Calculus I

Become acquainted with calculus principles, gain experience using and applying calculus methods in practical applications, and become familiar with topics like limits, derivatives, and computational techniques.

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Calculus I

$79

Plus membership

3 Credits

All courses include:

eTextbooks

2 to 3-day turnaround for grading

Multiple chances to improve your grade

On-demand tutoring & writing center

Student support 7 days a week

$79

Plus membership

3 Credits

All courses include:

eTextbooks

2 to 3-day turnaround for grading

Multiple chances to improve your grade

On-demand tutoring & writing center

Student support 7 days a week

Calculus I

$79

Plus membership

3 Credits

About This Course

|
ACE Approved 2024

General Calculus I acquaints you with calculus principles such as derivatives, integrals, limits, approximation, applications and integration, and curve sketching.

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What You'll Learn

Demonstrate the continuity or discontinuity of the function.

Solve the limit problems by using various limit laws.

Demonstrate various rules of derivatives.

Compute derivatives.

Demonstrate derivatives for trigonometric, exponential, and logarithmic functions

Apply Implicit differentiation

Apply L’Hôpital’s Rule to find the limit of indeterminate forms.

Sketch the graphs using the derivatives.

Compute area between the curves using integration.

Solve vertical motion problems.

Illustrate the Fundamental Theorem of Calculus.

Demonstrate convergence and divergence of improper integrals.

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Course Details

MAT250

|

General Calculus I

This course is designed to acquaint students with calculus principles such as derivatives, integrals, limits, approximation, applications and integration, and curve sketching. During this course, students will gain experience in the use of calculus methods and learn how calculus methods may be applied to practical applications. Topics covered include Special Functions, Limits, Derivatives, Computational Techniques, Applications of Differentiations, and Applications of Integration.

Prerequisites

Precalculus is a required prerequisite for General Calculus I. If you enroll, the assumption is made that you have previously completed Precalculus for credit with a passing score.

