VCE Maths Methods Units 1-4: Calculus
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# VCE Maths Methods Units 1-4: Calculus

Includes differentiation and antidifferentiation (integration)
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Created by Aaron Ng
Last updated 1/2017
English
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Includes:
• 4 hours on-demand video
• Access on mobile and TV
• Certificate of Completion
What Will I Learn?
• Understand what limits are and evaluate limits [as an introduction to calculus]
• Differentiate using first principles
• Differentiate the following functions: x^n, e^x, loge(x), sin(x), cos(x), tan(x)
• Differentiate using the chain rule, product rule and quotient rule
• Calculate the average and instantaneous rate of change of a function
• Find the equation of a tangent and normal line
• Find the maximum and minimum of a function
• Apply the concepts on differentiation on worded problems (including maximum and minimum problems, rate of change problems and motion graphs)
• Sketch the derivative and antiderivative of a given graph
• Antidifferentiate the following functions: x^n, e^x, 1/x, sin(x), cos(x)
• Integrate by recognition
• Evaluate definite integrals
• Calculate the approximate and exact area beneath a graph and between two graphs
• Calculate the average value of a function for a specified domain
• Apply the concepts on antidifferentiation on worded problems (including rate of change problems and motion graphs)
View Curriculum
Requirements
• Be able to sketch the graphs of the basic functions, including linear, quadratic, cubic, hyperbola, truncus, square root, exponential, logarithmic, sine, cosine and tangent graphs.
• Be able to solve equations algebraically, including polynomial, exponential, logarithmic and trigonometric equations.
• Be familiar with the formulae to calculate the area of basic 2D shapes (e.g. triangle), and the volume and total surface area of basic 3D shapes (e.g. sphere). If unsure, please refer to the formula sheet on the VCAA website: http://www.vcaa.vic.edu.au/.
• NOTE: The course requirements for unit 1/2 students may not be as extensive. For instance, they are generally only required to sketch the graphs of polynomial functions, and solve polynomial equations. However, they should be familiar with the formulae for the areas and volumes.
Description

After going through this course, you will be able to understand how calculus (differentiation and antidifferentiation/integration) works at an Australian VCE Maths Methods Units 1-4 level, and apply such knowledge on exam questions. Each lecture includes many clearly annotated diagrams to make mathematical concepts easier to understand, and will be followed by a quiz to test your understanding.

The lectures are designed to cater for both unit 1/2 students and unit 3/4 students, with unit 1/2 and unit 3/4 content indicated in the ‘lecture description’ and the beginning of each lecture. Unit 1/2 students only need to watch the unit 1/2 content of each lecture, although you may go on to watch the unit 3/4 content if you want to get a head start. Unit 3/4 students may find the unit 1/2 content a good revision for them. You are encouraged to go through the lectures in order since the content from the earlier lectures is often required in the later lectures.

Who is the target audience?
• This course mainly caters for students doing the Australian VCE subject, Mathematical Methods unit 1/2 and 3/4.
• However, students from other countries or Australian states at a level equivalent to year 11 and 12 in Australia may also find this course useful.
• This course will clearly outline the content from unit 1/2 and unit 3/4, although sometimes there is a slight overlap between the two. For unit 1/2 students, you can also watch the unit 3/4 content to get a head start for the future; while for unit 3/4 students, you can also watch the unit 1/2 content if you need some revision on the previously learnt material.
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Curriculum For This Course
23 Lectures
04:06:35
+
Introduction to calculus - Limits
2 Lectures 17:38

This lecture provides an introduction to the course.

Preview 03:12

Content: All covered in unit 1/2

- What are limits?

- When do limits exist?

- Theorem on limits

- Calculations of limits

Introduction to calculus - Limits
14:26

Introduction to calculus - Limits
3 questions
+
Differentiation
11 Lectures 02:07:31

Content: All covered in unit 1/2

- Introduction to differentiation (including notation)

- The derivative of a function using first principles

- Examples

Differentiation using first principles
06:03

Differentiation using first principles
2 questions

The content in this lecture is covered in unit 1/2. You will learn how to differentiate x^n, and the more complicated form of this rule which is (ax+b)^n.

Preview 11:22

Differentiation by rule – x^n
3 questions

The content in this lecture is covered in unit 3/4. You will learn how to differentiate e^x, log(x), sin(x), cos(x) and tan(x).

Differentiation by rule – e^x, log(x), sin(x), cos(x), tan(x)
08:11

Differentiation by rule – e^x, log(x), sin(x), cos(x), tan(x)
2 questions

The content in this lecture is covered in unit 1/2. You will learn how to find the average rate of change and (instantaneous) rate of change of a graph. Take note that although the concept is taught in unit 1/2, the second worked example provided in the lecture is suitable only for unit 3/4 students since it assumes knowledge from the lecture entitled “Differentiation by rule – e^x, log(x), sin(x), cos(x), tan(x)”. This means that unit 1/2 students should only watch the first 4 minutes and 52 seconds of the lecture, while unit 3/4 students can watch the entire lecture.

Average vs. instantaneous rate of change
07:56

Average vs. instantaneous rate of change
4 questions

Content: All covered in unit 3/4

- When do we use chain rule?

- What is chain rule?

