# Mochi does Maths!

## Ms Carolina's Math class

### Office Hours

Office hours will happen on a rotation basis during a cycle.

Week A - Tuesday 7:00 - 8:00 PM JST
Week B - Wednesday 7:00 - 8:00 PM JST

### Video Lessons

The structure of the lessons is as follows:

2. Video
3. Video and classroom examples
4. Class work.

Videos may  contain additional examples or content that may not be in the textbook. However, if you can't watch the videos, you should not worry, you still have access to all the textbooks and tasks via our classroom and mochidoesmaths.com.
Essential understandings:
Calculus describes rates of change between two variables and the accumulation of limiting areas. Understanding these rates of change and accumulations allow us to model, interpret and analyze real-world problems and situations. Calculus helps us to understand the behaviour of functions and allows us to interpret the features of their graphs.

Suggested concepts embedded in this topic:
Change, patterns, relationships, approximation, generalization, space, modelling.
AHL: Systems, quantity.

Content-specific conceptual understandings:
• The derivative may be represented physically as a rate of change and geometrically as the gradient or slope function.
• Areas under curves can be can be approximated by the sum of the areas of rectangles which may be calculated even more accurately using integration.
• Examining rates of change close to turning points helps to identify intervals where the function increases/decreases, and identify the concavity of the function.
• Numerical integration can be used to approximate areas in the physical world.
• Mathematical modelling can provide effective solutions to real-life problems in optimization by maximizing or minimizing a quantity, such as cost or profit.
• Derivatives and integrals describe real-world kinematics problems in two and three-dimensional space by examining displacement, velocity and acceleration.
AHL
• Some functions may be continuous everywhere but not differentiable everywhere.
• A finite number of terms of an infinite series can be a general approximation of a function over a limited domain.
• Limits describe the output of a function as the input approaches a certain value and can represent convergence and divergence.
• Examining limits of functions at a point can help determine continuity and differentiability at a point.

### Lesson 1: Limits

HL1
September 15th

HL2

September 16th

For this lesson you will need:

- Thorough knowledge and understanding of a diverse range of families of functions and their graphs.

- Factorization

- Properties of exponents and radicals.

In this lesson you will learn:

- the concept of limit.

- to calculate a limit using a table of values.

- to calculate a limit by direct evaluation.

- to calculate a limit with algebraic manipulation.

INSTRUCTIONS

Before you come to class

- Read: This week's reading will be from this chapter. This is a pre-calculus textbook that goes deeper into the concept of limit than IB textbooks do. It is very important that you get this concept clearly and quickly. Pages 897 - 911

- Watch: Please watch this session's video lessons:

Part I -  Limits using tables

Part II - Limits with algebraic manipulation

Part III - Limits from graphs

In class:

- Do: If there are any questions regarding the materials, please ask them at this time.

- Further examples

- Do: From the chapter shared above, complete from exercise 13.1questions 5-16.

- Additional practice: From MAAHL textbook by Oxford Exercise 4A, page 223.

### Lesson 2: Limits and continuity

September 17th

For this lesson you will need:

- Thorough knowledge and understanding of a diverse range of families of functions and their graphs.

- Factorization

- Properties of exponents and radicals.

In this lesson you will learn:

- the concept of lateral limit.

- the concept of continuity.

- to use limits to assess continuity of a function.

- Limits to infinity

- Convergence

INSTRUCTIONS

Before you come to class

- Read: - Read: This week's reading will be from this chapter. This is a pre-calculus textbook that goes deeper into the concept of limit than IB textbooks do. It is very important that you get this concept clearly and quickly. Pages 911-913

- Watch: Please watch this session's video lesson Lesson 2 - Calculus - Limits and continuity.

Part I - Lateral limits and continuity

Part II - Limits to infinity and convergence

In class:

- Do: If there are any questions regarding the materials, please ask them at this time.

- Do: Kahoot! Let's all go to https://kahoot.it/ for a quick game!

- Do: From the chapter shared above, complete from exercise 13.1 questions 17-20, 19-32.

- Additional practice: From MAAHL textbook by Oxford Exercise 4B, page 227.

### Lesson 3 - Rates of change and derivatives

September 22nd

For this lesson you will need:

- Thorough understanding of limits.

In this lesson you will learn:

- to find average rates of change.

- to find instantaneous rates of change.

- the equation of the tangent line to a function at some value of x.

- to differentiate a function from first principles.

