Reading Comprehension- 1G1

🚚 The Sharing Truck Story

Max brought his new red truck to the park. He wouldn't let anyone play with it. "My truck!" he said. Mia asked nicely, but Max shook his head.

Then... OH NO! The truck got stuck in a tree! 🎄 Mia helped with her jump rope. Max smiled and said, "Let's play together!" They built an awesome ramp and shared the truck all afternoon. 🌞

Vectors Explained

MCV4U: Complete Vector Guide

1. Vector Fundamentals

1.1 Geometric vs Algebraic Vectors

Geometric: Directed line segment \( \overrightarrow{AB} \)

Algebraic: \( \mathbf{v} = \langle v_x, v_y, v_z \rangle \)

From \( A(1,2) \) to \( B(4,6) \): \( \mathbf{v} = \langle 3,4 \rangle \)

2. Vector Operations

How to apply function transformations?

Complete Lesson: Function Transformation Order of Operations

1. General Rule: Inside vs. Outside Transformations

Transformations are split into two groups:

• Inside the function (horizontal transformations): Affect \( x \)-values
• Outside the function (vertical transformations): Affect \( y \)-values

2. Order of Operations

Horizontal Transformations (applied first, in reverse order)

1. Stretch/compression by \( \frac{1}{|b|} \)

   \( f(2x) \) compresses horizontally by \( \frac{1}{2} \)

2. Reflection if \( b < 0 \)

   \( f(-x) \) reflects over \( y \)-axis

3. Horizontal shift by \( h \)

   \( f(x - 3) \) shifts right 3 units

Left and Right hand Limit

Mastering Left-Hand and Right-Hand Limits: A Step-by-Step Guide with Examples

Understanding limits is a fundamental concept in calculus. Evaluating left-hand limits (\( \lim_{x \to c^-} \)) and right-hand limits (\( \lim_{x \to c^+} \)) is essential to determine if a limit exists at a given point. This guide walks you through the steps to evaluate these limits and explains when they are equal or unequal, supported by examples to solidify your understanding.

How to Evaluate Left and Right Limits: Step-by-Step Guide

Step 1: Understand the Function and Point of Interest (\(c\))

Identify the function \(f(x)\) and the value \(c\) at which you want to evaluate the limit. Determine if \(f(x)\) is continuous, piecewise-defined, or has discontinuities near \(c\).

Step 2: Find the Left-Hand Limit (\( \lim_{x \to c^-} f(x) \))

Approach the value of \(c\) from the left (values less than \(c\)). Substitute \(x\)-values slightly less than \(c\) into \(f(x)\) and observe the behavior.

Step 3: Find the Right-Hand Limit (\( \lim_{x \to c^+} f(x) \))

Approach \(c\) from the right (values greater than \(c\)). Substitute \(x\)-values slightly greater than \(c\) into \(f(x)\) and observe the behavior.

Step 4: Compare the Two Limits

If \( \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) \), the limit exists, and \(\lim_{x \to c} f(x)\) equals this common value. If \( \lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x) \), the limit does not exist.

Limit Laws

Limit Rules in Calculus with Examples

Basic Limit Rules

1. Constant Rule

Rule: \( \lim_{{x \to c}} k = k \)

Example:

\( \lim_{{x \to 4}} 7 = 7 \)

2. Identity Rule

Rule: \( \lim_{{x \to c}} x = c \)

Example:

\( \lim_{{x \to 5}} x = 5 \)

Limits

Guide to Limits in Calculus

Comprehensive guide to Limits in Calculus, designed for students in the MCV4U course.

1. Introduction to Limits

A limit describes the value that a function f(x) approaches as the input x approaches a particular value. Limits are crucial for understanding continuity, derivatives, and integrals in calculus.

Notation:

\[ \lim_{x \to c} f(x) = L \]

This means as \(x\) approaches \(c\), \(f(x)\) approaches \(L\).

2. Types of Limits

a) Finite Limits at Finite Points

If \(f(x)\) approaches a specific value \(L\) as \(x \to c\), the limit exists.

Example:
\[ \lim_{x \to 2} (3x + 1) = 7 \]
Solution: Substitute \(x = 2\): \[ f(2) = 3(2) + 1 = 7 \]

Finding roots of complex numbers

How to Find the \(n\)-th Roots of Complex Numbers

Finding the \(n\)-th root of a complex number involves converting the number to its polar form and using De Moivre’s Theorem. Below is a detailed, step-by-step guide on how to find both fourth roots and cube roots of complex and real numbers with examples.

Steps to Find the \(n\)-th Roots

Let’s break down the process of finding the \(n\)-th roots of a complex number into three main steps:

Step 1: Convert the Complex Number to Polar Form

A complex number \( z = x + yi \) can be written in polar form as:

\[ z = r(\cos \theta + i \sin \theta) \]

Where:

• r is the modulus: \( r = |z| = \sqrt{x^2 + y^2} \)
• \(\theta\) is the argument: \( \theta = \arg(z) \) found using trigonometry based on the quadrant.