• Long method
• Short method (recommended)

- More examples

Differentiation by rule – chain rule
11:55

Differentiation by rule – chain rule
3 questions

Content: All covered in unit 3/4

- When do we use product and quotient rule?

- What is product rule and quotient rule?

- Mixed examples (including chain rule, product rule and quotient rule)

Differentiation by rule – product rule and quotient rule
11:40

Differentiation by rule – product rule and quotient rule
2 questions

Content: All covered in unit 1/2

- Features of a derivative graph

- Sketching the derivative graph of a polynomial function

- Determining the domain of a derivative graph

- Sketching the derivative graph of a hybrid function

Sketching the derivative graph of a function
12:35

Sketching the derivative graph of a function
2 questions

The content in this lecture is covered in unit 1/2. You will learn how to find the equation of the tangent and normal line of a graph at a particular point. Take note that although the concept is taught in unit 1/2, the second worked example provided in the lecture is suitable only for unit 3/4 students since it assumes knowledge from a maths methods unit 3/4 level. This means that unit 1/2 students should only watch the first 8 minutes and 36 seconds of the lecture, while unit 3/4 students can watch the entire lecture.

Preview 11:53

Equation of the tangent and normal line
3 questions

Content: All covered in unit 1/2

- Finding the stationary points of a function

- Sign test: Verifying the nature of a stationary point

- More complicated examples (recommended for unit 3/4 students to watch as well! – from 8 minutes and 32 seconds onwards)

Finding the maximum and minimum of a function
14:01

Finding the maximum and minimum of a function
3 questions

This lecture goes through three worded problems, which involve deriving a function and using calculus to find the maximum or minimum value of the function. The main focus in this lecture is on how to derive the function from the worded problem. Examples one and two are suitable for unit 1/2 students, while example three is suitable for unit 3/4 students (although unit 1/2 students are encouraged to watch example three as well to improve their problem solving skills).

Applications (maximum and minimum problems)
17:15

Applications (maximum and minimum problems)
2 questions

This lecture goes through two worded problems, with example one relating to motion graphs (unit 1/2) and example two relating to rate of change problems (unit 3/4). Just like in the previous lecture, the main focus in this lecture is on how to derive the function from the worded problem.

Applications (motion graphs and rate of change problems)
14:40

Applications (motion graphs and rate of change problems)
4 questions
+
Antidifferentiation (integration)
10 Lectures 01:41:26

The content in this lecture is covered in unit 1/2. You will learn how to antidifferentiate x^n, and the more complicated form of this rule which is (ax+b)^n. You will also learn how to find the value of the constant ‘c’ in the antidifferentiated equation given the additional information provided.

Antidifferentiation by rule – x^n
13:40

Antidifferentiation by rule – x^n
4 questions

The content in this lecture is covered in unit 3/4. You will learn how to antidifferentiate e^x, 1/x, sin(x) and cos(x).

Antidifferentiation by rule – e^x, 1/x, sin(x), cos(x)
09:00

Antidifferentiation by rule – e^x, 1/x, sin(x), cos(x)
2 questions

Content: All covered in unit 3/4

- Features of an antiderivative graph

- Sketching the antiderivative graph of a polynomial function

Sketching the antiderivative graph of a function
07:58

Sketching the antiderivative graph of a function
2 questions

The concept of integration by recognition is covered in unit 3/4.

Integration by recognition
08:17

Integration by recognition
2 questions

Content: All covered in unit 3/4

- Indefinite integrals vs. definite integrals

- Evaluating definite integrals

- Properties of definite integrals

- More complicated examples

Definite integrals
11:51

Definite integrals
3 questions

This lecture is covered in unit 1/2, and covers the left-rectangle and right-rectangle methods in approximating the area beneath a graph. Note that in the final example in this lecture, a knowledge of exponential functions and graphs is required.

Approximate area calculations
09:05

Approximate area calculations
2 questions

Content: All covered in unit 3/4

- Fundamental theorem of integral calculus (in part one)

- Area beneath a graph (in part one)

- Area between two graphs (in part two)

- More complicated area calculations (in part two)

Warning: This lecture is quite content-heavy! So make sure you give yourself enough time to go through this lecture. This lecture also assumes a knowledge on the sketching of basic graphs, including quadratic graphs, exponential graphs, hyperbolic graphs etc.

Exact area calculations (part one)
10:28

Content: All covered in unit 3/4

- Fundamental theorem of integral calculus (in part one)

- Area beneath a graph (in part one)

- Area between two graphs (in part two)

- More complicated area calculations (in part two)

Warning: This lecture is quite content-heavy! So make sure you give yourself enough time to go through this lecture. This lecture also assumes a knowledge on the sketching of basic graphs, including quadratic graphs, exponential graphs, hyperbolic graphs etc.

Exact area calculations (part two)
11:37

Exact area calculations
6 questions

The content of this lecture is covered in unit 3/4.

Average value of a function
04:37

Average value of a function
2 questions

This lecture goes through three worded problems, with example one relating to motion graphs (unit 1/2), example two relating to rate of change problems (unit 3/4), and example three relating to area calculations (unit 3/4). Note, unit 3/4 students may find example three highly useful as an exam-style question, as it combines concepts from both differentiation and antidifferentiation.

Applications on antidifferentiation
14:53

Applications on antidifferentiation
3 questions