Limits

Evaluation of Limits

Limits are foundational in calculus, describing the behavior of functions as inputs approach specific values. Understanding limits is essential for defining derivatives and integrals.

• Indeterminate Forms: When direct substitution in a limit yields forms like or , algebraic simplification or L'Hôpital's Rule is used to evaluate the limit.

• L'Hôpital's Rule: If yields an indeterminate form, then (if the latter limit exists).

• Vertical and Horizontal Asymptotes: Vertical asymptotes occur where a function grows without bound as approaches a certain value. Horizontal asymptotes describe the behavior as approaches or .

• Continuity: A function is continuous at if .

Definition of Instantaneous Rate of Change

• The derivative at a point is defined as .

Derivatives

Rules and Techniques

Derivatives measure the rate at which a function changes. Several rules simplify the process of differentiation.

• Power Rule: For , .

• Product Rule: For , .

• Quotient Rule: For , .

• Chain Rule: For , .

Differentiation of ,

Implicit Differentiation

• Used when functions are not explicitly solved for in terms of .

Tangent Line to a Curve at a Point

• The equation of the tangent line at is .

Marginal Analysis

• Marginal cost, profit, and revenue are approximated by the derivative of the respective function.

• For example, marginal cost estimates the cost of producing one more unit.

Elasticity of Demand

• Elasticity measures the responsiveness of demand to changes in price: .

Graphing and Optimization

Critical Numbers

• Values of where or does not exist.

Intervals of Increase/Decrease

• Where , the function is increasing; where , it is decreasing.

Concavity and Inflection Points

• Where , the function is concave up; where , concave down.

• Inflection points occur where concavity changes.

Optimization

• Finding maximum or minimum values of functions, often subject to constraints.

Related Rates

• Problems involving rates at which related variables change with respect to time.

Integrals

Antiderivatives and General Antiderivatives

• The antiderivative of is a function such that .

• For , .

Method of Substitution (u-substitution)

• Used to simplify integrals by substituting , then .

Exponential Growth/Decay Differential Equations

• Solutions to are , modeling growth or decay.

Definite Integrals and Properties

• The definite integral represents the net area under from to .

Fundamental Theorem of Calculus

• If is an antiderivative of , then .