Learn to Sketch Curves using Calculus
4.7 (12 ratings)
200 students enrolled

# Learn to Sketch Curves using Calculus

Build up a strong toolbox of techniques, including differentiation, to enable you to sketch a range of functions.
4.7 (12 ratings)
200 students enrolled
Created by Jack Brown
Last updated 7/2015
English
English [Auto-generated]
Current price: \$9.99 Original price: \$34.99 Discount: 71% off
30-Day Money-Back Guarantee
This course includes
• 13.5 hours on-demand video
• Full lifetime access
• Access on mobile and TV
• Certificate of Completion
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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.

Who this course is for:
• 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).
Course content
Expand all 136 lectures 13:28:25
+ Introductions
5 lectures 10:32
Preview 01:16

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
+ 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
+ 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
+ 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
+ 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
+ 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
+ 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
+ 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
+ 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
+ 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