CalculusLimits & Continuity

Study Notes

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

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 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

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

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

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

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