Limits

Introduction to Limits

Formal Definition of Limits Part 1

Formal Definition of Limits Part 2

Determining Limits

Ex 1: Determine a Limit Numerically

Ex 2: Determine a Limit Numerically

Ex 3: Determine a Limit Numerically

Examples: Determining Basic Limits Graphically

Ex 1: Determine Limits from a Given Graph

Ex 2: Determine Limits from a Given Graph

Ex 1: Determine Limits from a Graph Using Function Notation

Ex 2: Determine Limits from a Graph Using Function Notation (Challenging)

Ex: Determining Basic Limits Using Direct Substitution

Ex: Determining Limits Involving an Absolute Value Function Graphically and Algebraically

Ex 1: Determining Limits and One-Sided Limits Graphically

Ex 2: Determining Limits and One-Sided Limits Graphically

Ex 1: One-Sided Limits and Vertical Asymptotes (Rational Function)

Ex 2: One-Sided Limits and Vertical Asymptotes (Rational Function)

Ex 3: One-Sided Limits and Vertical Asymptotes (Rational Function)

Ex 4: One-Sided Limits and Vertical Asymptotes (Tangent Function)

Ex 5: One-Sided Limits and Vertical Asymptotes (Cosecant Function)

Ex 1: Determine a Limit Analytically

Ex: Limits Involving the Greatest Integer Function

Ex: Determining Limits of Rational Functions by Factoring

Ex 2: Determine a Limit of a Piece-Wise Defined Function Analytically

Ex 3: Determine a Limit Analytically by Factoring

Ex 4: Determine Limits of a Rational Function Analytically

Ex 1: Determine a Limit of a Rational Function by Expanding or Factoring

Ex 2: Determine a Limit of a Rational Function by Factoring and Simplifying

Ex 3: Determine a Limit of a Rational Function by Factoring and Simplifying

Ex 1: Find a Limit by Rationalizing or Factoring

Ex 2: Find a Limit by Rationalizing or Factoring

Ex: Find a Limit Requiring Rationalizing

Ex: Determine Limits of a Piecewise Defined Function

Limits at Infinity

Limits at Infinity – Additional Examples

Ex: Determining Limits at Infinity Graphically

Ex: Limits at Infinity of a Polynomial Function

Ex: Limits at Infinity of a Rational Function (DNE)

Ex: Limits at Infinity of a Rational Function (Zero)

Ex: Limits at Infinity of Rational Function (Ratio of Leading Coefficients)

Ex: Limits at Infinity of a Function Involving a Square Root

Ex: Limits at Infinity of a Function Involving an Exponential Function

Limits involving Trigonometric Functions

Ex: Find Limits of Composite Function Graphically

Squeeze Theorem and Special Limits

Determining Limits Using Special Limits

Continuity Using Limits

Continuity

Intermediate Value Theorem

Ex: Determine Which Rule of Continuity at a Point is Violated

Ex: Continuity at a Point Concept Check

Ex 1: Find the Value of Constant to Make a Piecewise Defined Function Continuous Everywhere

Ex 2: Find the Value of Constant to Make a Piecewise Defined Function Continuous Everywhere

Ex 3: Find the Value of c to Make a Piecewise Defined Function Continuous Everywhere

Asymptotes: Part 1, Part 2

Average Rate of Change

Ex: Determine Average Rate of Change

Ex: Find the Average Rate of Change From a Table - Temperatures

Ex: Find the Average Rate of Change from a Graph

Ex: Find the Average Rate of Change Given a Function Rule

Ex: Average Rate of Change Application - Hot Air Balloon Function

Ex: Find the Average Rate of Change Given a Function on [2,t]

Ex: Find the Average Rate of Change Given a Function on [3, 3+h]

Ex: Use Average Velocity to Predict Instantaneous Velocity

Formal Definition of the Derivative

Introduction to the Derivative

Ex 1: Estimate the Value of a Derivative at a Point on a Graph Using a Tangent Line

Ex 2: Estimate the Value of a Derivative at a Point on a Graph Using a Tangent Line

Ex 3: Estimate the Value of a Derivative at a Point on a Graph Using a Tangent Line

