EXERCISE: Sketching Linear Graphs

06:44

Introducing the "Linear Graphs" Quiz

00:20

Linear Graphs

8 questions

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Sketching Quadratic Graphs
10 Lectures
53:48

Introducing Sketching Quadratic Graphs

02:21

Methods for Factorising Quadratics

04:29

Using the Quadratic Formula

05:36

Using the Discriminant

02:43

Transformations of y = x^2

05:24

Completing the Square

04:56

An Alternative Method for Finding the Vertex of a Parabola

02:53

Examples of Sketching Quadratic Graphs

13:20

EXERCISE: Sketching Quadratic Graphs

11:46

Introducing the "Quadratic Graphs" Quiz

00:20

Quadratic Graphs

8 questions

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Sketching Cubics and Higher Polynomials
14 Lectures
01:36:37

Introducing Sketching Cubics and Higher Polynomials

03:04

Transformations of y = x^3

04:26

Shapes of Cubics and Higher Polynomials

05:37

Introducing the Remainder Theorem and the Factor Theorem

02:35

Using the Remainder Theorem and the Factor Theorem

08:10

Polynomial Division Method 1

06:50

Polynomial Division Method 2

04:06

Examples of Sketching Cubic Graphs

11:29

EXERCISE: Sketching Cubic Graphs

14:15

An Example of Sketching a Higher Polynomial

09:29

EXERCISE: Sketching Higher Polynomials Part 1

07:30

EXERCISE: Sketching Higher Polynomials Part 2

10:03

EXERCISE: Sketching Higher Polynomials Part 3

08:53

Introducing the "Cubics and Higher Polynomials" Quiz

00:10

Cubics and Higher Polynomials

8 questions

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Sketching Rational Functions
13 Lectures
02:15:14

Introducing Asymptotes

09:02

Translating y = 1/x

05:30

Examples of a Higher-order Polynomial in the Denominator

18:11

EXERCISE 1: Sketching Rational Functions

15:31

Examples of Same-ordered Polynomial in both the Numerator and Denominator

18:30

EXERCISE 2: Sketching Rational Functions

11:52

Further Polynomial Division

04:59

Examples of a Higher-ordered Polynomial in the Numerator

18:56

EXERCISE 3: Sketching Rational Functions Part 1

06:51

EXERCISE 3: Sketching Rational Functions Part 2

09:22

EXERCISE 3: Sketching Rational Functions Part 3

10:44

An Example of a Rational Function with no Vertical Asymptotes

05:35

Introducing the "Rational Functions" Quiz

00:11

Rational Functions

8 questions

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Sketching Trigonometric Functions
10 Lectures
39:36

The Unit Circle

05:10

Sketching y = sin(x)

02:25

Sketching y = cosec(x)

02:14

Sketching y = cos(x)

02:06

Sketching y= sec(x)

01:33

Sketching y = tan(x)

02:21

Sketching y = cot(x)

02:47

Examples of Sketching Transformations of Trigonometric Functions

11:09

EXERCISE: Sketching Transformations of Trigonometric Functions

09:32

Introducing the "Trigonometric Functions" Quiz

00:19

Trigonometric Functions

5 questions

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Sketching Exponential and Logarithmic Functions
10 Lectures
01:23:26

Introducing Exponentials and Logarithms

06:07

Introducing the Exponential and Logarithmic Functions

07:04

The Laws of Logarithms

03:15

Solving equations involving the Exponential Function

08:33

Solving equations involving the Logarithmic Function

05:47

Examples of Sketching Exponential Functions

13:37

EXERCISE: Sketching Exponential Functions

18:20

Examples of Sketching Logarithmic Functions

09:03

EXERCISE: Sketching Logarithmic Functions

11:19

Introducing the "Exponential and Logarithmic Functions" Quiz

00:21

Exponential and Logarithmic Functions

8 questions

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