Limits

Evaluation of Limits

Limits are foundational in calculus, describing the behavior of functions as inputs approach a specific value. Evaluating limits often involves recognizing indeterminate forms and applying algebraic simplification or L'Hôpital's Rule.

• Indeterminate Forms: Expressions like or that require special techniques to evaluate.

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

Asymptotes

• Vertical Asymptotes: Occur where a function grows without bound as approaches a certain value.

• Horizontal Asymptotes: Describe the end behavior of a function as approaches or .

Continuity and Instantaneous Rates

• Continuity: A function is continuous at if .

• Instantaneous Rate of Change: Defined as the derivative at a point, representing the slope of the tangent line.

Derivatives

Basic Rules and Techniques

• Power Rule: For , .

• Product Rule: .

• Quotient Rule: .

Derivatives of Exponential and Logarithmic Functions

Chain Rule

• For composite functions:

Implicit Differentiation

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

Tangent Lines

• The derivative at a point gives the slope of the tangent line to the curve at that point.

Marginal Analysis

• Marginal Cost, Profit, Revenue: The derivative of cost, profit, or revenue functions with respect to quantity, representing the rate of change per unit.

Elasticity of Demand

• Measures the responsiveness of quantity demanded to changes in price.

• Elasticity formula:

Graphing and Optimization

• Critical Numbers: Values where or is undefined.

• Intervals of Increase/Decrease: Determined by the sign of the first derivative.

• Concavity and Inflection Points: Determined by the second derivative; 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 over time.

Integrals

Antiderivatives and General Antiderivatives

• Finding a function whose derivative is the given function.

• For (with ):

• For :

Method of Substitution (u-substitution)

• Used to simplify integrals by substituting part of the integrand with a new variable .

Exponential Growth/Decay Differential Equations

• General solution: , where is the growth/decay rate.

Definite Integrals and Properties

• Represents the net area under a curve from to .

Fundamental Theorem of Calculus

• Connects differentiation and integration: If is an antiderivative of , then .

• Represents the net amount of change over an interval.