뒤로Calculus I Study Notes: Functions, Limits, and Continuity
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Functions
Definition and Types of Functions
Functions are fundamental objects in calculus, describing relationships between variables. A function assigns each input exactly one output.
Domain: The set of all possible input values (x-values).
Range: The set of all possible output values (y-values).
Types of Functions: Linear, quadratic, polynomial, rational, trigonometric, exponential, logarithmic.
Piecewise Functions: Defined by different expressions over different intervals.
Example: is a polynomial function.
Graphing Functions
Graphing helps visualize the behavior of functions, including intercepts, asymptotes, and intervals of increase/decrease.
x-intercept: Where .
y-intercept: Where .
Vertical Asymptote: Line where approaches infinity.
Horizontal Asymptote: Line where approaches as goes to infinity.
Example: The graph of has a vertical asymptote at and a horizontal asymptote at .
Limits and Continuity
Definition of a Limit
The limit of a function describes its behavior as the input approaches a particular value.
Notation:
Left-hand limit:
Right-hand limit:
Infinite Limits: When increases or decreases without bound as approaches .
Example: does not exist (infinite limit).
Continuity
A function is continuous at a point if its limit exists and equals the function value at that point.
Continuous at :
Types of Discontinuity: Removable, jump, infinite.
Example: is discontinuous at but the discontinuity is removable.
Trigonometric Functions
Basic Trigonometric Functions
Trigonometric functions relate angles to ratios of sides in right triangles and are periodic.
Sine:
Cosine:
Tangent:
Unit Circle: Used to define trigonometric functions for all real numbers.
Example: On the unit circle, is the y-coordinate, is the x-coordinate.
Applications of Functions and Limits
Projectile Motion (Physics Context)
Functions and limits are used to model real-world phenomena such as projectile motion.
Position Function:
Velocity Function:
Acceleration: for gravity
Example: The height of a ball thrown upward can be modeled by .
Summary Table: Types of Discontinuity
Type | Description | Example |
|---|---|---|
Removable | Hole in the graph; limit exists but function is undefined | at |
Jump | Function "jumps" to a different value | Piecewise function with different values at |
Infinite | Function approaches infinity | at |
Additional info:
Some notes included physics applications (projectile motion) to illustrate function modeling.
Graph sketches and unit circle diagrams were referenced for trigonometric functions and limits.
Key formulas and definitions were expanded for clarity and completeness.