Ex 4: Estimate the Value of a Derivative at a Point on a Graph Using a Tangent Line

Ex: Determine the Open Intervals Where the First Derivative is Positive or Negative

Ex: Determine the Sign of the First Derivative at a Point on the Graph of a Function

Finding Derivatives using the Limit Definition

Example 1: Determine a Derivative using The Limit Definition

Example 2: Determine a Derivative using The Limit Definition

Example 3: Determine a Derivative using The Limit Definition

Differentiation of Basic Functions and Using the Power Rule

Finding Derivatives Using the Power Rule

Ex: Derivatives and Derivative Values of a Linear and Constant Function

Ex: Derivative of a Quotient Function By Simplifying

Ex: Find the Equation of a Tangent Line to a Quadratic Function at a Given value of x

Ex 1: Basic Derivatives Using the Power Rule

Ex: Find the Derivative Function and Derivative Function Value of a Quadratic Function

Ex: Find the Derivative of a Function Containing Radicals

Ex 2: Derivatives Using the Power Rule with Negative and Decimal Exponents

Ex 3: Derivatives Using the Power Rule with Radicals

Ex 4: Derivative Using the Power Rule Involving a Variety of Terms

Ex: Find a Derivative using the Power Rule with Negative Exponents

Ex: Determine Where a Function has Tangent Lines Parallel to a Given Line

The Derivatives of Sine and Cosine

Ex: Derivative and Derivative Value of Basic Cosine and Sine Functions

Ex: Find the Derivative and Equation of Tangent Line for a Basic Trig Function

Ex: Find a Derivative and Derivative Function Value (Cosine and Cosecant)

Ex: Find a Derivative of a Function Involving Radicals Using the Power Rule (Rational Exponents)

Ex: Determine the Points Where a Function Has Horizontal Tangent Lines

Ex: Determine the Equation of a Tangent Line to a Function Using the Power Rule

Ex: Determine the Points on a Function When the Tangents Lines Have a Given Slope

Determine the value of the derivative function on the graphing calculator

Find the Value of a Derivative Function at a Given Value of x

Applications of the Derivatives Using the Power Rule

Ex: Sketch the Graph of a Derivative Function Given the Graph of a Function

Ex 1: Determine the Graph of the Derivative Function Given the Graph of a Quadratic Function

Ex 2: Determine the Graph of the Derivative Function Given the Graph of a Cubic Function

Ex 1: Derivative of Trigonometric Functions – Simplify Before Differentiating

Ex: Find the Velocity and Acceleration Function from the Position Function

Differentiation Using the Product Rule

The Product Rule of Differentiation (Introduction)

Proof: The Product Rule of Differentiation

Ex: Find the Equation of a Tangent Line Using the Product Rule

The Product Rule (old)

Ex: Find a Derivative Using Product Rule (Basic Example)

Ex: Find a Derivative Using Product Rule (Polynomial*Exponential)

Ex 1: Determine a Derivative Using the Product Rule

Ex 2: Determine a Derivative Using the Product Rule

Ex: Find a Derivative Function Value - Product Rule Concept Check

Ex 1: Determine a Derivative Using the Product Rule Involving a Trig Function

Ex 2: Determine a Derivative Using the Product Rule Involving a Trig Function

Ex: Determine the Equation of a Tangent Line Using the Product Rule

Ex: Find a Derivative Using the Product Rule (Linear*Trig) and Find Equation of Tangent Line

Ex: Find a Derivative and Equation of Tangent Line Using Product and Chain Rule (Exp*Trig)

Ex: Find a Derivative Function and Derivative Value Using the Product Rule (3 products)

Ex 2: Derivative of Trigonometric Functions Using Product Rule – Simplify Before Differentiating

Differentiation Using the Quotient Rule

The Quotient Rule

Ex: Use the Quotient Rule to Find the Derivative and Derivative Value (Basic)

Ex 1: Quotient Rule or Power Rule to Find a Derivative (Comparison)

Ex 2: Quotient Rule or Power Rule to Find a Derivative (Comparison)