INSTRUCTIONS

Before you come to class

- Read: MAAHL by Oxford. Page 236-243.

- Watch: Please watch this session's video lesson Lesson 3 - Rates of change and derivatives.

Part I - Average and Instantaneous Rates of change

Part II - Derivation from first principles

In class:

- Do: If there are any questions regarding the materials, please ask them at this time.

- Do: Kahoot! Let's all go to https://kahoot.it/ for a quick game!

- Do: From MAAHL textbook by Oxford Exercise 4E, page 241 and Exercise 4F, page 243.

### Paper 1

September 23rd.

This day you will sit a cumulative test Paper 1.

The Exam will be 70~75 minutes long.

You will not be allowed a calculator for this test.

Section B: Long questions. Please answer on a separate piece of paper.

### Lesson 4 - Differentiation rules Part I

HL1: September 24th

HL2: September 25th

For this lesson you will need:

- Thorough understanding of limits.

- Thorough understanding of the concept of derivative.

In this lesson you will learn:

- learn the rule to differentiate power functions.

- learn the derivatives of common functions.

- understand the concept of differentiability.

- learn the product and quotient rules.

INSTRUCTIONS

Before you come to class

- Read: MAAHL by Oxford. Page 244-251 and 255-258.

- Watch: Please watch this session's video lesson Lesson 4 - Differentiation rules Part I.

Part I - Derivatives of common functions

Note - Differentiability

Part II - The product and quotient rules for differentiation

In class:

- Do: If there are any questions regarding the materials, please ask them at this time.

- Do: From MAAHL textbook by Oxford Exercise 4G, page 248; Exercise 4H, page 252; Exercise 4J page 256 and Exercise 4K page 258.

### Lesson 5 - Differentiation rules Part II

September 29th

For this lesson you will need:

- Thorough understanding of the concept of derivative.

- Thorough understanding of the derivation rules

- Thorough understanding of composite and inverse functions.

In this lesson you will learn:

- the chain rule to differentiate composite functions

- to calculate and write higher order derivatives using appropriate notation.

INSTRUCTIONS

Before you come to class

- Read: MAAHL by Oxford. Page 522-255, page 259-261.

- Watch: Please watch this session's video lesson Lesson 5 - Differentiation rules II

Part III - Chain Rule

Part IV - Higher order derivatives and examples

Note - Proof of chain rule

In class:

- Do: If there are any questions regarding the materials, please ask them at this time.

- Do: From MAAHL textbook by Oxford Exercise 4J, page 256; Exercise 4L page 261

### Paper 2

September 30th.

This day you will sit a cumulative* test Paper 2.

The Exam will be 70~75 minutes long.

You will be allowed a calculator for this test.

Section B: Long questions. Please answer on a separate piece of paper.

*Chain rule may be included

### IA

October 2nd

Please use the time of this lesson to work on the first draft of your Math IA.

Remember the deadline to submit this work is October 9th.

### Paper 1 & Paper 2 Correction

HL1:

October 6th

HL 2:

October 7th

For this lesson you will need:

- Paper 1 and Paper 2

- Formula booklet

- Calculator

In this lesson you will be given time to work on questions:

Paper 1: 1, 4, 6, 7

Paper 2: TBD

In addition to these problems, you might want to practice with this set.

### Lesson 6: Increasing and decreasing functions and other derivatives

October 8th

In this lesson you will learn:

- the derivative of inverse trigonometric functions

- to determine growth of a function using the first derivative

INSTRUCTIONS

Before you come to class

- Read: MAAHL by Oxford. Page 262-265.

- Watch: Please watch this session's video lesson Lesson 6.

I - Further differentiation and normal lines

Part II - Increasing and decreasing functions

In class:

- Do: If there are any questions regarding the materials, please ask them at this time.

- Do: From MAAHL textbook by Oxford Exercise 4M page 264

### Differentiation Review

October 27th

In this lesson you will review:

- The concept of derivative.

- The rules for differentiation.

- The derivatives of basic families of functions.

- The derivative as the gradient of a function.

- Tangent lines to a function.

- the relation between the graphs of the function, first derivative and second derivative.

INSTRUCTIONS

Before you come to class

- Watch: If needed go through any of the past videos you need reviewing.

- Write: write down any questions, items that need clarifying in class.

In class:

- Do: If you brought any prepared questions, please share them at this time.

### Paper 3

On Wednesday October 28th, we will have the first Paper 3.

