Chapter 1: Functions

Definition and Properties of Functions

Functions are foundational objects in calculus, describing relationships between sets of inputs and outputs. Understanding their properties is essential for further study in calculus.

• Function: A rule that assigns to each element in a set (domain) exactly one element in another set (range).

• Domain: The set of all possible input values for the function.

• Range: The set of all possible output values.

• Function Notation: If f is a function, then f(x) denotes the output when the input is x.

• Evaluating Functions: Substitute the input value into the function rule to find the output.

Example: For , the domain is all real numbers, and .

Types of Functions

• Polynomial Functions: Functions of the form .

• Rational Functions: Ratios of polynomials, , where .

• Radical Functions: Functions involving roots, such as .

• Absolute Value Functions: , which outputs the non-negative value of .

Graphs and Transformations

Understanding how basic graphs change under various transformations is crucial for visualizing functions.

• Shifts: Moving the graph horizontally or vertically.

• Stretches: Expanding or compressing the graph vertically or horizontally.

• Reflections: Flipping the graph over a line, such as the x-axis or y-axis.

Example: The graph of is the graph of shifted right by 2 units and up by 3 units.

Function Operations and Composition

• Operations: Functions can be added, subtracted, multiplied, or divided (where defined).

• Composition: The composition means applying first, then to the result.

Example: If and , then .

Inverse Functions

• Inverse Function: If is one-to-one, its inverse satisfies and .

Example: If , then .

Chapter 2: Limits & Continuity

Limits and One-Sided Limits

Limits describe the behavior of a function as the input approaches a particular value.

• Limit: means as approaches , approaches .

• One-Sided Limits: (from the left), (from the right).

Example: .

Limit Laws

• Limits can be combined using algebraic rules, such as:

• , if

Infinite Limits and Limits at Infinity

• Infinite Limits: When increases or decreases without bound as approaches a value.

• Limits at Infinity: Describes the behavior as approaches or .

Example: ; .

Continuity and Types of Discontinuity

• Continuity at a Point: is continuous at if .

• Types of Discontinuity:

  • Removable: Limit exists, but function is not defined or not equal to the limit at that point.

  • Jump: Left and right limits exist but are not equal.

  • Infinite: Function approaches infinity at the point.

Example: is not defined at , but (removable discontinuity).

Chapter 3: Derivatives

Definition of the Derivative

The derivative measures the instantaneous rate of change of a function with respect to its variable.

• Definition:

• Geometric Meaning: The slope of the tangent line to the curve at a point.

• Normal Line: The line perpendicular to the tangent at a point.

• Rate of Change: The derivative represents how a quantity changes with respect to another.

Example: For , .

Rules of Differentiation

• Power Rule:

• Sum Rule:

• Constant Rule:

• Product Rule:

• Quotient Rule:

Chain Rule and Implicit Differentiation

• Chain Rule:

• Implicit Differentiation: Used when is defined implicitly by an equation involving and .

Example: If , then leads to .

Trigonometric Derivatives

Applications of Derivatives

• Related Rates: Problems involving rates at which related variables change.

• Linearization: Approximating a function near a point using its tangent line.

• Differentials: Small changes in variables, .

Example: If a balloon's radius increases at 2 cm/s, how fast is the volume increasing when cm? Use related rates with .

Study Plan

• Practice daily and solve 10–15 problems per topic to reinforce understanding.

• Focus on mastering limits, derivatives, the chain rule, and their applications.

• Review mistakes carefully and simulate exam conditions for effective preparation.