The Product and Quotient Rule With Trigonometric Functions

Ex 1: Determine a Derivative Using the Quotient Rule

Ex 2: Determine a Derivative Using the Quotient Rule

Ex 3: Determine a Derivative Using the Quotient Rule

Ex: Find a Derivative Function Value Using the Quotient Rule and by Interpreting a Graph

Ex: Find a Derivative and Derivative Function Value Using the Quotient Rule (square roots)

Ex: Find a Derivative and Derivative Function Value Using the Quotient Rule (linear/trig)

Ex: Find a Derivative and Using the Quotient Rule (trig/poly)

Ex: Find the X-values Where a Function has Derivative Function Value (Quotient Rule)

Ex: Determine the Slope of a Tangent Line Using the Quotient Rule

Ex: Derivative with The Quotient Rule Involving Trig Functions - Equation of Tangent Line

Ex:Derivative and Derivative Function Value Using the Quotient Rule (Tangent)

Ex: Determine the Equation of a Tangent Line to Using the Quotient Rule Involving a Trig Function

Ex 1: Determine a Derivative Using the Quotient Rule Involving a Trig Function

Ex 2: Determine a Derivative Using the Quotient Rule Involving a Trig Function

Average Revenue, Cost, Profit Functions and their Derivatives

Differentiation Using the Chain Rule

The Chain Rule: Part 1, Part 2

The Chain Rule with Transcendental Functions

Ex 1: Chain Rule Concept Check

Ex 2: Power Rule with Chain Rule Concept Check

Ex 3: Power Rule with Chain Rule Concept Check

Ex 4: Power Rule with Chain Rule Concept Check

Ex: Derivatives Using the Chain Rule - Quadratic Raised to a Power

Ex: Derivatives Using the Chain Rule - Negative Exponent

Ex 1: Determine a Derivative Using the Chain Rule

Ex 2: Determine a Derivative Using the Chain Rule

Ex 3: Determine a Derivative Using the Chain Rule

Ex 4: Determine a Derivative Using the Chain Rule Involving an Exponential Function

Ex 5: Determine a Derivatives using The Chain Rule Involving Trig Functions

Ex: Derivatives Using the Chain Rule Involving a Trigonometric Functions

Ex: Derivatives Using the Chain Rule Involving an Exponential Function with Base e

Ex: Derivative using the Product Rule and Chain Rule – Product of Polynomials to Powers

Ex 1: Determine a Derivative Using the Chain Rule and Product Rule

Ex 2: Determine a Derivative Using the Chain Rule and Product Rule Involving a Radical

Ex 3: Determine a Derivative Using the Chain Rule and Product Rule With a Trig Function

Ex: Determine a Derivative Using the Chain Rule and Quotient Rule

Ex: Derivative Using the Chain Rule Twice - Trig Function Raised to Power

Ex: Derivative Using the Chain Rule Twice - Exponential and Trig Functions

Differentiation of Exponential Functions

Graphing Exponential Functions

Derivatives of Exponential Functions with base e

Ex 1: Derivatives Involving the Exponential Function with Base e

Ex 2: Derivatives Involving the Exponential Function with Base e and the Product Rule

Ex 3: Derivatives Involving the Exponential Function with Base e and the Power Rule

Ex 4: Derivatives Involving the Exponential Function with Base e and the Quotient Rule

Ex 5A: Derivatives Involving the Exponential Function with Base e and the Quotient Rule

Ex 5B: Derivatives Involving the Exponential Function with Base e and the Quotient Rule

Ex 1: Derivatives of Exponential Functions

Ex 2: Derivatives of Exponential Functions With Chain Rule

Ex 3: Derivatives of Exponential Functions with the Product Rule

Ex 4: Derivatives of Exponential Functions with the Quotient Rule

Ex: Find the Equation of a Tangent Line at a Given Point – Linear and Exponential Function

Ex: Application of the Derivative of an Exponential Function (Rate of Depreciation)

Differentiation of Hyperbolic Functions

Introduction to Hyperbolic Functions

Prove a Property of Hyperbolic Functions: (sinh(x))^2 - (cosh(x))^2 = 1

Prove a Property of Hyperbolic Functions: (tanh(x))^2 + (sech(x))^2 = 1

Prove a Property of Hyperbolic Functions: sinh(x+y)=sinh(x)cosh(y)+cosh(x)sinh(y)

