BackMAT135 Differential Calculus: Introduction and Functions Study Notes
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Course Introduction
Course Format and Expectations
This course, MAT135 – Differential Calculus, emphasizes active learning, where students learn by doing with guidance rather than passive observation. The course structure includes in-class activities, assessments, and collaborative work to strengthen understanding of calculus concepts.
In-class activities: Recaps of key theorems, worked examples, questions and polls, individual and group work, and discussion of challenging material.
Assessments: Weekly assignments, feedback opportunities, and office hours for support.
Collaboration: Students are encouraged to participate actively, ask and answer questions, and work respectfully with peers.
Chapter 1: Functions and Graphs
Section 1.1: Review of Functions
A function is a fundamental concept in calculus, representing a rule that assigns each input from a set called the domain to exactly one output in a set called the range.
Definition: A function f consists of a set of inputs (domain), a set of outputs (range), and a rule assigning each input to one output.
Example: The function has domain (all real numbers) and range .
Acceptable Function Values: For a function, each input must correspond to exactly one output. For example, if and , this is acceptable. However, if and , this is not acceptable, as one input cannot have two outputs.
Vertical Line Test
The vertical line test is a graphical method to determine if a curve represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function.
Key Point: A graph passes the vertical line test if and only if it is the graph of a function.
Example: Consider three curves. Only those where every vertical line crosses at most once are functions.
Domain of a Function
The domain of a function is the set of all possible input values for which the function is defined.
Example: For , the domain is , since the square root is only defined for non-negative arguments.
Composite Functions
A composite function is formed when the output of one function becomes the input of another. If and are functions, the composition is defined by .
Definition: The composition is only defined when the range of is a subset of the domain of .
Example: If and is defined by a table, then .
Summary Table: Function Properties
Property | Description | Example |
|---|---|---|
Function | Assigns each input to exactly one output | |
Vertical Line Test | Graph passes if no vertical line intersects more than once | Parabola passes; circle fails |
Domain | Set of all valid input values | for |
Composite Function | If , , then |
Additional info:
Active learning and collaboration are emphasized throughout the course to enhance understanding of calculus concepts.
Students are encouraged to participate, ask questions, and work respectfully with peers both in-person and online.
