Math Doctor Bob

Calculus - Part 1: Limits and Derivatives

Limits 1a - Definitions and Concepts

Limits 1b - Delta-Epsilon Formulation

Limits 1c - Limit Failure

Limits 1d - Polynomials and Rational Functions

Limits 1e - Compositions and the Squeeze Theorem

Limits 1f - Trigonometric Functions

Examples of Limits

Continuity 1a - Definition and Basic Concepts

Continuity 1b - Polynomial/Rational Functions and The Extreme Value Theorem

Examples of Continuity

Fast Solution of Inequality Using Continuity

Bisection Method 1

Bisection Method 2

Vertical Asymptotes 1a

Vertical Asymptotes 1b

Definition of Tangent Line

Example of Tangent Line

Definition of Derivative

Power Rule for Derivatives

Derivatives of Polynomials

Tangent Line to x^2 - 4x

Horizontal Tangent Lines to a Polynomial

Derivative of sin(x) and cos(x)

Tangent Lines to sin(x)

Motion in a Line

The Product Rule

General Product Rule

Power Rule for Rational Exponents

The Quotient Rule

Trig Derivatives

Examples of Trig Derivatives

Tangent Lines to sec(x)

Tangent Lines to cot(x)

The Chain Rule

Example of Chain Rule 1 - Basic examples

Example of Chain Rule 2 - Approximation with Tangent Line

Example of Chain Rule 3 - Trig Functions

Example of Chain Rule 4 - Triple Chain Rule

Higher Order Derivatives

Graphs and Higher Order Derivatives

Motion of a Spring

Implicit Differentiation 1 - Definition and Basic Concepts

Implicit Differentiation 2 - Basic Example

Implicit Differentiation 3 - Approximation with Tangent Line

Implicit Differentiation 4 - Example with Trig Functions

Implicit Differentiation 5 - Higher Derivatives

Related Rates

Example of Related Rates 1

Example of Related Rates 2

Extreme Value Theorem

Example of Extreme Value Theorem 1

Example of Extreme Value Theorem 2

Example of Extreme Value Theorem 3

Rolle's Theorem

Mean Value Theorem

Increasing/Decreasing and Derivatives 1

Increasing/Decreasing and Derivatives 2

Example of Increasing/Decreasing 1

Example of Increasing/Decreasing 2

Example of Increasing/Decreasing 3

First/Second Derivative Test for f(x) = x^4 - 12x^3

First/Second Derivative Test for f(x) = sin(x)

First/Second Derivative Test for f(x) = x^2 - 6x^{4/3}

Concavity and the Second Derivative

Concavity for f(x) = sin(x)

Concavity for f(x) = (x^2 - 36)/(x-2)

Concavity for f(x) = |x^2 - 4x - 12|

Example of Limit at Infinity 1

Example of Limit at Infinity 2

Example of Limit at Infinity 3

Checklist for Sketching Functions

Graph of f(x) = x^4 - 8x^3

Graph of f(x) = (x-2)/(x-1)

Graph of f(x) = sin(x) + cos(x)

Graph of f(x) = sin(x)/(1+cos(x))

Graph of f(x) = x^{4/3} - 8x^{2/3}

Optimization 1

Optimization 2

Optimization 3

Optimization - Maximizing Profit

Newton's Method 1

Newton's Method 2

Differentials 1

Differentials 2

Calculus - Part 2: Basic Integration

Definition of Antiderivative

Antiderivative of a Polynomial

Antiderivative of (x-1)(x-2)/sqrt(x^3)

Basic Trig Antiderivatives

Antiderivative of tan^2(x)

Antiderivative of sin(x)/[1-sin^2(x)]

Visualizing an Antiderivative

Solving Differential Equations

Equations of Motion

Graphing f(x) from f'(x)

