Functions

Basic Properties of Functions

Understanding functions is fundamental in calculus. A function relates each input (domain) to exactly one output (range).

• 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) that the function can produce.

• Finding Function Values: Function values can be determined using a graph, table, or equation.

• Example: For , the domain is all real numbers, and the range is all non-negative real numbers.

Lines

Linear functions are represented by straight lines and are characterized by their slope and intercepts.

• Slope: Measures the steepness of the 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: .

• Slope-Intercept Form: , where is the slope and is the y-intercept.

• Example: Find the equation of a line passing through (2,3) with slope 4: .

Quadratic Functions

Quadratic functions have the form and their graphs are parabolas.

• Vertex: The point where and .

• y-intercept: The value of .

• x-intercepts: Solve for .

• Example: For , vertex at , y-intercept at $3x=1x=3$.

Exponential Functions and Compound Interest

Exponential functions model growth and decay, including financial applications like compound interest.

• Continuous Compound Interest Formula: , where is principal, is rate, is time.

• Example: If , , , then .

Limits

Existence of Limits

Limits describe the behavior of functions as inputs approach a specific value.

• Limit at a Point: exists if approaches a single value as approaches .

• Polynomial and Rational Functions: Limits exist at all points except where the denominator is zero.

• Piecewise Functions: Check limits from both sides at points where the formula changes.

• Example: .

Limits at Infinity and Horizontal Asymptotes

Limits at infinity help identify horizontal asymptotes in rational functions.

• Horizontal Asymptote: If , then is a horizontal asymptote.

• Example: .

One-Sided and Two-Sided Limits

Limits can be approached from the left () or right ().

• One-Sided Limit: or .

• Two-Sided Limit: exists if both one-sided limits are equal.

• Example: For defined differently on each side of , check both limits.

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:

• Example: For , .

Derivatives

Computing Derivatives

Derivatives measure the instantaneous rate of change of a function.

• Polynomial Functions: Use the power rule: .

• Exponential Functions: ; .

• Example: ; .

Definition of Derivative

The derivative at a point is the instantaneous rate of change, given by the limit definition.

• Instantaneous Rate of Change:

• Example: For , .

Non-Existence of Derivative

The derivative may not exist at points where the function is not continuous or has a sharp corner.

• Discontinuity: If the function is not continuous at a point, the derivative does not exist.

• Sharp Corners: At points where the graph has a corner or cusp, the derivative is undefined.

• Example: at has no derivative.

Applications

Cost, Revenue, Profit, and Marginal Analysis

Calculus is used in business to analyze cost, revenue, and profit functions.

• Cost Function (): Total cost to produce units.

• Revenue Function (): Total revenue from selling units.

• Profit Function (): .

• Marginal Cost/Revenue/Profit: The derivative of each function, representing the rate of change per unit.

• Example: If , then marginal cost is .

Position, Velocity, and Acceleration

In physics, derivatives describe motion.

• Position (): Location at time .

• Velocity (): , the rate of change of position.

• Acceleration (): , the rate of change of velocity.

• Example: If , then , .

Compound Interest Applications

Exponential functions are used to model compound interest in finance.

• Continuous Compound Interest:

• Example: , , years: