Calculus Lectures in Videos

UNIT 0 - FUNCTIONS: A Review of Precalculus 
Beginning 

• Definition of a Function [7.5 min.]
• Visualizing Functions: Graphs [10.5 min.]
• Domain (& Range) of Functions [7.5 min.]
• Some Exercises [11 min.]

Graphing Technology 

• Viewing Windows [3.5 min.]
• Zooming In or Out [2.5 min.]
• Errors in Resolution [5.5 min.]

New Functions From Old 

• Operations on Functions [10 min.]
• How Operations Affect Function Graphs [6.5 min.]
• Functions with Symmetric Graphs [2 min.]
• Some Exercises [8 min.]

Families of Functions 

• The Power Function Family y = x^p [7 min.]
• The Polynomial Function, and Rational Function Families [4 min.]

Trigonometry for Calculus 

• Right Triangle Trigonometry [11.5 min.]
• Trigonometric Graphs [5 min.]
• Handy Trigonometric Identities [5 min.]
• Laws of Sine and Cosine [4 min.]
• Trigonometric Families [4 min.]

Inverse Functions 

• A Function Inverse to Another Function [6 min.]
• When do Inverse Functions (& Their Graphs) Exist? [4 min.]
• Inverse Trigonometric Functions [10 min.]

Exponential & Logarithmic Functions 

• The Exponential Function Family [6 min.]
• The Logarithmic Function Family [6.5 min.]
• Solving Exponential & Logarithmic Equations [6 min.]

UNIT 1 - LIMITS of Functions: Approach & Destination 
Intuitive Beginning 

• A New Tool: The "Limit" [24 min.]
• Some Limit Examples [6.5 min.]
• Two-sided & One-sided Limits [17 min.]
• Limits that Fail to Exist: When f(x) grows without bound [9 min.]
• Limits at Infinity: When x grows without bound [11.5 min.]
• More Limits that Fail to Exist: Infinity & Infinite Indecision [6 min.]
• An Exercise on Limits [5 min.]

The Algebra of Limits as x -> a 

• Basic Limits [7.5 min.]
• Limits of Sums, Differences, Products, Quotients, & Roots [11 min.]
• Limits of Polynomial Functions [7 min.]
• Limits of Rational Functions & the Apparent Appearance of 0/0 [17 min.]
• Limits of Piecewise-Defined Functions: When One-sided Limits Matter! [12 min.]
• Some Exercises [11 min.]

The Algebra of Limits as x -> +/- inf : End Behavior 

• Basic Limits [4.5 min.]
• Limits of Sums, Differences, Products, Quotients, & Roots [10 min.]
• Limits of Polynomial Functions: Two End Behaviors [9 min.]
• Limits of Rational Functions: Three Types of End Behavior [16 min.]
• Limits of Functions with Radicals [9.5 min.]
• Some Exercises [6.5 min.]
• Limits of ln(x), ex, and More [7 min.]

 Continuous Functions 

• Functions Continuous (or not!) at a Single Point x=c [14.5 min.]
• Functions Continuous on an Interval [6 min.]
• Properties & Combinations of Continuous Functions [14 min.]
• The Intermediate Value Theorem & Approximating Roots: f(x) = 0 [13 min.]
• Some Exercises [13.5 min.]

Trigonometric Functions 

• The 6 Trigonometric Functions: Continuous on Their Domains [4 min.]
• When Inverses are Continuous [3 min.]
• Finding a Limit by "Squeezing" [6 min.]
• Sin(x)/x -> 1 as x -> 0, and Other Limit Tales [11 min.]
• Some Exercises [7.5 min.]

UNIT 2 - The DERIVATIVE of a Function 
Measuring Rates of Change 

• Slopes of Tangent Lines [14 min.]
• One-Dimensional Motion [7 min.]
• Average Velocity [6.5 min.]
• Instantaneous Velocity [8 min.]
• General Rates of Change [8.5 min.]
• Some Exercises [10.5 min.]

What is a Derivative? 

• Definition of the Derived Function: The "Derivative", & Slopes of Tangent Lines [11 min.]
• Instantaneous Velocity [2 min.]
• Functions Differentiable (or not!) at a Single Point [9 min.]
• Functions Differentiable on an Interval [3 min.]
• A Function Differentiable at a point is Continuous at that point [6 min.]
• Other Derivative Notations [4 min.]
• Some Exercises [11 min.]

