BackMAT135 – Differential Calculus: Functions and Graphs Study Notes
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Introduction to Differential Calculus
Course Overview
This course introduces the foundational concepts of differential calculus, focusing on functions, their properties, and graphical representations. Active learning and collaborative problem-solving are emphasized throughout the course.
Active Learning: Students are encouraged to learn by doing, participating in discussions, and working through examples.
Course Structure: Includes recaps of key theorems, worked examples, polls, and group work.
Assessments: Weekly assignments and feedback to strengthen understanding.
Functions and Their Properties
Definition of a Function
A function is a rule that assigns each input from a set called the domain to exactly one output in a set called the range.
Domain (D): The set of all possible input values.
Range (R): The set of all possible output values.
Example: For , the domain is all real numbers (), and the range is .
Acceptable Functions
If a function assigns the same output to different inputs (e.g., and ), it is still a valid function.
If a function assigns different outputs to the same input (e.g., and ), it is not a valid function.
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.
Application: Only graphs that pass the vertical line test are functions.
Example: Parabolas and lines typically pass the test, while circles do not.
Domain of Functions
Finding the Domain
To find the domain of a function, identify all input values for which the function is defined.
Example: For , the domain is .
Composite Functions
Definition and Notation
Given two functions and , the composition is defined by .
Requirement: The range of must be a subset of the domain of .
Example: If is defined by a table and , then is found by first evaluating , then applying to that result.
Summary Table: Function Properties
Property | Valid Function? | Reason |
|---|---|---|
Same output for different inputs | Yes | Each input has one output |
Different outputs for same input | No | Violates definition of function |
Passes vertical line test | Yes | Graph represents a function |
Fails vertical line test | No | Graph does not represent a function |
Key Terms and Definitions
Function: A rule assigning each input to exactly one output.
Domain: Set of all possible inputs.
Range: Set of all possible outputs.
Vertical Line Test: A method to determine if a graph represents a function.
Composite Function: A function formed by applying one function to the result of another.
Examples and Applications
Example 1: has domain and range .
Example 2: has domain .
Example 3: If and , then .
Additional info: These notes are based on introductory slides for a differential calculus course, focusing on the concept of functions, their domains and ranges, and graphical representations. The content is foundational for further study in calculus and related mathematical fields.
