# Calculus I Videos-Delaware

Course- Calculus I with Professor Delaware –available at http://www.youtube.com/user/umkc#p/p Dr. Delaware is one of the faculty members in the Department of Mathematics and Statistics. His video course posted on YouTube covers all topics in your class. You can use them as a reinforcement of the lectures.  Strongly recommended!

UNIT 0 – FUNCTIONS: A Review of Precalculus

Lecture 1

Definition of a Function

Visualizing Functions: Graphs

Domain (& Range) of Functions

Some Exercises

Graphing Technology

Viewing Windows

Zooming In or Out

Errors in Resolution

Lecture 2

New Functions From Old

Operations on Functions

How Operations Affect Function Graphs

Functions with Symmetric Graphs

Some Exercises

Families of Functions

The Power Function Family y = x^p

The Polynomial Function, and Rational Function Families

Lecture 3

Trigonometry for Calculus

Right Triangle Trigonometry

Trigonometric Graphs

Handy Trigonometric Identities

Laws of Sine and Cosine

Trigonometric Families

Inverse Functions

A Function Inverse to Another Function

When do Inverse Functions (& Their Graphs) Exist?

Inverse Trigonometric Functions

Exponential & Logarithmic Functions

The Exponential Function Family

The Logarithmic Function Family

Solving Exponential & Logarithmic Equations

Lecture 4

UNIT 1 – LIMITS of Functions: Approach & Destination 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

More Limits that Fail to Exist: Infinity & Infinite Indecision

An Exercise on Limits

Lectures 5-6

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: When One-sided Limits Matter!

Some Exercises

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

Some Exercises

Limits of ln(x), e^x, and More

Lecture 7

Continuous Functions

Functions Continuous on an Interval

Properties & Combinations of Continuous Functions

The Intermediate Value Theorem & Approximating Roots: f(x) = 0

Some Exercises

Lecture 8

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

Some Exercises

Lecture 9

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

Slopes of Tangent Lines

One-Dimensional Motion

Average Velocity

Instantaneous Velocity

General Rates of Change

Some Exercises

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

A Function Differentiable at a point is Continuous at that point

Other Derivative Notations

Some Exercises

Lecture 11

Finding Derivatives I: Basic Rules

The Power Rule

Constant Multiple, Sum, & Difference Rules

Notation for Derivatives of Derivatives

Some Exercises

Finding Derivatives II:

The Product Rule

The Quotient Rule

Some Exercises

Lecture 12

Finding Derivatives III:

The Sine Function

The Other Trigonometric Functions

Some Applications

Finding Derivatives IV:

The Chain Rule: Derivatives of Compositions of Functions

Generalized Derivative Formulas

Some Exercises

Lecture 13

When Rates of Change are Related

Differentiating Equations to “Relate Rates”

A Strategy

An Exercise

More on Derivatives

Local Linear Approximations of Non-Linear Functions

Defining “dx” and “dy” Alone

Lecture 14

UNIT 3 – Some Special DERIVATIVES Implicit Differentiation

Functions Defined Implicitly

Derivatives of Functions Defined Implicitly

Some Exercises

Derivatives Involving Logarithms

Derivatives of Logarithmic Functions

The “Logarithmic Differentiation” Technique

The Derivative of Irrational Powers of x

Some Exercises

Lecture 15

Derivatives Involving Inverses

Derivatives of Inverse Functions

Derivatives of Exponential Functions

Derivatives of Inverse Trigonometric Functions

Some Exercises

Finding Limits Using Differentiation

Limits of Quotients that appear to be “Indeterminate”: The Rule of L’Hopital

Some Examples

Finding Other “Indeterminate” Limits

Lecture 16

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

Increasing & Decreasing Functions: The 1st Derivative Applied

Functions Concave Up or Concave Down: The 2nd Derivative Applied

When Concavity Changes: Inflection Points

Logistic Growth Curves: A Brief Look

Some Exercises

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

Some Exercises

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

Some Exercises

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

Some Exercises

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: Cost, Revenue, Profit, & Marginal Analysis

Some Exercises

Lecture 21

Newton’s Method for Approximating Roots of Equations

Development of the Method

Strength & Weaknesses of the Method

The Mean Value Theorem for Derivatives

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

Some Exercises

Lecture 22

One-Dimensional Motion & the Derivative

Rectilinear Motion Revisited

Velocity, Speed, & Acceleration

Analyzing a Position Graph

An Exercise

UNIT 5 – The INTEGRAL of a Function  The Question of Area

Brief History and Overview [17.5 min.]

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

Some Exercises

Lecture 24

Indefinite Integration by Substitution

The Substitution Method of Indefinite Integration: A Major Technique

Straightforward Substitutions

More Interesting Substitutions

Some Exercises

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

Some Exercises

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

Some Exercises

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

Some Exercises

Lecture 28

One-Dimensional Motion & the Integral

Position, Velocity, Distance, & Displacement

Uniformly Accelerated Motion

The Free Fall Motion Model

Definite Integration by Substitution

Extending the Substitution Method of Integration to Definite Integrals

Some Exercises

Lecture 29

UNIT 6 – The DEFINITE INTEGRAL Applied

Plane Area

Area Between Two Curves [One Floor, One Ceiling]

Area Between Two Curves [One Left, One Right]

An Exercise

Lecture 29 Part II

Volumes I

Volumes by Slicing

Volumes of Solids of Revolution: Disks

Volumes of Solids of Revolution: Washers

Some Exercises

Lecture 30

Volumes II

Volumes of Solids of Revolution: Cylindrical Shells

An Exercise

Length of a Plane Curve

Finding Arc Lengths [11.5 min.]

Finding Arc Lengths of Parametric Curves [6.5 min.]

Lecture 31

Average (Mean) Value of a Continuous Function

Work

Work Done by a Constant Force

Work Done by a Variable Force

Do-It-Yourself Integrals: Pumping Fluids

Work as Change in Kinetic Energy