Antiderivative of a Piecewise-Defined Function

Overview of Rectangular Approximation of Area

Rectangular Approximation of Area

Overview of Summation Formulas

General Method for Integer Sum Formula

Riemann Sum with Signed Area

Limit Summation Formula

Limit Process for Area

Definition of Definite Integral

Definite Integral as Area 1 - Using the Graph

Definite Integral as Area 2 - Breaking Up the Region

Definite Integral as Area 3 - Area Below the x-axis

The First Fundamental Theorem of Calculus

Area Under a Curve 1

Area Under a Curve 2

The Mean Value Theorem for Integrals

Example of Mean Value Theorem for Integrals

The Second Fundamental Theorem of Calculus

Example of 2nd Fundamental Theorem of Calculus 1

Example of 2nd Fundamental Theorem of Calculus 2

Integration By Substitution: Antiderivatives

Integration by Substitution: Definite Integrals

Integration by Substitution 1

Integration by Substitution 2

Integration by Substitution 3

Integration by Substitution 4

Integration by Substitution 5

Trapezoid Rule

Simpson's Rule

Calculus - Part 3: Log, Exp, and Inverse Trig Integrals

Definition of ln(x) Using Integration

Properties of ln(x) Using Integration

End Behavior of ln(x)

Derivative of ln(x) as a Slope

The Chain Rule for ln(x) and ln|x|

Derivative of f(x) = ln(ln(x^2 + 1))

Graph of f(x) = 3ln(x-2) + 4

Graph of f(x) = ln(x^2 - x)

Graph of f(x) = ln|x^2-x|

Implicit Differentiation with ln(x)

Logarithmic Differentiation

Antiderivative of tan(x) Using ln(x)

Antiderivative of ln(x^4)/x (HD Version)

Antiderivative involving 1/x 1

Antiderivative involving 1/x 2

Log Integral for (x^2+5x+1)/(x^2+1)

Trig Antiderivatives involving 1/x

Area under f(x) = ln(x)/x

Inverse Functions

Inverse Functions: ln(x) and exp(x)

Graph of f(x) = 2 - e^{2x-1}

Inverse Function for f(x) = 2 e^{x-2}

No Inverse Function for f(x)=ln(x)-exp(x)

Derivative of an Inverse Function

Properties of Derivative of exp(x)

Graph of f(x) = x^2 e^{-x}

Graph of Normal Distribution

Examples of Derivatives with e^x

Implicit Differentiation with e^x

Tangent Line to x^2 e^x at x=1

Indefinite Integrals with e^x

Definite Integral of exp(-3x+2) (HD Version)

Definite Integral of (e^x - 1)(e^x - 2)/e^2x (HD Version)

Graphing with 2^x

Equations with Logarithms

Derivatives of log_b(x) and b^x

Tangent Lines to log_b(x) and b^x

Logarithmic Differentiation

Indefinite Integrals with b^x and log_b(x)

Definite Integrals with b^x and log_b(x)

Compound Interest

Evaluating Inverse Trig Functions 1

Evaluating Inverse Trig Expressions 2

Evaluating Inverse Trig Expressions 3

Derivatives of Inverse Trig Functions

Graph of f(x) = tan^{-1}(1-x^2)

Tangent Line to f(x) = xsin^{-1}(x/2)

Integrals with Inverse Trig Functions 1

Integrals with Inverse Trig Functions 2

Integrals with Inverse Trig Functions 3

Calculus - Part 4: Applied Integration

The Area between Two Curves 1

The Area between Two Curves 2

Example of Volume Using Cross Sectional Area

The Disk/Washer Method for Volume 1

The Disk/Washer Method for Volume 2

The Disk/Washer Method for Volume 3

The Shell Method for Volume 1

The Shell Method for Volume 2

The Shell Method for Volume 3

Formula for Arc Length 1

Formula for Arc Length 2

Formula for Arc Length 3

Arc Length Along Parabola 1: Base Case

Arc Length Along Parabola 2: Sinh Formula

Arc Length Along Parabola 3: Log Formula

Area of a Surface of Revolution

Moments and Center of Mass 1

Moments and Center of Mass 2

Moments and Center of Mass 3

Moments and Center of Mass 4

Moments and Center of Mass 5

Calculus - Part 5: Advanced Integration Techniques

Integration by Parts 1

Integration by Parts 2

Integration by Parts 3

Integration by Parts 4 - Antiderivative of e^(3x)cos(4x) (Double IBP)