### Week 10: Nov 2nd ~ Nov 6th (Week B)

HL1 Nov 3rd / HL 2 Nov 4th

In this lesson you will learn:

- The concept of implicit differentiation.

INSTRUCTIONS

Before you come to class

- Read: Oxford. Mathematics Analysis and Approaches HL. Pages 288 - 293

- Watch: Lesson 7 - Implicit differentiation Part I, Part II

In class:

- Quiz

- Worked examples

- Practice questions

### Lesson 8: Derivatives of logarithms and inverse trigonometric functions

HL1 Nov 4th / HL2 Nov 5th

In this lesson you will learn:

- To deduce the derivatives of logarithmic and inverse trigonometric functions.

INSTRUCTIONS

Before you come to class

- Read: Oxford. Mathematics Analysis and Approaches HL. Pages 483 - 488

- Watch: Video Lesson part III, part IV

In class:

- Worked examples

- Practice questions

### Lesson 9: Related rates of change

HL1 Nov 5th / HL2 Nov 6th

In this lesson you will learn:

- To apply implicit differentiation to real world problems

- To identify ligatures and related rates

INSTRUCTIONS

Before you come to class

- Read: Oxford. Mathematics Analysis and Approaches HL. Pages 288 - 293

- Watch: Part I

### Related Rates further practice

HL2 :November 9th

HL1: November 10th

In this lesson you will apply the concept of related rates practicing with past IB questions.

INSTRUCTIONS

Before you come to class

- Review the materials for lesson 9.

During class:

### Lesson 10: Optimization I

HL1: November 11th

HL2: November 12th

In this lesson you will learn:

- To use differentiation in optimization problems

INSTRUCTIONS

Before you come to class

- Read: Pearson. Mathematics Analysis and Approaches HL page 677-682

- Watch: Video Lesson

### Optimization - Practice

HL1: November 17th

HL2: November 18th

In this lesson you will practice:

- Optimization and differentiation problems

INSTRUCTIONS

Before you come to class

- Review our previous lesson

During class

### Lesson 11 - L'Hôpital's Rule

HL1: November 18th

HL2: November 19th

In this lesson you will learn:

- L'Hôpital's rule for undetermined limits

INSTRUCTIONS

Before you come to class

- Review differentiation methods

- Read: Mathematics Analysis and Approaches HL. Oxford. Page 550-553

- Video lesson: Part I, Part II

During class

- Investigation 9, MAA HL Oxford page 550.

- MAA HL Oxford page 553. Exercise 8J.

- If you are interested in a more formal proof please read this document.

### Lesson 12 - Maclaurin Series I

HL1: November 19th

HL2: November 20th

In this lesson you will learn:

- To expand functions using Maclaurin Series

INSTRUCTIONS

Before you come to class

- Review differentiation methods

- Read: Mathematics Analysis and Approaches HL. Oxford. Page 554-562

- Video lesson: Part I, part II

During class

- Investigation 10, MAA HL Oxford page 554.

- MAA HL Oxford page 556. Exercise 8K.

- Kignity Assignment - Maclaurin series practice

### Lesson 13 - Maclaurin Series II

HL1: November 24th

HL2: November 26th

In this lesson you will learn:

- Maclaurin expansion of composite functions

INSTRUCTIONS

Before you come to class

- Review differentiation methods

- Watch: Video lesson

During class

- MAA HL Cambridge page 112, exercise 4C.

### Paper 1 (2)

On november 25th you will sit a paper 1 exam.

This is a non-calculator test.

Use black or blue ink to write, you may use pencil for graphs and diagrams only.

If English is not your first language, you can bring a paper copy of a your-language/English dictionary.

### Lesson 14 - Binomial Series and Applications

HL1: November 26th

HL2: November 27th

In this lesson you will learn:

- To find the binomial series

- To apply Maclaurin series to calculate limits.

INSTRUCTIONS

Before you come to class

- Review differentiation methods

- Video Lesson: part I

- Read: MAA HL Oxford page 558-561

During class

MAA HL Oxford page 562 Exercise 8L

### Lesson 15 - Antiderivatives

HL2: November 30th

In this lesson you will learn:

- The concept of antiderivative/anti-differentiation

- How to use initial values to find integration constants

INSTRUCTIONS

Before you come to class

- Review differentiation methods

- Video Lesson: part I

- Read: MAAHL Pearson page 698 - 701.

MAAHL Oxford page 444 - 450.

During class

- MAA HL Oxford page 562 Exercise 7A.

- Kognity