Limits and Average Rate of Change

Secant Lines and Average Rate of Change

The average rate of change of a function over an interval measures how much the function's output changes per unit change in input. In the context of motion, it represents the average speed over a time interval.

• Secant Line: A line passing through two points on a curve. Its slope gives the average rate of change between those points.

• Formula: where and are function values at and .

• Example: If a runner covers 1200 meters in 10 minutes and 2700 meters in 30 minutes, the average speed between 10 and 30 minutes is: meters/minute.

Instantaneous Rate of Change and Tangent Lines

The instantaneous rate of change at a point is the slope of the tangent line to the curve at that point. It is found using derivatives.

• Tangent Line: A line that touches the curve at one point and has the same slope as the curve at that point.

• Derivative: The limit of the average rate of change as the interval shrinks to zero.

• Formula:

Limits and Continuity

Definition of a Limit

The limit of a function as approaches is the value that gets closer to as gets closer to .

• Notation:

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

• Existence: The limit exists if both one-sided limits are equal.

• Example: For for , for , if both sides approach 1.

Evaluating Limits

• Direct Substitution: Plug in the value of if the function is continuous at that point.

• Factoring: Factor numerator and denominator to cancel common terms.

• Rationalization: Multiply by conjugate to simplify expressions with roots.

• Special Limits: Use known limits such as .

• Example:

Continuity

A function is continuous at if:

• is defined

• exists

Asymptotes and Holes

Vertical and Horizontal Asymptotes

An asymptote is a line that a graph approaches but never touches.

• Vertical Asymptote: Occurs when the denominator of a rational function is zero and the numerator is not zero at that point. Example: For , vertical asymptotes at and .

• Horizontal Asymptote: Determined by the degrees of numerator and denominator. Example: If degrees are equal, horizontal asymptote at .

Holes in Graphs

A hole occurs when both numerator and denominator are zero at the same -value, and the factor cancels.

• Example: has a hole at .

Derivatives and Tangent Lines

Definition of Derivative

The derivative of a function at a point measures the instantaneous rate of change, or the slope of the tangent line.

• Limit Definition:

• Power Rule:

• Sum Rule:

• Example: ,

Finding the Equation of a Tangent Line

• Find the derivative

• Evaluate at the point

• Use point-slope form:

• Example: For at , slope is $1y + 2 = 1(x + 1)$

Intermediate Value Theorem

Statement and Application

The Intermediate Value Theorem states that if is continuous on and is between and , then there exists in such that .

• Used to show existence of roots in an interval.

• Example: If and , then has a root in .

Domain of Functions

Finding the Domain

The domain of a function is the set of all input values for which the function is defined.

• Exclude values that make the denominator zero or result in negative values under even roots.

• Example: For , domain is .

Summary Table: Types of Asymptotes

Evaluating Limits Numerically

Numerical Substitution

• Plug in values close to the point of interest to estimate the limit.

• Example: , try

Key Formulas and Rules

• Power Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Practice Problems and Applications

• Find the slope of the tangent line to at

• Find all asymptotes and holes for

• Use the Intermediate Value Theorem to show a root exists for in

Additional info: Some explanations and examples have been expanded for clarity and completeness.