Prove a Property of Hyperbolic Functions: (sinh(x))^2=(-1+cosh(2x))/2

Ex 1: Derivative of a Hyperbolic Function

Ex 2: Derivatives of Hyperbolic Functions with the Chain Rule

Ex 3: Derivative of a Hyperbolic Function Using the Product Rule

Ex 4: Derivative of a Hyperbolic Function Using the Quotient Rule

Ex 5: Derivatives of Hyperbolic Functions with the Chain Rule Twice

Ex 1: Derivative of an Inverse Hyperbolic Function with the Chain Rule

Ex 2: Derivative of an Inverse Hyperbolic Function with the Chain Rule

Ex 3: Derivative of an Inverse Hyperbolic Function with the Chain Rule

Differentiation of Logarithmic Functions

Logarithms

Derivatives of Logarithmic Functions

Ex 1: Derivatives of the Natural Log Function

Ex 2: Derivatives of the Natural Log Function with the Chain Rule

Ex 3: Derivatives of the Natural Log Function with the Chain Rule

Ex 4: Derivatives of the Natural Log Function with the Chain Rule

Ex 5: Derivatives of the Natural Log Function with the Product Rule

Ex 6: Derivatives of the Natural Log Function using Log Properties

Ex 7: Derivatives of the Natural Log Function using Log Properties

Ex 8: Derivatives of the Natural Log Function using Log Properties

Ex 9: The derivative of f(x) = ln(ln(5x))

Derivatives of a^x and logax

Ex 1: Derivative of the Log Function, not base e

Ex 2: Derivative of the Log Function using the Product Rule

Logarithmic Differentiation

Ex: Logarithmic Differentiation

Ex 1: Logarithmic Differentiation

Ex 2: Logarithmic Differentiation and Slope of a Tangent Line

Ex 3: Logarithmic Differentiation and Slope of a Tangent Line

Differentiation of Inverse Trigonometric Functions

The Derivatives of the Inverse Trigonometric Functions

Ex 1: Derivatives of Inverse Trig Functions

Ex 2: Derivatives of Inverse Trig Functions

Ex 3: Derivatives of Inverse Trig Functions

Higher Order Differentiation

Higher-Order Derivatives: Part 1, Part 2

Higher Order Derivatives of Transcendental Functions

Ex 1: Determine Higher Order Derivatives

Ex 2: Determine Higher Order Derivatives

Ex 3: Determine Higher Order Derivatives

Ex 4: Determine Higher Order Derivatives Requiring the Chain Rule

Ex 5: Determine Higher Order Derivatives Requiring the Product Rule and Chain Rule

Ex 6: Determine Higher Order Derivatives Requiring the Quotient Rule

Ex: Higher Order Derivatives Using the Product Rule

Ex 1: First and Second Derivatives Using the Chain Rule - f(x)=tan(2x)

Ex 2: First and Second Derivatives Using the Chain Rule - f(x)=ln(cos(x))

Ex: Determine the Velocity Function and Acceleration Function from the Position Function

Ex: Find the First and Second Derivative Functions and Function Value (Exponential and Polynomial)

Applications of Differentiation – Relative Extrema

Ex: Find the Critical Numbers of a Cubic Function

Ex: Concavity of a Degree 5 Polynomial - Irrational Critical Numbers

Increasing and Decreasing Functions

Ex: Determine Increasing or Decreasing Intervals of a Function

Ex 1: Determine the Intervals for Which a Function is Increasing and Decreasing

Ex 2: Determine the Intervals for Which a Function is Increasing and Decreasing

Ex: Determine Increasing/Decreasing Intervals and Relative Extrema

Ex: Determine Increasing/Decreasing Intervals and Relative Extrema (Product Rule with Exponential)

Ex: Determine Increasing/Decreasing Intervals and Absolute Extrema (Product Rule)

Determine where a trig function is increasing/decreasing and relative extrema

Ex 1: First Derivative Concept - Given Information about the First Derivative, Describe the Function