Finding Derivatives I: Basic Rules 

• The Power Rule [13.5 min.]
• Constant Multiple, Sum, & Difference Rules [5.5 min.]
• Notation for Derivatives of Derivatives [6 min.]
• Some Exercises [7 min.]

Finding Derivatives II:

• The Product Rule [14.5 min.]
• The Quotient Rule [9 min.]
• Some Exercises [10 min.]

Finding Derivatives III: 

• The Sine Function [5 min.]
• The Other Trigonometric Functions [9 min.]
• Some Applications [11 min.]

Finding Derivatives IV: 

• The Chain Rule: Derivatives of Compositions of Functions [14.5 min.]
• Generalized Derivative Formulas [5.5 min.]
• Some Exercises [8 min.]

When Rates of Change are Related 

• Differentiating Equations to "Relate Rates" [10 min.]
• A Strategy [14.5 min.]
• An Exercise [11 min.]

More on Derivatives 

• Local Linear Approximations of Non-Linear Functions [7 min.]
• Defining "dx" and "dy" Alone [5 min.]

UNIT 3 - Some Special DERIVATIVES 
Implicit Differentiation 

• Functions Defined Implicitly [7 min.]
• Derivatives of Functions Defined Implicitly [7.5 min.]
• The Derivative of Rational Powers of x [6 min.]
• Some Exercises [6 min.]

Derivatives Involving Logarithms 

• Derivatives of Logarithmic Functions [14.5 min.]
• The "Logarithmic Differentiation" Technique [5.5 min.]
• The Derivative of Irrational Powers of x [3.5 min.]
• Some Exercises [5 min.]

Derivatives Involving Inverses 

• Derivatives of Inverse Functions [10 min.]
• Derivatives of Exponential Functions [7.5 min.]
• Derivatives of Inverse Trigonometric Functions [6 min.]
• Some Exercises [5 min.]

Finding Limits Using Differentiation 

• Limits of Quotients that appear to be "Indeterminate": The Rule of L'Hopital [14 min.]
• Some Examples [11 min.]
• Finding Other "Indeterminate" Limits [16 min.]

UNIT 4 - The DERIVATIVE Applied 
Analyzing the Graphs of Functions I 

• Increasing & Decreasing Functions: The 1st Derivative Applied [16 min.]
• Functions Concave Up or Concave Down: The 2nd Derivative Applied [14 min.]
• When Concavity Changes: Inflection Points [19 min.]
• Logistic Growth Curves: A Brief Look [3 min.]
• Some Exercises [14 min.]

Analyzing the Graphs of Functions II 

• Local Maximums & Minimums [11 min.]
• The 1st Derivative Test for Local Maximums & Minimums [7.5 min.]
• The 2nd Derivative Test for Local Maximums & Minimums [9.5 min.]
• Polynomial Function Graphs [15 min.]
• Some Exercises [17.5 min.]

Analyzing the Graphs of Functions III 

• What to Look For in a Graph [2 min.]
• Rational Function Graphs [19.5 min.]
• Functions Whose Graphs have Vertical Tangents or Cusps [16 min.]
• Some Exercises [18.5 min.]

Analyzing the Graphs of Functions IV 

• Global Maximums & Minimums [4.5 min.]
• Global Extrema on (finite) Closed Intervals [10 min.]
• Global Extrema on (finite or infinite) Open Intervals [13 min.]
• When a Single Local Extremum must be Global [4.5 min.]
• Some Exercises [12 min.]

Optimization Problems 

• Applied Maximum & Minimum Problems [4 min.]
• Optimization over a (finite) Closed Interval: 
  Maximizing Area or Volume, Minimizing Cost [21.5 min.]
• Optimization over Other Intervals: Minimizing Materials or Distance [11 min.]
• An Economics Application: Cost, Revenue, Profit, & Marginal Analysis [9 min.]
• Some Exercises [17 min.]

Newton's Method for Approximating Roots of Equations 

• Development of the Method [12 min.]
• Strength & Weaknesses of the Method [5.5 min.]

The Mean Value Theorem for Derivatives 

• A Special Case of the Mean Value Theorem: Rolle's Theorem [8.5 min.]
• The (Full) Mean Value Theorem for Derivatives [20 min.]
• Direct Consequences of This Mean Value Theorem [13 min.]
• Some Exercises [7 min.]

