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 **

Graphing Technology

**Lecture 2 **

New Functions From Old

How Operations Affect Function Graphs

Functions with Symmetric Graphs

Families of Functions

The Power Function Family *y *= *x^p
*

The Polynomial Function, and Rational Function Families

**Lecture 3 **

Trigonometry for Calculus

Handy Trigonometric Identities

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 **

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

**Lectures 5-6 **

The Algebra of Limits as *x *→ *a *

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!

The Algebra of Limits as *x *→ ± ∞: End Behavior

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

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

**Lecture 7 **

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: *f*(*x) = *0

**Lecture 8 **

Trigonometric Functions

The 6 Trigonometric Functions: Continuous on Their Domains

Finding a Limit by “Squeezing”

Sin(*x*)/*x *→ 1 as *x *→ 0, and Other Limit Tales

**Lecture 9 **

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

**Lecture 10 **

What is a Derivative?

Definition of the Derived Function: The “Derivative”, & Slopes of Tangent Lines

Functions Differentiable (or not!) at a Single Point

Functions Differentiable on an Interval

A Function Differentiable at a point is Continuous at that point

**Lecture 11 **

Finding Derivatives I: Basic Rules

Constant Multiple, Sum, & Difference Rules

Notation for Derivatives of Derivatives

Finding Derivatives II:

**Lecture 12 **

Finding Derivatives III:

The Other Trigonometric Functions

Finding Derivatives IV:

The Chain Rule: Derivatives of Compositions of Functions

Generalized Derivative Formulas

**Lecture 13 **

When Rates of Change are Related

Differentiating Equations to “Relate Rates”

More on Derivatives

Local Linear Approximations of Non-Linear Functions

**Lecture 14**

UNIT 3 – Some Special DERIVATIVES Implicit Differentiation

Derivatives of Functions Defined Implicitly

The Derivative of Rational Powers of *x*

Derivatives Involving Logarithms

Derivatives of Logarithmic Functions

The “Logarithmic Differentiation” Technique

The Derivative of Irrational Powers of x

**Lecture 15**

Derivatives Involving Inverses

Derivatives of Inverse Functions

Derivatives of Exponential Functions

Derivatives of Inverse Trigonometric Functions

Finding Limits Using Differentiation

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

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

**Lecture 17 **

Analyzing the Graphs of Functions II

The 1st Derivative Test for Local Maximums & Minimums

The 2nd Derivative Test for Local Maximums & Minimums

**Lecture 18**

Analyzing the Graphs of Functions III

Functions Whose Graphs have Vertical Tangents or Cusps

**Lecture 19 **

Analyzing the Graphs of Functions IV

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: Cost, Revenue, Profit, & Marginal Analysis

**Lecture 21**

Newton’s Method for Approximating Roots of Equations

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

**Lecture 22 **

One-Dimensional Motion & the Derivative

Velocity, Speed, & Acceleration

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

The Graphs of Antiderivatives: Integral Curves & the Slope Field Approximation

The Antiderivative as Solution of a Differential Equation

**Lecture 24**

Indefinite Integration by Substitution

The Substitution Method of Indefinite Integration: A Major Technique

More Interesting Substitutions

**Lecture 25**

Area Defined as a Limit

Summation Properties & Handy Formulas

Definition of Area “Under a Curve”

Approximating Area Numerically

**Lecture 26 **

The Definite Integral

The Definite Integral of a Continuous Function = Net “Area” Under a Curve

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

Definite Integration by Substitution

Extending the Substitution Method of Integration to Definite Integrals

**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]

**Lecture 29 Part II **

Volumes I

Volumes of Solids of Revolution: Disks

Volumes of Solids of Revolution: Washers

**Lecture 30 **

Volumes II

Volumes of Solids of Revolution: Cylindrical Shells

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

Do-It-Yourself Integrals: Pumping Fluids

Work as Change in Kinetic Energy