CalculusLimits & Continuity

Division by Zero

• Division by zero is undefined in mathematics
• This concept is essential to understand in calculus and other mathematical operations

Limits

• Evaluating limits is a crucial concept in calculus
• One-sided limits are used to evaluate limits from the left or right
• Limit theorems provide rules for evaluating limits
• Vertical asymptotes occur when a function approaches a specific value as the input increases or decreases without bound
• Horizontal asymptotes occur when a function approaches a specific value as the input increases or decreases without bound

Continuity

• Continuity is defined as a function being continuous at a point if the function is defined at that point, the limit exists, and the limit equals the function value
• Types of discontinuities include removable and essential discontinuities
• Theorems on continuity provide rules for determining continuity
• Composite functions can be continuous if the individual functions are continuous

Differentiation

• The derivative is defined as the rate of change of a function with respect to its input
• Differentiability is a necessary condition for a function to be differentiable
• Properties of derivatives include linearity, product rule, and chain rule
• Derivatives of basic functions include power rule, product rule, and quotient rule
• The chain rule is used to differentiate composite functions
• Higher-order derivatives can be used to find the acceleration of a function
• Implicit differentiation is used to find the derivative of an implicitly defined function
• Related rates are used to find the rate at which one quantity changes with respect to another

Extreme Values and Optimization

• Maximum and minimum values can be found using the first derivative test
• Critical numbers are points where the derivative is zero or undefined
• The Extreme Value Theorem states that a continuous function on a closed interval has a maximum and minimum value
• Applications of optimization include modeling real-world problems
• The first and second derivative tests are used to determine the maximum or minimum value of a function
• Concavity and inflection points are used to determine the shape of a function
• Curve sketching is used to visualize the shape of a function

Integration

• The definite integral is defined as the area under a curve between two points
• Properties of definite integrals include linearity and additivity
• The Fundamental Theorem of Calculus relates the derivative of a function to the area under its curve
• The average value of a function can be found using the definite integral
• Sigma notation is used to write the sum of a series
• Areas and volumes can be found using definite integrals

Logarithmic and Exponential Functions

• The natural logarithmic function is defined as the inverse of the exponential function
• The derivative of the natural logarithmic function is 1/x
• Logarithmic differentiation is used to differentiate logarithmic functions
• Logarithmic equations can be solved using logarithmic properties
• Exponential decay is used to model real-world problems
• The integration of exponential functions involves using the natural logarithmic function
• Inverse trigonometric functions can be used to solve trigonometric integrals