BackMAT1339 Calculus and Vectors: Structured Study Notes
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Introduction to Calculus and Vectors
Course Overview
This course introduces the foundational concepts of calculus and vectors, including rates of change, limits, derivatives, optimization, and vector operations. Applications span polynomial, rational, trigonometric, exponential, and logarithmic functions, as well as geometric interpretations in two and three dimensions.
Key Topics: Rate of change, limits, derivatives, optimization, curve sketching, vectors, equations of lines and planes, intersections.
Prerequisites: Ontario 4U Functions (MHF4U) or MAT1318 or equivalent.
Precalculus Review
Real Numbers and Intervals
Understanding interval notation and set notation is essential for describing domains and ranges in calculus.
Interval Notation | Set Notation |
|---|---|
[a, b] | {x ∈ ℝ : a ≤ x ≤ b} |
(a, b) | {x ∈ ℝ : a < x < b} |
[a, b) | {x ∈ ℝ : a ≤ x < b} |
(a, b] | {x ∈ ℝ : a < x ≤ b} |
[a, ∞) | {x ∈ ℝ : x ≥ a} |
(a, ∞) | {x ∈ ℝ : x > a} |
(−∞, b] | {x ∈ ℝ : x ≤ b} |
(−∞, b) | {x ∈ ℝ : x < b} |
Solving Inequalities: Techniques for solving linear and quadratic inequalities are reviewed.
Absolute Value:
Triangle Inequality: For real numbers and , .
Properties of Exponents
,
Factoring Polynomials
Rationalizing Denominator or Numerator
If denominator is , multiply by $\sqrt{a}$.
If denominator is , multiply by .
Topic 1. Rate of Change (1.1, 1.2)
Average and Instantaneous Rate of Change
Rates of change describe how a quantity varies over time or with respect to another variable.
Average Rate of Change: Slope between two points and :
Instantaneous Rate of Change: Slope of the tangent at :
Example: Find a line whose graph passes through points (1,2) and (3,6).
Topic 2. Limits and Continuity (1.3, 1.4)
Limits of Sequences
The limit of a sequence as is the value the sequence approaches.
Definition: if approaches as increases.
Divergence: If no such exists, the sequence diverges.
Example:
Limits of Functions
The limit of a function as approaches is the value $f(x)$ gets close to as $x$ gets close to $a$.
Definition:
Properties:
, constant
, if
Example:
Continuity
A function is continuous at if .
Types of Discontinuity:
Jump Discontinuity: Left and right limits exist but are not equal.
Infinite Discontinuity: One or both one-sided limits do not exist (often due to vertical asymptotes).
Removable Discontinuity: Limits exist and are equal, but is undefined.
Example: Determine the continuity of at .
Topic 3. Derivatives (1.5, 2.1-2.6)
First Principles Definition of the Derivative
The derivative of at is defined as:
Represents the instantaneous rate of change or the slope of the tangent line at .
Example: For , .
Some Differentiation Rules
Constant Rule:
Constant Multiple Rule:
Sum/Difference Rule:
Power Rule:
Example: ,
The Product Rule
Example:
The Quotient Rule
Example:
The Chain Rule
General Power Rule:
Example:
Velocity, Acceleration, and Second Derivatives
Derivatives can describe physical quantities such as velocity and acceleration.
Velocity: , where is position.
Acceleration:
Example: If , then , .
*Additional info: These notes cover the foundational topics for a first-year Calculus course, including all major differentiation rules and their applications.*
