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Learn to Sketch Curves using Calculus

Build up a strong toolbox of techniques, including differentiation, to enable you to sketch a range of functions.
Instructor:
Jack Brown
229 students enrolled
English [Auto]
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

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.

Introductions

1
Who is this course for?
2
What will be covered on this course and what do I need to know?
3
How will my progress be assessed?
4
Sketching vs Plotting - what is the difference?
5
Introducing the "Algebra Skills Practice" Quiz
6
Algebra Skills Practice

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.

Transformations

1
Introducing Function Notation
2
EXERCISE: Using Function Notation
3
Introducing the "Using Function Notation for Substitution" Quiz
4
Using Function Notation for Substitution
5
Introducing Transformations
6
Introducing Translations
7
Introducing Stretches in the x and y-direction
8
Introducing Reflections in the x and y-axes
9
EXERCISE: Describing Transformations
10
Introducing the "Transformations" Quiz
11
Transformations

Introducing Differentiation

1
Differentiation: before we begin...
2
Introducing Differentiation
3
Differentiating Linear and Constant Terms
4
Differentiating ax^n
5
Differentiating Polynomials
6
EXERCISE: Differentiating Polynomials
7
Introducing the "Differentiating Polynomials" Quiz
8
Differentiating Polynomials
9
Differentiating sin(x) and cos(x)
10
EXERCISE: Differentiating sin(x) and cos(x)
11
Differentiating exp(x) and ln(x)
12
EXERCISE: Differentiating exp(x) and ln(x)
13
Introducing the "Differentiation so far" Quiz
14
Differentiation so far
15
Introducing the Chain Rule
16
Basic Examples of using The Chain Rule
17
More Examples of using the Chain Rule
18
EXERCISE: The Chain Rule
19
Introducing The Product Rule
20
Examples of using the Product Rule
21
Examples of using the Product Rule with the Chain Rule
22
EXERCISE: The Product Rule
23
Introducing The Quotient Rule
24
Examples of using the Quotient Rule
25
Examples of using the Quotient Rule with the Chain Rule
26
EXERCISE: The Quotient Rule
27
Introducing the "Identifying which method to use" Quiz
28
Identifying which method to use: Chain, Product or Quotient Rule?

Using Differentiation

1
Introducing Stationary Points
2
An Example of finding Stationary Points for a Polynomial
3
EXERCISE 1: Stationary Points
4
Examples of finding Stationary Points using the Chain Rule
5
EXERCISE 2: Stationary Points
6
An Example of finding Stationary Points using the Product Rule
7
EXERCISE 3: Stationary Points
8
An Example of finding Stationary Points using the Quotient Rule
9
EXERCISE 4: Stationary Points
10
Introducing the Second Derivative
11
Examples of finding the Second Derivative
12
EXERCISE: Finding the Second Derivative
13
Local Minimums and Local Maximums
14
Example of determining the Type of Stationary Point
15
EXERCISE 1: Finding and Determining Types of Stationary Points
16
EXERCISE 2: Finding and Determining Types of Stationary Points
17
EXERCISE 3: Finding and Determining Types of Stationary Points
18
EXERCISE 4: Finding and Determining Types of Stationary Points

Sketching Linear Graphs

1
Introducing Sketching Linear Graphs
2
Some important straight lines we need to know
3
Transformations and the line y = x
4
Finding where a Linear Graph crosses the coordinate axes
5
Examples of Sketching Linear Graphs
6
EXERCISE: Sketching Linear Graphs
7
Introducing the "Linear Graphs" Quiz
8
Linear Graphs

Sketching Quadratic Graphs

1
Introducing Sketching Quadratic Graphs
2
Methods for Factorising Quadratics
3
Using the Quadratic Formula
4
Using the Discriminant
5
Transformations of y = x^2
6
Completing the Square
7
An Alternative Method for Finding the Vertex of a Parabola
8
Examples of Sketching Quadratic Graphs
9
EXERCISE: Sketching Quadratic Graphs
10
Introducing the "Quadratic Graphs" Quiz
11
Quadratic Graphs

Sketching Cubics and Higher Polynomials

1
Introducing Sketching Cubics and Higher Polynomials
2
Transformations of y = x^3
3
Shapes of Cubics and Higher Polynomials
4
Introducing the Remainder Theorem and the Factor Theorem
5
Using the Remainder Theorem and the Factor Theorem
6
Polynomial Division Method 1
7
Polynomial Division Method 2
8
Examples of Sketching Cubic Graphs
9
EXERCISE: Sketching Cubic Graphs
10
An Example of Sketching a Higher Polynomial
11
EXERCISE: Sketching Higher Polynomials Part 1
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