Ex 2: First Derivative Concept - Given Information about the First Derivative, Describe the Function

Ex 1: Interpret the Graph of the First Derivative Function – Degree 2

Ex 2: Interpret the Graph of the First Derivative Function - Degree 3

The First Derivative Test to Find Relative Extrema

Ex: Critical Numbers / Relative Extrema / First Derivative Test

Determining Relative Extrema on the Graphing Calculator

Ex 1: Determine Relative Extrema Using The First Derivative Test

Ex 2: Determine Relative Extrema Using The First Derivative Test Involving a Rational Function

Ex 3: Determine Relative Extrema Using The First Derivative Test Involving a Trig Function

Ex 1: Sketch a Graph Given Information About a Function's First Derivative

Ex 2: Sketch a Graph Given Information About a Function's First Derivative

Finding Max and Mins Applications: Part 1, Part 2

Ex: Optimization - Maximized a Crop Yield (Calculus Methods)

Ex: Profit Function Applications – Average Profit, Marginal Profit, Max Profit

Ex: Profit Function Application - Maximize Profit

Elasticity of Demand: Part 1, Part 2

Ex: Elasticity of Demand Application Problem

Ex: Elasticity of Demand - Quadratic Demand Function

Exponential Growth Models Part 1, Part 2

Exponential Decay Models: Part 1, Part 2

Marginals

Ex: Marginals and Marginal Profit

Ex: Marginals and Marginal Average Cost

Applications of Differentiation – Concavity

Determining the Concavity of a Function

Concavity of Transcendental Functions (Additional Examples)

Ex: Determine Concavity and Points of Inflection

Ex: Determine Concavity and Absolute Extrema (Product and Quotient Rule)

Ex: Determine Increasing/Decreasing/Concavity Intervals of a Function

Ex: Determine Increasing/Decreasing/Concavity Intervals of a Rational Function

Ex: Find the Intervals a Function is Increasing/Decreasing/Concave Up or Down - Rational Exponent

Ex: Determine Increasing / Decreasing / Concavity by Analyzing the Graph of a Function

The Second Derivative Test to Determine Relative Extrema

Ex 1: The Second Derivative Test to Determine Relative Extrema

Ex 2: The Second Derivative Test to Determine Relative Extrema

Ex: Critical Numbers / Relative Extrema / Second Derivative Test

The Second Derivative Test using Transcendental Functions

Example: Increasing/Decreasing / Concavity / Relative Extrema / Points of Inflection

Ex 1: Sketch a Function Given Information about Concavity

Ex 2: Sketch a Function Given Information about Concavity

Ex: Determine the Sign of f(x), f'(x), and f''(x) Given a Point on a Graph

Ex 1: Intervals for Which the First and Second Derivative Are Positive and Negative Given a Graph

Ex 2: Intervals for Which the First and Second Derivative Are Positive and Negative Given a Graph

Applications of Differentiation – Maximum/Minimum/Optimization Problems

Ex 1: Max / Min Application Problem - Derivative Application

Ex 2: Max / Min Application Problem - Derivative Application

Ex 3: Max / Min Application Problem - Derivative Application

Ex: Optimization - Maximum Area of a Rectangle Inscribed by a Parabola

Ex: Optimization - Minimize the Surface Area of a Box with a Given Volume

Ex: Optimization - Minimize the Cost to Make a Can with a Fixed Volume

Ex: Derivative Application - Maximize Profit

Ex: Derivative Application: Maximize Area

Ex: Derivative Application - Minimize the Cost of a Fenced Area

Optimization - Maximize the Area of a Norman Window

Ex 1: Cost Function Applications - Marginal Cost, Average Cost, Minimum Average Cost

Ex 2: Cost Function Applications - Marginal Cost, Average Cost, Minimum Average Cost

Ex: Find a Demand Function and a Rebate Amount to Maximize Revenue and Profit

Ex: Given the Cost and Demand Functions, Maximize Profit

Animation: The graphs of f(x), f’(x), f’’(x)

