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.

2.3 Circles

Equation of a Circle: MPM2D Grade 10 Mathematics

1. Introduction to the Equation of a Circle

A circle is a set of all points in a plane that are at a fixed distance (called the radius) from a fixed point (called the center). The standard form of the equation of a circle depends on the location of its center.

1.1. Equation of a Circle with Center at the Origin

If the center of the circle is at the origin (0,0), the equation of the circle is:

\[ x^2 + y^2 = r^2 \]

Example:

Consider a circle with a radius of 5 units centered at the origin.

The equation is:

\[ x^2 + y^2 = 5^2 \]

\[ x^2 + y^2 = 25 \]