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VCE Maths Methods Units 1-4: Calculus

Includes differentiation and antidifferentiation (integration)
Instructor:
Aaron Ng
15 students enrolled
English [Auto]
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)

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.

Introduction to calculus - Limits

1
Before we get started on calculus

This lecture provides an introduction to the course.

2
Introduction to calculus - Limits

Content: All covered in unit 1/2

- What are limits?

- When do limits exist?

- Theorem on limits

- Calculations of limits

3
Introduction to calculus - Limits

Differentiation

1
Differentiation using first principles

Content: All covered in unit 1/2

- Introduction to differentiation (including notation)

- The derivative of a function using first principles

- Examples

2
Differentiation using first principles
3
Differentiation by rule – x^n

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.

4
Differentiation by rule – x^n
5
Differentiation by rule – e^x, log(x), sin(x), cos(x), tan(x)

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).

6
Differentiation by rule – e^x, log(x), sin(x), cos(x), tan(x)
7
Average vs. instantaneous rate of change

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.

8
Average vs. instantaneous rate of change
9
Differentiation by rule – chain rule

Content: All covered in unit 3/4

- When do we use chain rule?

- What is chain rule?

  • Long method
  • Short method (recommended)

- More examples

10
Differentiation by rule – chain rule
11
Differentiation by rule – product rule and quotient rule

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)


12
Differentiation by rule – product rule and quotient rule
13
Sketching the derivative graph of a function

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

14
Sketching the derivative graph of a function
15
Equation of the tangent and normal line

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.

16
Equation of the tangent and normal line
17
Finding the maximum and minimum of a function

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)

18
Finding the maximum and minimum of a function
19
Applications (maximum and minimum problems)

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).

20
Applications (maximum and minimum problems)
21
Applications (motion graphs and rate of change problems)

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.

22
Applications (motion graphs and rate of change problems)

Antidifferentiation (integration)

1
Antidifferentiation by rule – x^n

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.

2
Antidifferentiation by rule – x^n
3
Antidifferentiation by rule – e^x, 1/x, sin(x), cos(x)

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).

4
Antidifferentiation by rule – e^x, 1/x, sin(x), cos(x)
5
Sketching the antiderivative graph of a function

Content: All covered in unit 3/4

- Features of an antiderivative graph

- Sketching the antiderivative graph of a polynomial function

6
Sketching the antiderivative graph of a function
7
Integration by recognition

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

8
Integration by recognition
9
Definite integrals

Content: All covered in unit 3/4

- Indefinite integrals vs. definite integrals

- Evaluating definite integrals

- Properties of definite integrals

- More complicated examples

10
Definite integrals
11
Approximate area calculations

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.

12
Approximate area calculations
13
Exact area calculations (part one)

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.

14
Exact area calculations (part two)

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.

15
Exact area calculations
16
Average value of a function

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

17
Average value of a function
18
Applications on antidifferentiation

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.

19
Applications on antidifferentiation
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