Absolute Extrema

Absolute Extrema of Transcendental Functions

Ex 1: Absolute Extrema on an Closed Interval

Ex 2: Absolute Extrema on an Open Interval

Ex 1: Determine Asymptotes and Graph a Rational Function

Ex 2: Determine Asymptotes and Graph a Rational Function

Ex 3: Determine Asymptotes and Graph a Rational Function

Ex 4: Determine Asymptotes and Graph a Rational Function (Slant)

Ex: Absolute Extrema of a Quadratic Function on a Closed Interval

Ex: Absolute Extrema of a Trigonometric Function on a Closed Interval

Differentials

Ex: Use a Tangent Line to Approximate a Square Root Value

Ex: Use a Tangent Line to Approximate a Quotient

Ex: Use a Tangent Line to Approximate a Cube Root Function Value – Chain Rule

Ex 1: Determine Differential y (dy)

Ex 2: Differentials: Determine dy given x and dx

Ex: Differentials to Approximate Propagated Error and Relative Error

Ex: Using Differentials to Approximate the Value of a Cube Root.

Ex: Differentials: Compare delta y and dy

Ex: Find dy Given a Tangent Function - Requires the Chain Rule

Ex: Differentials - Approximate Delta y Using dy Using a Sine Function and Find Error Percent

Ex: Use Differentials to Approximate Possible Error for the Surface Area of a Sphere

Rolle’s Theorem and the Mean Value Theorem

Rolle’s Theorem

The Mean Value Theorem

Ex 1: Mean Value Theorem – Quadratic Function

Ex 2: Mean Value Theorem – Cubic Function

Ex 3: Mean Value Theorem – Rational Function

Ex 4: Mean Value Theorem – Quadratic Fomula Needed

Implicit Differentiation

Implicit Differentiation of Equations containing Transcendental Functions

Ex 1: Implicit Differentiation

Ex 2: Implicit Differentiation Using the Product Rule

Ex 3: Implicit Differentiation Using the Product Rule and Factoring

Ex 4: Implicit Differentiation Involving a Trig Function

Ex: Implicit Differentiation - Equation of Tangent Line

Ex: Implicit Differentiation Involving a Trig Function

Ex: Implicit Differentiation to Determine a Second Derivative

Related Rates

Ex 1: Related Rates: Determine the Rate of Change of Profit with Respect to Time

Ex 2: Related Rates: Determine the Rate of Change of the Area of a Circle With Respect to Time

Ex 3: Related Rates: Determine the Rate of Change of Volume with Respect to Time

Ex 4: Related Rates: Ladder Problem

Ex: Related Rates - Area of Triangle

Ex: Related Rates - Right Circular Cone

Ex: Related Rates - Rotating Light Projecting on a Wall

Ex: Related Rates - Volume of a Melting Snowball

Ex: Related Rates - Air Volume and Pressure

Ex: Related Rates Problem – Rate of Change of a Shadow from a Light Pole

Ex 2: Related Rates Problem -- Rate of Change of a Shadow from a Light Pole

Ex: Related Rates Problem -- Rate of Change of Distance Between Ships

Ex: Related Rates - Find the Rate of Change of Revenue

Ex: Related Rates - Find the Rate of Change of Revenue (Quotient Rule)

Newton’s Method and L’Hopital’s Rule

Newton’s Method

Ex: Newton’s Method to Approximate Zeros – 2 Iterations

L’Hopital’s Rule: Part 1, Part 2

Ex 1: L'Hopitals Rule Involving Trig Functions

Ex 2: L'Hopitals Rule Involving Trig Functions

Ex 3: L'Hopitals Rule Involving Exponential Functions

Ex: Use L'Hopital's Rule to Determine a Limit Approaching Infinity

Ex: Use L'Hopital's Rule to Determine a Limit Approaching Zero

Ex 1: Use L'Hopital's Rule to Determine a Limit Approaching Zero with Trig Function

Ex 2: Use L'Hopital's Rule to Determine a Limit Approaching Zero with Trig Function

Graphing Calculator

Determining the value of a definite integral on the graphing calculator

Sequences on the TI84 Graphing Calculator

Sequences and Series on the TI84

Graph Partial Sums of an Infinite Series on the TI84

Graphing Parametric Equations in the TI84