Calculus 1 Tutorials and Help

Introduction to Limits

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The Limit Laws

Limits and Continuity

Continuous Functions

The Continuity of Polynomial Functions

The Continuity of Rational Functions

A Survey of Continuous Functions

Checking for Continuity

Discontinuity

Visualizing Continuity and Discontinuity

One-Sided Limits and Continuity on an Interval

Continuity and Composite Functions

Continuity and the Intermediate Value Theorem

Finding Extrema on an Interval

Infinity

Diverging Limits

Visualizing Infinite Limits

Infinite Limits and Graphing guidelines

Undefined and Indeterminate Limit Values

Defining Limits At Infinity

Limits at Infinity and Infinite Limits

Limits at Infinity and the Limit Laws

Limits at Infinity and Rational Functions

Limits at Infinity in Rational Functions

Limits at Infinity and the Exponential Function

Taking a Limit at Infinity

Taking a Limit at Negative Infinity

Taking a Limit at Positive Infinity

Trigonometric Limits

Taking a Trigonometric Limit

Limits and the Equation of a Tangent Line

Direct Substitution and Continuous Functions

The Definition of the Sum and Difference Laws

Caveats when Determining Limits

Determining the Existence of Limits

Left-Hand and Right-Hand Limits

Proof of the Constant Multiple Law

Proof of the Product Law

Proof of the Quotient Law

Proving that a Limit Exists

Proving that a One-Sided Limit Exists

The Constant Multiple Law

Proof of the Sum/Difference Law

The Definition of a Limit.

