Limits

Introduction to Limits

Properties of Limits

Formal Definition of Limits Part 1

Formal Definition of Limits Part 2

Ex: Limit Definition - Find Delta Values, Given Epsilon For a Limit

Ex 1: Limit Definition - Determine Delta for an Arbitrary Epsilon (Linear)

Ex 2: Limit Definition - Determine Delta for an Arbitrary Epsilon (Quadratic)

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: Limits of the Floor Function (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 and Special Limits

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)

Determine a Limit at Infinity: Rational Function in Factored Form

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

Special Limits in the Form sin(x)/x

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

Determine a Slope to Make a Piecewise Function Continuous Everywhere (Lin/Quad)

Asymptotes: Part 1, Part 2

Limit and Continuity Concept Check with Piecewise Defined Function 1

Limit and Continuity Concept Check with Piecewise Defined Function 2

Average Rate of Change

Average Rate of Change

Graphical Approach to Average and Instantaneous 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 - Miles Per Hour

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]

Function Notation: Building to the Difference Quotient (Quadratic)

Building the Difference Quotient: Quadratic Function

Ex: Use the Slope to Secant Lines to Predict the Slope of a Tangent Line

Ex: Use Average Velocity to Predict Instantaneous Velocity

Estimate Instantaneous Velocity from Average Velocity

Ex: Determine the Intervals for Which the Slope of Tangent Lines is Positive, Negative, and Zero

Ex: Determine the Sign the Slope of a Tangent Line at Point on a Function

Ex: Approximate the Slope of a Tangent Line at a Point on a Function

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

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

Ex : Determine The Value of a Derivative using the Limit Definition (Quadratic)

Ex : Determine The Value of a Derivative using the Limit Definition (Rational)

Use the Limit Definition of the Derivative to Find a Derivative Function Value

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

Ex: Determine the Derivative of a Function Using the Limit Definition (ax^2+bx+c)

Use the Limit Definition of the Derivative to Find the Equation of a Tangent Line

Determine Derivative Function Values Using Desmos

Determine Derivative Function Values Using a Free Online Calculator (MathAS)

Differentiation of Basic Functions and Using the Power Rule

Derivative Flashcards Without The Chain Rule

Differentiation Techniques: 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

Equation of Tangent Line and Normal Line to a Cubic Function

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

Determine Basic Derivatives and Derivative Function Values (No Chain)

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

Ex: Find the x-intercept of a Tangent 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

Determine a Derivative Function Value on the TI84 (Newer Software)

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

Interpret the Meaning of a Derivative Function Value (Cost)

Interpret the Meaning of a Derivative Function Value (Population)

Interpret the Meaning of a Derivative Function Value (Temperature)

Determine a Function Value and Derivative Value Using Tangent Line

Why is the Derivative of the Area of a Circle Equal to the Circumference?

Why is the Derivative of the Volume of a Sphere Equal to the Surface Area?

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 1: Derivative of Trigonometric Functions – Simplify Before Differentiating

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

Find the First and Second Derivative Using the Quotient Rule (Mono Over Squared Binomial)

Find a Derivative using the Quotient Rule (Exponential Over Monomial)

The Derivative of a Rational Function Using the Quotient Rule (Powers of Binomials)

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

Derivative Flashcards With The Chain Rule

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

Determine a Derivative Function Using the Chain Rule: e to a Product

Determine a Derivative Function Using the Chain Rule (Ln of Linear Function)

Determine a Derivative Function Using the Chain Rule (Cosine of Ln)

Determine a Derivative Function Using the Chain Rule (Ln of a Sum)

Determine a Derivative Function Using the Chain Rule (Cosine)

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

Find a Derivative Using the Product Rule (Exponential Functions)

Find the Derivative of the Square Root of a Log Function

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

Determine a Derivative Function Using the Chain Rule: Comp of Three Functions

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

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)

Derivative App: Rate of Growth of People Infected by Flu y=ae^(kt)

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

Logarithmic Differentiation

Logarithmic Differentiation: sin(x) to the power of x

Logarithmic Differentiation: 2+x to the power of 2/x

Find a Derivative Using Logarithmic Differentiation

The Derivative of a Rational Function Using the Log Differentiation (Powers of Binomials)

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

Ex: Find an Inverse Derivative Function Value (Cubic)

Ex: Find an Inverse Derivative Function Value (Cubic + Rational)

Ex: Find an Inverse Derivative Function Value (Sine)

Ex: Find an Inverse Derivative Function Value (Square Root)

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

Find the Derivative of a Composite Function Involving Natural Log and Hyperbolic 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: Find Higher Order Derivatives of Sine

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)

The Second Derivative of 5ln(sec(x)+tan(x))

Applications of Differentiation – Relative Extrema

Ex: Find the Critical Numbers of a Cubic Function

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)

Ex: Find the Intervals Incr/Decr and Relative Extrema Using the First Derivative

Ex: Find the Intervals Incr/Decr and Relative Extrema (Quad Formula Used)