Integration By Parts 5 - Antiderivative for e^{3x}cos(4x) (Fast Solution)

Integration by Parts 6

Fast Antiderivative of x^2 exp(3x)

Reduction Formula for x^n exp(ax)

Integrals with cos^m(x) sin^n(x)

Integral of cos(mx)cos(nx)

Integral of tan^m(x) sec^n(x)

Integral of tan^6(x)

Integral of sec^5(x)

Trig Substitution 1 - Basic Inverse Trig Integrals

Trig Substitution 2 - Integral for (1+x^2)^{5/2}

Trig Substitution 3 - Integral of x^2/sqrt(1-4x^2)

Trig Substitution 4 - Integral of sqrt(e^{2x} - 1)

Trig Substitution Integral of sqrt(4-36x^2)/x^2

Integration with Partial Fractions 1

Integration with Partial Fractions 2

Integration with Partial Fractions 3

Integration with Partial Fractions 4

Integration with Partial Fractions 5

Partial Fraction Integral with Linear and Quadratic Factors

Partial Fraction Integral of x^2/(4x^-36)^2

Trig Sub Integral of x^2/(4x^2-36)^2

L'Hopital's Rule 1

L'Hopital's Rule 2

L'Hopital's Rule 3

L'Hopital's Rule 4

Growth of Functions at Infinity

Improper Integrals 1

Improper Integrals 2

Improper Integral of 1/x^p

Improper integral of exp(-1/x)/x^2

Mean of the Exponential Distribution

Calculus - Part 6: Sequences and Series

Sequences 1a

Examples of Sequences

Examples of Recursive Sequences

Sequences 1b

Sequences 2

Sequences 3

Sequences 4

Infinite Series 1a

Infinite Series 1b - Geometric Series

Infinite Series 1c

Infinite Series 2

Infinite Series 3

Fractals

The Integral Test for Series 1a

The Integral Test for Series 1b

The Integral Test for Series 2

Estimating Sums with the Integral Test

Direct Comparison Test for Series 1

Divergence of Series for 1/ln(n)

Limit Comparison Test for Series 1

Limit Comparison Test for Series 2

Rational Function Test for Series

Alternating Series 1a

Alternating Series 1b - Estimating the Remainder

Alternating Series 1c

Absolute Convergence Test

The Ratio Test for Series

Series Convergence for n!/n^n

The Root Test for Series

Root Test for Series Sum (1-1/n^2)^{n^3}

Series Test Round-Up 1

Series Test Round-Up 2

Series Test Round-Up 3

Motivating Taylor Polynomials 1

Motivating Taylor Polynomials 2

Re-centering Polynomials

Approximating with Maclaurin Polynomials

Fast Maclaurin Polynomial for Rational Function

Approximating with Taylor Polynomials

Taylor's Theorem for Remainders

Taylor's Theorem : Remainder for 1/(1-x)

Power Series 1a

Power Series 1b

Power Series 1c

Power Series 1d

Example of Interval of Convergence Using Ratio Test

Power Series with Squares

Derivative/Antiderivative of a Power Series 1a

Derivative/Antiderivative of a Power Series 1b

Derivative/Antiderivative of a Power Series 1c

Increasing the Interval of Convergence

Constructing Power Series from Functions 1a

Constructing Power Series from Functions 1b

Constructing Power Series from Functions 1c

The Taylor Series for f(x) = ln(x) at x = 1

The Maclaurin Series for f(x) = 1/(1-x)^2

The Maclaurin Series for f(x) = e^x

The Maclaurin Series for sin(x), cos(x), and tan(x)

The Maclaurin Series of f(x) = (1+x)^{1/2} 1a

The Maclaurin Series for f(x) = (1+x)^{1/2} 1b