Math 210 - Calculus I

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This is a collection of video lectures for Math 210: Calculus I from UMKC (University of Missouri-Kansas city). Consisting of 31 lectures taught by Professor Richard Delaware, this course introduces the concepts and techniques of differential calculus and integral calculus. The topics covered here include a review of precalculus, limits of functions, the derivative of a function, applications of differential calculus, the integral of a function, and applications of integral calculus.

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Listing of the videos and their contents


Unit 0 - Functions: A Review of Precalculus

Lecture 01 - Beginning, Graphing Technology
Definition of a Function. Visualizing Functions: Graphs. Domain (& Range) of Functions. Viewing Windows. Zooming In or Out. Errors in Resolution.

Lecture 02 - New Functions from Old, Families of Functions
Operations on Functions. How Operations Affect Function Graphs. Functions with Symmetric Graphs. The Power Function Family. The Polynomial Function, and Rational Function Families.

Lecture 03 - Trigonometry for Calculus, Inverse Functions, and Exponential & Logarithmic Functions
Right Triangle Trigonometry. Trigonometric Graphs. Handy Trigonometric Identities. Laws of Sine and Cosine. A Function Inverse to Another Function. Inverse Trigonometric Functions. The Exponential Function Family. The Logarithmic Function Family.


Unit 1 - Limits of Functions: Approach & Destination

Lecture 04 - Intuitive Beginning
A New Tool: The "Limit". Some Limit Examples. Two-sided & One-sided Limits. Limits that Fail to Exist: When f(x) grows without bound. Limits at Infinity: When x grows without bound.

Lecture 05 - The Algebra of Limits as x -> a
Basic Limits. Limits of Sums, Differences, Products, Quotients, & Roots. Limits of Polynomial Functions. Limits of Rational Functions & the Apparent Appearance of 0/0. Limits of Piecewise-Defined Functions.

Lecture 06 - The Algebra of Limits as x -> +/- ∞ : End Behavior
Basic Limits. Limits of Sums, Differences, Products, Quotients, & Roots. Limits of Polynomial Functions: Two End Behaviors. Limits of Rational Functions: Three Types of End Behavior. Limits of Functions with Radicals.

Lecture 07 - Continuous Functions
Functions Continuous (or not!) at a Single Point x=c. Functions Continuous on an Interval. Properties & Combinations of Continuous Functions. The Intermediate Value Theorem & Approximating Roots.

Lecture 08 - Trigonometric Functions
The 6 Trigonometric Functions: Continuous on Their Domains. When Inverses are Continuous. Finding a Limit by "Squeezing". Sin(x)/x -> 1 as x -> 0, and Other Limit Tales.


Unit 2 - The Derivative of a Function

Lecture 09 - Measuring Rates of Change
Slopes of Tangent Lines. One-Dimensional Motion. Average Velocity. Instantaneous Velocity. General Rates of Change.

Lecture 10 - What is a Derivative?
Definition of the Derived Function: The "Derivative", & Slopes of Tangent Lines. Instantaneous Velocity. Functions Differentiable (or not!) at a Single Point. Functions Differentiable on an Interval.

Lecture 11 - Finding Derivatives I & II
The Power Rule. Constant Multiple, Sum, & Difference Rules. Notation for Derivatives of Derivatives. The Product Rule. The Quotient Rule.

Lecture 12 - Finding Derivatives III & IV
The Sine Function. The Other Trigonometric Functions. The Chain Rule: Derivatives of Compositions of Functions. Generalized Derivative Formulas.

Lecture 13 - When Rates of Change are Related
Differentiating Equations to "Relate Rates". A Strategy. Local Linear Approximations of Non-Linear Functions. Defining "dx" and "dy" Alone.


Unit 3 - Some Special Derivatives

Lecture 14 - Implicit Differentiation, Derivatives Involving Logarithms
Functions Defined Implicitly. Derivatives of Functions Defined Implicitly. The Derivative of Rational Powers of x. Derivatives of Logarithmic Functions.

Lecture 15 - Derivatives Involving Inverses, Finding Limits Using Differentiation
Derivatives of Inverse Functions. Derivatives of Exponential Functions. Derivatives of Inverse Trigonometric Functions. Limits of Quotients that appear to be "Indeterminate": The Rule of L'Hopital. Finding Other "Indeterminate" Limits.