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

Determine Local (Relative) Extrema of a Polynomial Function Using the TI-84

Determine Local (Relative) Extrema of a Polynomial Function Using Desmos

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

Business Applications of Differentiation and Relative Extrema

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

Elasticity of Demand: D(x)=sqrt(300-0.5x^3)

Ex: Elasticity of Demand - Quadratic Demand Function

Determine Elasticity of Demand and Unit Elasticity Price (Linear Demand)

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: Given the First Derivative, Describe the Function (Incr/Decr/CCU/CCD)

Ex: Determine Concavity and Points of Inflection

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

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: Determine Concavity and Points of Inflection - f(x)=x^2*e^(4x)

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 - Maximized a Crop Yield (Calculus Methods)

Ex: Derivative Application - Minimize Cost

Ex: Derivative Application - Maximize Profit

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: Find the Average Cost Function and Minimize the Average Cost

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

Absolute Extrema on a Close Interval: Cubic Function

Absolute Extrema on a Close Interval: Quartic Function

Absolute and Relative Extrema From a Graph (Closed Interval)

Absolute and Relative Extrema From a Graph (Open Interval)

Absolute Extrema of Transcendental Functions

Ex 1: Absolute Extrema on an Closed Interval

Ex 2: Absolute Extrema on an Open Interval

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

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

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

Differentials

Introduction to Differentials

Tangent Line Approximation Given a Function and Derivative Function Value

Make a Tangent Line Approximation for a Square Root Function Value

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

Differentials

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

Determine Absolute Error and Percent Error

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

Proof of Rolle's Theorem

Ex 1: Rolle's Theorem

Ex 2: Rolle's Theorem with Product Rule

The Mean Value Theorem

Proof of 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

Introduction to Basic Implicit Differentiation

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

Ex: Perform Implicit Differentiation and Find the Equation of a Tangent Line

Ex: Find dy/dx Using Implicit Differentation and the Product Rule - e^(2xy)=y^n

Ex: Find dy/dx Using Implicit Differentation and the Product Rule - ax-bxy-cy^n=d

Related Rates

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

Solve a Related Rates Problem Using the Product Rule

Related Rates Applicate: Leaking Conical Tank

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

Determine if L'Hopital's Rule Can Be Applied to a Limit (Ex 1)

Determine if L'Hopital's Rule Can Be Applied to a Limit (Ex 2)

Determine if L'Hopital's Rule Can Be Applied to a Limit (Ex 3)

L'Hopital's Rule - Justification Using Tangent Lines (Form 0/0)

Partial Proof of L'Hopital's Rule (Only Form 0/0)

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

Proofs

The Squeeze Theorem

Prove the Limit as x Approaches 0 of sin(x)/x

Prove the Limit as x Approaches 0 of (1-cos(x))/x

Prove the Limit as x Approaches 0 of (e^x-1)/x

Prove the Derivative of a Constant: d/dx[c]

Proof - the Derivative of a Constant Times a Function: d/dx[cf(x)]

Proof - the Derivative of Sum and Difference of Functions: d/dx[f(x)+g(x)]

Proof - The Derivative of Sine: d/dx[sin(x)]

Proof - The Derivative of Cosine: d/dx[cos(x)]

Proof - The Power Rule of Differentiation

Proof - The Product Rule of Differentiation

Proof - The Quotient Rule of Differentiation

Proof - The Chain Rule of Differentiation

Proof - The Derivative of f(x) = e^x: d/dx[e^x]=e^x (Limit Definition)

Proof - The Derivative of f(x) = e^x: d/dx[e^x]=e^x (Implicit Differentiation)

Proof - The Derivative of f(x)=ln(x): d/dx[ln(x)]=1/x (Implicit Diff)

Proof - The Derivative of f(x)=log_a(x): d/dx[log_a(x)]=1/((ln a)x)

Proof - The Derivative of f(x)=a^x: d/dx[a^x]=(ln a)a^x (Definition)

Proof - The Derivative of f(x)=a^x: d/dx[a^x]=(ln a)a^x (Using Logs)

Proof - The Derivative of Tangent: d/dx[tan(x)]

Proof - The Derivative of Cotangent: d/dx[cot(x)]

Proof - The Derivative of Secant: d/dx[sec(x)]

Proof - The Derivative of Cosecant d/dx[csc(x)]

Proof - The Derivative of f(x)=arcsin(x): d/dx[arcsin(x)]

Proof - The Derivative of f(x)=arccos(x): d/dx[arccos(x)]

Proof - The Derivative of f(x)=arctan(x): d/dx[arctan(x)]

Proof - The Derivative of f(x)=arccot(x): d/dx[arccot(x)]

Proof - The Derivative of f(x)=arccsc(x): d/dx[arccsc(x)]

Proof - The Derivative of f(x)=arcsec(x): d/dx[arcsec(x)]

Proof of Rolle's Theorem

The Mean Value Theorem

Proof of the Mean Value Theorem

L'Hopital's Rule - Justification Using Tangent Lines (Form 0/0)

Partial Proof of L'Hopital's Rule (Only Form 0/0)