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Mastering Continuity in Calculus: Key Concepts & Applications

Antiderivatives

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Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

Area Under Curves

Areas between curves

Average Value

Average value of a function

Basic Integration Techniques

U-Substitution

U-Substitution: Integrating Radical Functions

Not-So-Obvious U-Substitution

Concept of Limits

Finding limits from graphs

Finding limits algebraically - direct substitution

Finding limits algebraically - when direct substitution is not possible

Limit laws

Evaluating Limits of Functions

limit laws

Evaluating Limits with specific limits given

Introduction to Calculus - Limits

Continuity

Intermediate value theorem

Proving the Existence of Real Roots Using IVT

Intermediate Value Theorem

Application of Intermediate Value Theorem

Proving the Existence of a Real Root Using IVT

Discontinuities of Rational Functions (denominator=0)

Critical Points and Extrema

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

Optimization

Curve Analysis

Curve sketching

Derivative Concept

Estimating Derivatives from a table

Slope and equation of tangent line

Definition of derivative

Definition of Derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

Derivative Rules

Chain rule

Chain Rule

Differentiation of Trigonometric Function

Differentiate Radical Functions with Chain Rule

Differentiate: Trigonometric Functions

Chain rule: Differentiate Trigonometric Functions

Differentiation of Exponential Function

Differentiation of Exponential Functions

Differentiation: Chain Rule with Logarithmic Functions

Differentiation of Logarithmic Function

Product rule

Quotient rule

Differentiation using Quotient Rule

Differentiation using Chain Rule and Quotient Rule

Power rule

Power Rule

Application of the Power Rule

Power rule application

Applying Power Rule

Power Rule with Negative Exponents

Power rule with rational exponents

Power rule and rational exponents

Derivatives of Special Functions

Derivative of inverse trigonometric functions

Derivative of trigonometric functions

Derivative of Trigonometric Functions

Derivative of Composite Trig Functions

Derivative of exponential functions

Derivative of Exponential Functions

Differentiation of Composite Functions

Derivative of power and exponential functions

Derivative of logarithmic functions

Derivative of log with arbitrary base

Derivative of Natural Logarithm Functions

Derivative of Logarithmic Functions

Fundamental Theorem of Calculus

Fundamental theorem of calculus

Fundamental Theorem of Calculus Part II

Fundamental Theorem of Calculus Part I

Implicit Differentiation

Implicit differentiation

Limits at Infinity

Infinite limits - vertical asymptotes

Limits at infinity - horizontal asymptotes

Limit Evaluation

Limits at Infinity of Exponential Functions

Motion and Rates

Position velocity acceleration

Velocity of a Particle

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and Acceleration

Riemann Sums

Riemann sum

Riemann Sum

Evaluating integrals with a Riemann sum

Evaluating integrals with a Riemann Sum

Riemann Sum with Trapezoid Approximation

Trapezoidal Approximation of Riemann Sums

Tangent Lines

Linear approximation

Use of Linear Approximations in Estimating Square Roots

Linearization of Radical Functions

Linearization of Polynomial Functions

Linearization of Rational Functions

Linearization of Exponential Functions

Linearization of Logarithmic Functions

Linear Approximation of Trigonometric Functions

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Intro

The idea of continuity lies in many things we experience in our daily lives, for instance, the time it takes you to log into StudyPug and read this section. Limits and continuity are so related that we cannot only learn about one and ignore the other. We will learn about the relationship between these two concepts in this section.

Continuity in functions in calculus