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Mastering Continuity in Calculus: Key Concepts & Applications

Demonstrate an understanding of limits and limit notation

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Finding limits from graphs

Continuity

Discontinuities of Rational Functions (denominator=0)

Introduction to Calculus - Limits

Evaluate limits of functions and recognize 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

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

Determine limits of trigonometric functions

Squeeze theorem

Quadratic Function with Integer Vertex

l'Hospital's rule

l'Hopital's rule

l'Hospital's Rule

Evaluating the limit using L'Hospital's Rule

l"Hospital"s rule

Apply limits to define and understand derivatives

Quotient rule

Differentiation using Quotient Rule

Differentiation using Chain Rule and Quotient Rule

Infinite limits - vertical asymptotes

Limits at infinity - horizontal asymptotes

Limit Evaluation

Limits at Infinity of Exponential Functions

Definition of derivative

Definition of Derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

Apply the fundamental counting principle to solve problems

Fundamental counting principle

Fundamental Counting Principle Involving Restrictions

Determine the number of permutations of n elements taken r at a time

Factorial notation

Path counting problems

Permutations

Permutation vs. Combination

Determine the number of combinations of n elements taken r at a time

Combinations

Problems involving both permutations and combinations

Expand powers of a binomial using the binomial theorem

Pascal's triangle

Pascal's Triangle

Pascal"s Triangle

Pascal"s triangle

Binomial theorem

Demonstrate an understanding of the characteristics of quadratic functions

Characteristics of quadratic functions

Transformations of quadratic functions

Quadratic function in general form: y = ax^2 + bx + c

Converting from general to vertex form by completing the square

Shortcut: Vertex formula

Converting general form into vertex form by applying the vertex formula

Vertex Formula Application

Graphing quadratic functions: General form VS. Vertex form

Finding the quadratic functions for given parabolas

Applications of quadratic functions

Quadratic function in vertex form: y = a(x-p)^2 + q

Solve problems that involve quadratic equations

Solving quadratic equations by factoring

Solving quadratic equations by completing the square

Using quadratic formula to solve quadratic equations

Solving a quadratic equation with TWO REAL SOLUTIONS

Solving a quadratic equation with TWO COMPLEX SOLUTIONS

Nature of roots of quadratic equations: The discriminant

Applications of quadratic equations

Solving quadratic inequalities

Arithmetic mean vs. Geometric mean

Analyze arithmetic and geometric sequences and series to solve problems

Descartes&apos; rule of signs

Descartes' Rule of Signs

Descartes' rule of signs

Descartes" rule of signs

Polynomial long division

Polynomial synthetic division

Polynomial functions

Recognizing a Polynomial Function

Remainder theorem

Finding the Remainder Using Synthetic Division and the Remainder Theorem

Factor theorem

Using the Factor Theorem to Test For Factors of a Polynomial

Arithmetic sequences

Arithmetic series

Arithmetic series formula

Geometric sequences

Geometric series

Geometric Series

Infinite geometric series

Fundamental theorem of algebra

Sigma notation

Demonstrate an understanding of reciprocal functions

Characteristics of polynomial graphs

Determining the equation of a polynomial function

Applications of polynomial functions

Graphing reciprocals of linear functions

Multiplicities of polynomials

Imaginary zeros of polynomials

Solving polynomial inequalities

Graphing reciprocals of quadratic functions

Demonstrate an understanding of rational expressions and equations

Partial fraction decomposition

Solving rational equations

Simplifying rational expressions and restrictions

Adding and subtracting rational expressions

Adding and Subtracting Rational Expressions

Multiplying rational expressions

Multiplying Rational Expressions

Multiplying Rational Expressions in Factored Form

Dividing rational expressions

Dividing Rational Expressions

Dividing Rational Expressions in Factored Form

Applications of rational equations

Simplifying complex fractions

Demonstrate an understanding of absolute value functions

What is a rational function?

Point of discontinuity

Slant asymptote

Determining the Equation of a Slant Asymptote Using Long Division

Determining the Equation of a Slant Asymptote Using Synthetic Division

Solving rational inequalities

Solving Rational Inequalities With Two Fractions

Absolute value functions

Expressing an Absolute Value Linear Function as a Piecewise Function

Expressing an Absolute Value Quadratic Function as a Piecewise Function

Solving absolute value equations

Solving absolute value inequalities

Solving Absolute Value Inequalities Involving "less than"

Solving Absolute Value Inequalities Involving "greater than"

Recognizing Absolute Value Inequalities with "No Solution" or "All Real Numbers"

Vertical asymptote

Horizontal asymptote

Horizontal Asymptote

Graphs of rational functions

Analyze quadratic inequalities in one variable

Function notation (advanced)

Composite functions

Adding functions

Subtracting functions

Multiplying functions

Dividing functions

Operations of Functions – In a Nutshell

Operations with functions

Inequalities of combined functions

Inequalities of Combined Functions

Difference quotient: applications of functions

Solve systems of linear-quadratic and quadratic-quadratic equations graphically and algebraically

System of linear equations

System of linear-quadratic equations

System of quadratic-quadratic equations

Pre-Calculus B 120

Limits

Permutations, Combinations and Binomial Theorem

Relations and Functions

math

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