One-Dimensional Motion & the Derivative 

• Rectilinear Motion Revisited [4.5 min.]
• Velocity, Speed, & Acceleration [12 min.]
• Analyzing a Position Graph [8.5 min.]
• An Exercise [11.5 min.]

UNIT 5 - The INTEGRAL of a Function 
The Indefinite Integral 

• "Undo-ing" a Derivative: Antiderivative = Indefinite Integral [16 min.]
• Finding Antiderivatives [22 min.]
• The Graphs of Antiderivatives: Integral Curves & the Slope Field Approximation [16.5 min.]
• The Antiderivative as Solution of a Differential Equation [5 min.]
• Some Exercises [6.5 min.]

Indefinite Integration by Substitution 

• The Substitution Method of Indefinite Integration: A Major Technique [7.5 min.]
• Straightforward Substitutions [10.5 min.]
• More Interesting Substitutions [11.5 min.]
• Some Exercises [7 min.]

Area Defined as a Limit 

• The Sigma Shorthand for Sums [13.5 min.]
• Summation Properties & Handy Formulas [9 min.]
• Definition of Area "Under a Curve" [15 min.]
• Net "Area" [4 min.]
• Approximating Area Numerically [2.5 min.]
• Some Exercises [6.5 min.]

The Definite Integral 

• The Definite Integral Defined [11.5 min.]
• The Definite Integral of a Continuous Function = Net "Area" Under a Curve [6 min.]
• Finding Definite Integrals [10.5 min.]
• A Note on the Definite Integral of a Discontinuous Function [6 min.]
• Some Exercises [6.5 min.]

The Fundamental Theorem of Calculus 

• The Fundamental Theorem of Calculus, Part 1 [15 min.]
• Definite & Indefinite Integrals Related [7.5 min.]
• The Mean Value Theorem for Integrals [9.5 min.]
• The Fundamental Theorem of Calculus, Part 2 [7 min.]
• Differentiation & Integration are Inverse Processes [2 min.]
• Some Exercises [5 min.]

One-Dimensional Motion & the Integral 

• Position, Velocity, Distance, & Displacement [16 min.]
• Uniformly Accelerated Motion [12 min.]
• The Free Fall Motion Model [6.5 min.]
• An Exercise [5 min.]

Definite Integration by Substitution 

• Extending the Substitution Method of Integration to Definite Integrals [9 min.]
• Some Exercises [4 min.]

UNIT 6 - The DEFINITE INTEGRAL Applied 
Plane Area 

• Area Between Two Curves [One Floor, One Ceiling] [11 min.]
• Area Between Two Curves [One Left, One Right] [7.5 min.]
• An Exercise [5.5 min.]

Volumes I 

• Volumes by Slicing [12.5 min.]
• Volumes of Solids of Revolution: Disks [15.5 min.]
• Volumes of Solids of Revolution: Washers [12 min.]
• Some Exercises [8.5 min.]

Volumes II 

• Volumes of Solids of Revolution: Cylindrical Shells [14.5 min.]
• An Exercise [6 min.]

Length of a Plane Curve 

• Finding Arc Lengths [11.5 min.]
• Finding Arc Lengths of Parametric Curves [6.5 min.]

Average Value of a Function 

• Average (Mean) Value of a Continuous Function [13 min.]

Work 

• Work Done by a Constant Force [3 min.]
• Work Done by a Variable Force [13.5 min.]
• Do-It-Yourself Integrals: Pumping Fluids [8 min.]
• Work as Change in Kinetic Energy [6 min.]
• An Exercise [5 min.]

Calculus 2

Surfaces and Partial Derivatives
Tangent plane and error estimation
Chain rule for functions of two variables

Integrals of trig functions and reduction formulae
Integration by trig substitution and partial fractions
Integration + Partial Fractions
Integration via rationalizing substitutions and other substitutions

Partial derivatives and integration
Introduction to Separable Differential Equations
Linear and Exact Differential Equations

How to solve 2nd order differential equations

What is a Taylor polynomial? 
Sequences and their limits
Introduction to series + the integral test
Integration and differential equations
Series, comparison + ratio tests
Alternating series and absolute convergence
What is a Taylor series?
What is a power series?

Arc length + average value of a function.
Surface area of revolution of functions
Calculus revision lecture

Hydrostatic Force

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