Topic Subtopics
Preliminaries and Functions
  • Welcome to Calculus
  • The Two Questions of Calculus
  • Average Rates of Change
  • How to Do Math
  • Functions
  • Graphing Lines
  • Parabolas
  • Some Non-Euclidean Geometry
Limits and Continuity
  • Finding Rate of Change over an Interval
  • Finding Limits Graphically
  • The Formal Definition of a Limit
  • The Limit Laws, Part I
  • The Limit Laws, Part II
  • One-Sided Limits
  • The Squeeze Theorem
  • Continuity and Discontinuity
  • Evaluating Limits
  • Limits and Indeterminate Forms
  • Two Techniques for Evaluating Limits
  • An Overview of Limits
Derivatives
  • Rates of Change, Secants, and Tangents
  • Finding Instantaneous Velocity
  • The Derivative
  • Differentiability
  • The Slope of a Tangent Line
  • Instantaneous Rate
  • The Equation of a Tangent Line
  • More on Instantaneous Rate
  • The Derivative of the Reciprocal Function
  • The Derivative of the Square Root Function
Computational Techniques
  • A Shortcut for Finding Derivatives
  • A Quick Proof of the Power Rule
  • Uses of the Power Rule
  • The Product Rule
  • The Quotient Rule
  • An Introduction to the Chain Rule
  • Using the Chain Rule
  • Combining Computational Techniques
Implicit Differentiation
  • An Introduction to Implicit Differentiation
  • Finding the Derivative Implicitly
  • Using Implicit Differentiation
  • Applying Implicit Differentiation
Dealing with Indeterminate Forms
  • Indeterminate Forms
  • An Introduction to L’Hôpital’s Rule
  • Basic Uses of L’Hôpital’s Rule
  • More Exotic Examples of Indeterminate Forms
  • L’Hopital’s rule and Indeterminate Products
  • L'Hôpital's rule and Indeterminate Differences
  • L'Hôpital's rule and One to the Infinite Power
  • Another example of One to the Infinite Power
Applications of Differentiations
  • Acceleration and the Derivative
  • Solving Word Problems Involving Distance and Velocity
  • Higher-Order Derivatives and Linear Approximation
  • Using the Tangent Line Approximation Formula
  • Newton’s Method
  • The Connection Between Slope and Optimization
  • The Fence Problem
  • The Box Problem
  • The Can Problem
  • The Wire-Cutting Problem
  • The Ladder Problem
  • The Baseball Problem
  • The Blimp Problem
  • Math Anxiety
Curve Sketching
  • An Introduction to Curve Sketching
  • Three Big Theorems
  • Morale Moment
  • Critical Points
  • Maximum and Minimum
  • Regions Where a Function Increases or Decreases
  • The First Derivative Tests
  • Magic Math
  • Concavity and Inflection Points
  • Using the Second Derivative to Examine Concavity
  • The Mobius Band
  • Graphs of Polynomial Functions
  • Cusp Points and the Derivative
  • Domain-Restricted Functions and the Derivative
  • The Second Derivative Test
  • Vertical Asymptotes
  • Horizontal Asymptotes and Infinite Limits
  • Graphing Functions with Asymptotes
  • Functions with Asymptotes and Holes
  • Functions with Asymptotes and Critical Points
Introduction to Integrals
  • Antidifferentiation
  • Antiderivatives of Powers of x
  • Antiderivatives of Trigonometric and Exponential Functions
  • Undoing the Chain Rule
  • Integrating Polynomials by Substitution
  • Integrating Composite Trigonometric Functions by Substitution
  • Integrating Composite Exponential and Rational Functions by Substitution
  • More Integrating Trigonometric Functions by Substitution
  • Choosing Effective Function Decompositions
  • Approximating Areas of Plane Regions
  • Areas, Riemann Sums, and Definite Integrals
  • The Fundamental Theorem of Calculus, Part I
  • The Fundamental Theorem of Calculus, Part II
  • Illustrating the Fundamental Theorem of Calculus
  • Evaluating Definite Integrals
Applications of Integration
  • Antiderivatives and Motion
  • Gravity and Vertical Motion
  • Solving Vertical Motion Problems
  • The Area between Two Curves
  • Limits of Integration and Area
  • Common Mistakes to Avoid When Finding Areas
  • Regions Bound by Several Curves
  • Finding Areas by Integrating with Respect to y: Part One
  • Finding Areas by Integrating with Respect to y: Part Two
  • Area, Integration by Substitution, and Trigonometry
3 Techniques of Integration
  • Finding Partial Fraction Decompositions
  • Partial Fractions
  • Long Division
  • An Introduction to Integration by Parts
  • Applying Integration by Parts to the Natural Log Function
  • Inspirational Examples of Integration by Parts
  • Repeated Application of Integration by Parts
  • Algebraic Manipulation and Integration by Parts
Special Functions
  • A Review of Trigonometry
  • Graphing Trigonometric Functions
  • The Derivatives of Trigonometric Functions
  • The Number Pi
  • Graphing Exponential Functions
  • Derivatives of Exponential Functions
  • The Music of Math
  • Evaluating Logarithmic Functions
  • The Derivative of the Natural Log Function
  • Using the Derivative Rules with Transcendental Functions
  • The Exponential and Natural Log Functions
  • Differentiating Logarithmic Functions
  • Logarithmic Differentiation
  • The Basics of Inverse Functions
  • Finding the Inverse of a Function
  • Derivatives of Inverse Functions
  • The Inverse Sine, Cosine, and Tangent Functions
  • The Inverse Secant, Cosecant, and Cotangent Functions
  • Evaluating Inverse Trigonometric Functions
  • Derivatives of Inverse Trigonometric Functions
  • More Calculus of Inverse Trigonometric Functions
  • Defining the Hyperbolic Functions
  • Hyperbolic Identities
  • Derivatives of Hyperbolic Functions

Your score provides a percentage score and letter grade for each course. A passing percentage is 70% or higher.

Assignments for this course include:

  • 4 Graded Topic Reviews
  • 4 Graded Quizzes
  • 4 Graded Exams
  • 1 Graded Final

This course does not require a text.


General Calculus I students also take:

General Physics I

General Physics I

Introduction to Philosophy


Helpful resources:

Why AP Calculus is the Scariest AP Test of All

Do You Have to Take Math in College?

Tips to Help College Students to Succeed in an Online Math Course

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