The Definition of a One-Sided Limit

The Definition of the Constant Multiple Law

The Definition of the Product Law

The Definition of the Quotient Law

The Limit

The Limit of a Function At A Point

The Product Law

The Quotient Law

The Squeeze Theorem

The Sum and Difference Laws

Using the Limit Laws

Visualizing One Sided Limits

When Substitution Fails

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Tangent Lines and Limits

Secant Lines

Average Velocity and Secant Lines

Tangent Lines

Visualizing Tangent Lines

Tangent Lines as Limits

Visualizing a Tangent as the Limit of Secants

Deriving a Tangent Line

Determining the Equation of a Tangent Line

Instantaneous Velocity

Visualizing Instantaneous Velocity

Instantaneous Velocity and Tangent Lines

Instantaneous Velocity as a Limit

Calculating Instantaneous Velocity As A Limit

Calculating Average Rate of Change From Observational Data

Calculating Instantaneous Rate Of Change from Observational Data

Defining Instantaneous Rate Of Change

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Introduction to Derivatives

The Derivative

Limits and Derivatives

Derivatives minus Limits

Derivative Notations

Derivative and Slope

The Derivative At A Point

Visualizing the Derivative at a Point

The Magnitude of the Derivative at a Point

The Derivative As Instantaneous Rate Of Change

The Derivative as the Limit of Difference Quotients

Determining the Derivative From Data

Estimating Derivatives

The Derivative Function

The Graph of a Derivative Function

Differentiation on an Interval

Differentiability and Continuity

Finding a Horizontal Tangent

Finding a Vertical Tangent

First Form of a Derivative

Taking the Derivative Form One

Second Form of a Derivative

Taking the Derivative Form Two

The First and Second Derivatives

Notations for Higher Derivatives

Factoring to Find a Derivative

Finding the Nth Derivative

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Rules of Differentiation

Derivative of a Constant

Constant Function Test

Taking the Derivative of a Constant

Derivative of a Power Function

Proof of the Power Rule

Using the Power Rule

The Constant Multiple Rule

Using The Constant Multiple Rule

The Sum Difference Rule

Using the Sum Difference Rule

Introducing the Product Rule

The Product Rule

Proof of the Product Rule

Using the Product Rule

The Quotient Rule

The Proof of the Quotient Rule

Using the Quotient Rule

The Chain Rule

Proof of the Chain Rule

Applying the First Form of the Chain Rule

Applying the Second Form of the Chain Rule

Using Multiple Derivative Rules

Combining the Chain and Power Rules

Generalizing the Combination of Chain and Power Rules

Taking the Derivative of a Polynomial

Taking the Derivative of a Radical Function

Implicit Differentiation

Using Implicit Differentiation

Derivatives of Implicit and Explicit Functions

Derivatives for the Circle and Sphere

Derivative of the Natural Exponential

Derivatives of Exponential Functions

Derivatives of Logarithmic Functions

Derivatives of Trigonometric Functions

Logarithmic Differentiation

Proof of the Derivative of Hyperbolic Arccosecant

Proof of the Derivative of Hyperbolic Arccosine

Proof of the Derivative of Hyperbolic Arccotangent

Proof of the Derivative of Hyperbolic Arcsecant

Proof of the Derivative of Hyperbolic Arctangent

Proof of the Derivative of Hyperbolic Cosecant

Proof of the Derivative of Hyperbolic Cosine

Proof of the Derivative of Hyperbolic Secant

Proof of the Derivative of Hyperbolic Sine

Proof of the Derivative of Hyperbolic Tangent

Proof of the Derivative of the Cosecant Function

Proof of the Derivative of the Cosine Function

Proof of the Derivative of the Cotangent Function

Proof of the Derivative of the Sine Function

Proof of the Derivative of the Tangent Function

Proof of the Derivative Rule for exponential functions

Proof of the Rule for Derivatives of Logarithmic Functions

The Chain Rule and Exponential Functions

The Derivatives of Inverse Hyperbolic Trigonometric Functions

The Derivatives of Inverse Trigonometric Functions

Proof of the Derivative of Hyperbolic Arcsine

Proof of the Derivative of the Secant Function

The Derivative of the Natural Log Function

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Derivatives of Exponential & Logarithmic Functions

Derivatives of Exponential Functions

Proof of the Derivative Rule for exponential functions

Taking the Derivative of an Exponential Function

The Chain Rule and Exponential Functions

Derivatives of Logarithmic Functions

The Chain Rule and Logarithmic Functions

The Chain Rule in Logarithmic Functions

Proof of the Rule for Derivatives of Logarithmic Functions

Taking the Derivative of a Logarithmic Function

Logarithmic Differentiation

The Number e as a Limit

Exponential Derivatives and the Number e

Derivative of the Natural Exponential

The Derivative of the Natural Log Function

Proof of Derivative of Natural Logarithm

Taking the Derivative of Natural Logarithmic Functions

Ways to Simplify Exponential Equations

Simplifying Exponential Equations

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Derivatives of Trigonometric Functions

Proof of the Derivative of the Sine Function

Proof of the Derivative of the Cosine Function

Proof of the Derivative of the Tangent Function

Proof of the Derivative of the Cosecant Function

Proof of the Derivative of the Secant Function

Proof of the Derivative of the Cotangent Function

Taking Derivatives of Trigonometric Functions

The Derivatives of Inverse Trigonometric Functions

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Maximum and Minimum Values of a Function

Extrema

Visualizing Extrema

Local Extrema

Relative Extrema

Finding Relative Extrema

Graphing Relative Extrema

Global Extrema

Endpoint Extrema

Graphing Extrema

The Extreme Value Theorem

Fermat's Theorem

Proof of Fermat's Theorem

Seeing Fermat's Theorem In Action

When Fermat's Theorem Does Not Work

Critical Numbers

Visualizing Critical Numbers

Differentiation and Critical Numbers

Fermat's Theorem and Critical Numbers

Determining the Critical Numbers of a Function

Graphing Critical Numbers

The Closed Interval Method

The Steps to the Closed Interval Method

Finding Extrema with the Closed Interval Method

Rolle's Theorem

Examining Rolle's Theorem

Applications of Rolle's Theorem

The Mean Value Theorem

Proof of the Mean Value Theorem

Rolle's Theorem vs The Mean Value Theorem

Applying the Mean Value Theorem

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Determining the Shape of a Function

Derivatives and Function Intervals

Decreasing and Increasing Intervals

Interpreting the Increasing/Decreasing Test

Proof of the Increasing/Decreasing Test

The Increasing/Decreasing Test and the Mean Value Theorem

The First Derivative Test

The First Derivative Test and the Increasing/Decreasing Test

First Derivative Test and Local Extrema

Using the First Derivative Test

Interval Charts

Determining the Direction on an Interval

Finding the Intervals of a Function

Choosing a Test Number

Determining the Sign of a Test Number

Graphing Intervals

Determining Absolute Extreme Values

Concavity

Upward Concavity On an Interval

Downward Concavity On an Interval

The Concavity Test.

