Click to download the playlist as a pdf: Calculus I (Differential Calculus) Playlist

What is Calculus?

Limits

Introduction to 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

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

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]
Ex:  Use the Slope to Secant Lines to Predict the Slope of a Tangent Line
Ex:  Use Average Velocity to Predict Instantaneous 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)
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)

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
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

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
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: 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)

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
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

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)

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
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
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: 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 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
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
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
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

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

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