Chapter R: Functions, Graphs, and Models

R3: Functions, Domain, and Range

Understanding the concepts of domain and range is fundamental to working with functions in calculus. The domain is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values).

• Finding the Domain: To determine the domain of a function, identify all real numbers for which the function produces a real output. Pay special attention to denominators (cannot be zero) and even roots (radicands must be non-negative).

• Finding the Range: The range can often be found by analyzing the function's behavior or by considering its graph.

• Interval Notation: Express domains and ranges using interval notation, e.g., .

• Rational Functions: The domain excludes values that make the denominator zero.

• Radical Functions: For even roots, the radicand must be greater than or equal to zero.

Example: For , the domain is because .

R4: Lines and Applications

Linear equations and their applications are essential in modeling and solving real-world problems.

• Slope of a Line: The slope measures the steepness of a line and is calculated as .

• Equations of Lines: Common forms include:

  • Slope-intercept form:

  • Point-slope form:

  • Standard form:

• Special Lines:

  • Horizontal lines: (slope )

  • Vertical lines: (undefined slope)

• Applications: Linear equations are used to model cost, revenue, price, and to find break-even points (where cost equals revenue).

Example: If the cost to produce items is and the revenue is , the break-even point is found by solving .

R5: Factoring and Algebraic Skills

Factoring and manipulating algebraic expressions are foundational skills for calculus.

• Factoring: Expressing an expression as a product of its factors, e.g., .

• Quadratic Functions: Functions of the form ; can be solved by factoring, completing the square, or using the quadratic formula.

• Absolute Value Functions: returns the non-negative value of .

• Solving Equations: Includes equations with roots, powers, and radicals.

• Converting Forms:

  • Radical form:

  • Rational exponent form:

Example:

Chapter 1: Differentiation

Section 1.1: Limits from Graphs and Functions

The concept of a limit is central to calculus, describing the behavior of a function as the input approaches a particular value.

• Determining Limits from a Graph: Observe the value the function approaches as approaches a specific point from both sides.

• Determining Limits from a Function: Substitute values close to the point of interest or use algebraic simplification.

• Limit Notation: means as approaches , approaches .

Example: If approaches 3 as approaches 2, then .

Section 1.2: Limit Properties and Continuity

Limits have several properties that make them easier to evaluate, and the concept of continuity describes functions without breaks or jumps.

• Limit Properties: For functions and and constant :

  • , provided

• Limits of Rational Functions: If and are polynomials, if .

• Continuity: A function is continuous at if:

  • is defined

  • exists

Example: is continuous everywhere because it meets all three criteria at every .

Section 1.3: Average Rate of Change and Difference Quotients

The average rate of change measures how a function's output changes per unit change in input, and the difference quotient is foundational for defining the derivative.

• Average Rate of Change: For between and :

• Difference Quotient: The expression is called the difference quotient and is used to approximate the slope of the tangent line at .

Example: For , the difference quotient is .

Section 1.4: The Derivative Using the Limit Definition

The derivative of a function at a point measures the instantaneous rate of change, defined using limits.

• Limit Definition of the Derivative:

• Verifying a Derivative: Use the limit definition to confirm the derivative found by other means.

• Non-differentiable Points: Points where the function has a sharp corner, cusp, vertical tangent, or discontinuity are not differentiable.

Example: For , .

Exam Preparation Tips

• Review in-class notes, quizzes, and homework assignments, especially those covering the sections above.

• Practice problems involving all the key skills: finding domains/ranges, working with lines, factoring, evaluating limits, and using the limit definition of the derivative.

• Prepare a formula sheet according to the exam guidelines, focusing on formulas and procedures rather than worked examples.

• Show all work clearly and neatly on the exam.