The Concavity Test and the Second Derivative

Determining Concavity

Concavity and Increasing/Decreasing Functions

Concavity and Extremes

Using Concavity to Optimize

Inflection

The Second Derivative and Inflection Points

Finding Inflection Points

Curve Sketching

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L'Hopital's Rule

L'Hôpital's Rule

Why does L'Hôpital's Rule Work?

Indeterminate Forms

Evaluating Limits Approaching Zero over Zero

Evaluating Limits Approaching Infinity over Infinity

Evaluating Limits Approaching Infinity times Zero

Evaluating Indeterminate powers

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Applications of Differentiation

Constant Difference Test

Static Rate of Change

Using Static Rates of Change

Differentiation and Parameters

Changing Units Through Differentiation

Polynomial Parameters for Derivatives

Interpretation of Derivatives of Polynomials

The Differential in the Denominator

Derivative Ratios

Applications of Derivatives

The Derivative In Economics

The Derivative In Engineering

The Derivative In Physics

Higher Derivatives

Higher Derivatives of Position

Using Higher Derivatives of Position

Acceleration and the Second Derivative

A physical interpretation of the third derivative

Strategies for Solving related rates problems

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Introduction to Integrals

Riemann Sums

Finding a Left Riemann Sum

Finding a Right Riemann Sum

Computing a Middle Riemann Sum

Computing a Trapezoidal Riemann Sum

Anti-Differentiation

Integrals

Integral Notation

Integration and Parameters

Units and Integration

First Fundamental Theorem of Calculus

Taking the Derivative of Integral

Taking the Integral of a Derivative

Finding Extrema on an Interval

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Simple Methods for Evaluating Integrals

The Indefinite Integral

Integrals and Constants

Taking the Integral of a Constant

Integrals and Powers

Taking the Integral of a Power

Integrals with Addition or Subtraction

Taking the Integral of a Sum

Taking the Integral of a Difference

Trigonometric Integrals

Taking the Integral of Sine and Cosine

Integral of Tangent

Taking the Integral of Tangent

Integral of Secant

Taking the Integral of Secant

Integral of Cotangent

Taking the Integral of Cotangent

Integral of Cosecant

Taking the Integral of Cosecant

Integral of Inverse Trigonometric Functions

Integration Using Inverse Trigonometric Functions

Hyperbolic Trigonometric Integrals

Integration of Hyperbolic Trigonometric Functions

Integral of an Exponential Function

Taking the Integral of an Exponential Function

Integral of a Polynomial

Taking the Integral of a Polynomial

Taking the Integral of a Function

Definite Integrals

Limits of Integration

Second Fundamental Theorem of Calculus

Choosing Appropriate Limits of Integration

Applying Limits of Integration

Computing a Definite Integral

Improper Integrals

Computing an Improper Integral

L'Hopital's Rule for Improper Integration

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More Complex Methods for Evaluating Integrals

Reverse Chain Rule

Substitution

Matching Limits and Differential

Integration by Substitution

Trigonometric Substitution

Integration Using Trigonometric Substitution

Integration by Parts

Using Integration by Parts

Double Integration by Parts

Partial Fraction Decomposition

Integration by Partial Fraction Decomposition

Integration by Fraction Decomposition

Decomposing with Non-Repeated Linear Terms

Decomposing with Repeated Linear Terms

Decomposing with Non-Linear Terms

Choosing the Correct Method of Integration

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Tutorials

Calculus 1 Tutorials

Introduction to Limits

AVAILABLE PATHWAYS

The Limit Laws

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Limits and Continuity

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Tangent Lines and Limits

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Introduction to Derivatives

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Rules of Differentiation

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Derivatives of Exponential & Logarithmic Functions

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Derivatives of Trigonometric Functions

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Maximum and Minimum Values of a Function

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Determining the Shape of a Function

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L'Hopital's Rule

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Applications of Differentiation

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Introduction to Integrals

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Simple Methods for Evaluating Integrals

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More Complex Methods for Evaluating Integrals

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Applications of Integrations

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Sequences and Series

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