Functions

Basic Properties of Functions

Understanding functions is fundamental in calculus. Functions relate inputs (domain) to outputs (range), and can be represented in various forms.

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) the function can produce.

• Finding Function Values: Function values can be determined using a graph, table, or equation. For example, if f(x) is given, substitute the value of x to find f(x).

Lines

Linear functions are characterized by constant rates of change and are represented by straight lines.

• Slope: Measures the steepness of a line. Calculated as .

• Finding a Line Between Two Points: Use the slope formula and point-slope form .

• Finding a Line from Slope and Point: Use point-slope form as above.

• Slope-Intercept Form: The equation , where m is the slope and b is the y-intercept.

Quadratic Functions

Quadratic functions have the form and their graphs are parabolas.

• Vertex: The highest or lowest point of the parabola. Calculated as , .

• y-intercept: The point where the graph crosses the y-axis, found by evaluating .

• x-intercepts: Points where the graph crosses the x-axis, found by solving .

Exponential Functions and Compound Interest

Exponential functions model growth and decay processes, including financial applications.

• Continuous Compound Interest Formula: , where P is the principal, r is the rate, t is time, and A_t is the amount after time t.

• Application: Used to solve problems involving continuous growth, such as investments.

Limits

Understanding Limits

Limits describe the behavior of functions as inputs approach specific values.

• Existence of Limits: A limit exists at a point if the function approaches a single value from both sides.

• Polynomial and Rational Functions: Limits can be found by direct substitution if the function is continuous at the point.

• Piecewise Functions: Check the limit from both sides at points where the function's definition changes.

• Limits at Infinity (Horizontal Asymptotes): For rational functions, compare degrees of numerator and denominator to determine asymptotic behavior.

• One-sided and Two-sided Limits: One-sided limits ( or ) consider approach from one direction; two-sided limits consider both.

• Limits from Graphs: Analyze the graph to determine the value the function approaches.

Limit Definition of Instantaneous Rate of Change

The derivative is defined as the limit of the average rate of change as the interval shrinks to zero.

• Definition:

• Application: Used to calculate the instantaneous rate of change at a point.

Derivatives

Computing Derivatives

Derivatives measure how a function changes as its input changes.

• Polynomial Functions: Use the power rule:

• Exponential Functions: and

Definition and Interpretation

• Instantaneous Rate of Change: The derivative at a point gives the slope of the tangent line to the graph at that point.

• Non-existence of Derivative: The derivative may not exist at points where the function is not continuous or has a sharp corner (cusp).

Applications of Derivatives

Business Applications

Derivatives are used to analyze rates of change in economics and business.

• Cost, Revenue, Profit: Functions representing total cost, revenue, and profit.

• Marginal Cost/Revenue/Profit: The derivative of each function gives the marginal value, representing the rate of change with respect to quantity.

Physical Applications

Derivatives describe motion and change in physical systems.

• Position, Velocity, Acceleration: If s(t) is position, then velocity is and acceleration is .

Compound Interest Revisited

Exponential functions and their derivatives are used in financial modeling, such as continuous compound interest.

• Formula:

• Derivative: , representing the instantaneous rate of change of the investment.