## Math 210: Calculus I

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.

More details of the videos are found:

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