Unit 4 - The Derivative Applied

Lecture 16 - Analyzing the Graphs of Functions I
Increasing & Decreasing Functions. Functions Concave Up or Concave Down. When Concavity Changes: Inflection Points. Logistic Growth Curves: A Brief Look.

Lecture 17 - Analyzing the Graphs of Functions II
Local Maximums & Minimums. The 1st Derivative Test for Local Maximums & Minimums. The 2nd Derivative Test for Local Maximums & Minimums. Polynomial Function Graphs.

Lecture 18 - Analyzing the Graphs of Functions III
What to Look For in a Graph. Rational Function Graphs. Functions Whose Graphs have Vertical Tangents or Cusps.

Lecture 19 - Analyzing the Graphs of Functions IV
Global Maximums & Minimums. Global Extrema on (finite) Closed Intervals. Global Extrema on (finite or infinite) Open Intervals. When a Single Local Extremum must be Global.

Lecture 20 - Optimization Problems
Applied Maximum & Minimum Problems. Optimization over a (finite) Closed Interval: Maximizing Area or Volume, Minimizing Cost. Optimization over Other Intervals: Minimizing Materials or Distance. An Economics Application.

Lecture 21 - Newton's Method for Approximating Roots of Equations, The Mean Value Theorem for Derivative
Development of the Method. Strength & Weaknesses of the Method. A Special Case of the Mean Value Theorem: Rolle's Theorem. The (Full) Mean Value Theorem for Derivatives. Direct Consequences of This Mean Value Theorem.

Lecture 22 - One-Dimensional Motion & the Derivative
Rectilinear Motion Revisited. Velocity, Speed, & Acceleration. Analyzing a Position Graph.


Unit 5 - The Integral of a Function

Lecture 22-5 - The Question of Area
Brief History and Overview

Lecture 23 - The Indefinite Integral
"Undo-ing" a Derivative: Antiderivative = Indefinite Integral. Finding Antiderivatives. The Graphs of Antiderivatives: Integral Curves & the Slope Field Approximation. The Antiderivative as Solution of a Differential Equation.

Lecture 24 - The Indefinite Integration by Substitution
The Substitution Method of Indefinite Integration: A Major Technique. Straightforward Substitutions. More Interesting Substitutions.

Lecture 25 - Area Defined as a Limit
The Sigma Shorthand for Sums. Summation Properties & Handy Formulas. Definition of Area "Under a Curve". Net "Area". Approximating Area Numerically.

Lecture 26 - The Definite Integral
The Definite Integral Defined. The Definite Integral of a Continuous Function = Net "Area" Under a Curve. Finding Definite Integrals. A Note on the Definite Integral of a Discontinuous Function.

Lecture 27 - The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus, Part 1. Definite & Indefinite Integrals Related. The Mean Value Theorem for Integrals. The Fundamental Theorem of Calculus, Part 2. Differentiation & Integration are Inverse Processes.

Lecture 28 - One Dimensional Motion & the Integral
Position, Velocity, Distance, & Displacement. Uniformly Accelerated Motion. The Free Fall Motion Model.


Unit 6 - The Definite Integral Applied

Lecture 29a - Plane Area
Area Between Two Curves [One Floor, One Ceiling]. Area Between Two Curves [One Left, One Right].

Lecture 29b - Volumes I
Volumes by Slicing. Volumes of Solids of Revolution: Disks. Volumes of Solids of Revolution: Washers.

Lecture 30 - Volumes II
Volumes of Solids of Revolution: Cylindrical Shells

Lecture 30-03 - Length of a Plane Curve
Finding Arc Lengths

Lecture 30-04 - Length of a Plane Curve
Finding Arc Lengths of Parametric Curves

Lecture 31 - Average Value of a Function, Work
Average (Mean) Value of a Continuous Function. Work Done by a Constant Force. Work Done by a Variable Force. Do-It-Yourself Integrals: Pumping Fluids. Work as Change in Kinetic Energy.

Lecture 31-06 - Work: An Exercise
An Exercise