General Calculus I

Online Calculus Course Overview

This 3-credit General Calculus I course is designed to acquaint students with calculus principles such as derivatives, integrals, limits, approximation, applications and modeling, and sequences and series. 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 Functions & Graphs, Limits & Continuity, Derivatives, the Application of Derivatives, Antiderivatives and definite Integrals.

General Calculus I has been reviewed and authorized by The College Board as an AP Program course.

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Demos

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Calculus Course Text

• Larson, R., Hostetler, R. P., and Edwards, B.  Calculus, 8th edition, Brooks Cole, 2005.  ISBN: 9780618502981
• Larson, R., Hostetler, R. P., and Edwards, B.  Calculus – Early Transcendental Functions, 3rd edition.  Houghton Mifflin Company, 2003.  ISBN: 9780618223077
• Varberg, D., Purcell, E., and Rigdon, S.  Calculus, 9th Edition.  Prentice Hall, 2006.  ISBN 9780131429246 [buy a text]
  
  Do I have to buy the textbook?
  As in many college courses, purchasing the book is ultimately up to you. The StraighterLine courses use reading assignments and practice work from the textbook as supplements, but you will not be required to turn in anything from the book. StraighterLine recommends that you purchase the appropriate text so you are equipped with as many resources as possible (please note that e-books are generally less expensive but may have slight differences in page numbers and resources compared to a hard copy text).

Online Calculus Course Objectives

• work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal
• understand the connections among these representations
• understand the meaning of the derivative in terms of a rate of change and local linear approximationbe able to use derivatives to solve a variety of problems
• be able to use derivatives to solve a variety or problems
• understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change
• should be able to use integrals to solve a variety of problems
• understand the relationship between the derivative and the definite integral as expressed in both parts of the fundamental theorem of calculus

Calculus Course Prerequisites

StraighterLine suggests, though does not require, that students take Precalculus or its equivalent before enrolling in General Calculus I.

Important Terms

In this online General Calculus course, different terms are used to designate tasks:

• Practice Exercise: A non-graded assignment to assist you in practicing the skills discussed in a topic.
• Exam: A graded online test.

Online Course Evaluation Criteria

StraighterLine does not apply letter grades. Students earn a score as a percentage of 100%. A passing percentage is 70% or higher.

If you have chosen a Partner College to award credit for this course, your final grade will be based upon that college's grading scale. Only passing scores will be considered by Partner Colleges for an award of credit.

There are a total of 1000 points in the course:

Topic	Assessment	Points Available
11	Graded Exam - Unit 1	150
18	Graded Exam - Unit 2	150
30	Graded Exam - Unit 3	150
39	Graded Exam - Unit 4	150
51	Graded Exam - Unit 5	150
	Cumulative Final Exam	250
Total		1000

At the end of the course, each student will receive the number of points earned. The student's final letter grade is determined by the corresponding institution's grading scale.

Online Calculus Course Topics and Objectives

UNIT	lESSON	tOPIC
Unit 1: Functions and Graphs	Lesson 1: Functions and Function Notation	Topic 1: Definition of a function Topic 2: Vertical line test Topic 3: Function notation Topic 4: Finding input & output values Topic 5: Domain & range
	Lesson 2: Absolute Value and Piecewise Defined Functions	Topic 1: Piecewise defined functions Topic 2: The absolute value function Topic 3: Solving equations involving absolute values
	Lesson 3: Inequalities	Topic 1: Solving inequalities Topic 2: Solving inequalities involving absolute values
	Lesson 4: Composition and Combination of Functions	Topic 1: Composition of functions Topic 2: Inverse functions Topic 3: Non-invertible functions
	Lesson 5: Exponential and Logarithmic Functions	Topic 1: The family of exponential functions Topic 2: The number e Topic 3: Logarithmic functions Topic 4: Solving exponential and logarithmic functions Topic 5: Changing the base of logarithmic functions
	Lesson 6: Transformation of Functions	Topic 1: Horizontal & vertical shifts Topic 2: Reflections & symmetry Topic 3: Vertical stretches & compressions Topic 4: Horizontal stretches & compressions
	Lesson 7: Trigonometric Functions	Topic 1: Radians and arc length Topic 2: The sin and cosine functions Topic 3: Other trigonometric functions Topic 4: Inverse trigonometric functions Topic 5: Trigonometric identities
	Lesson 8: Polynomial and Rational Functions	Topic 1: Variations Topic 2: Power functions Topic 3: Polynomial functions Topic 4: Rational functions
	Lesson 9: Vectors and Vector-Valued Functions	Topic 1: Vectors Topic 2: Components of a vector Topic 3: Addition, subtraction, & scalar multiplication Topic 4: The zero and unit vectors
	Lesson 10: Polar Coordinates and Graphs	Topic 1: Polar coordinates Topic 2: Coordinate conversion Topic 3: Polar graphs Topic 4: Special polar graphs
	Lesson 11: Parametric Equations and Conic Sections	Topic 1: Parametric equations Topic 2: Eliminating the parameter Topic 3: Finding parametric equations
Unit 2: Limits and Continuity	Lesson 12: Intuitive Definition of a Limit	Topic 1: Definition Topic 2: Using tables & graphs to find limits Topic 3: Application: Using limits to find velocity
	Lesson 13: Algebraic Techniques for Finding Limits	Topic 1: Calculating limits using limit laws Topic 2: Direct substitution property Topic 3: Indeterminate forms
	Lesson 14: One-Sided Limits	Topic 1: Definition of one-sided limits Topic 2: Finding one-sided limits Topic 3: Existence of limits
	Lesson 15: Infinite Limits	Topic 1: Definition of infinite limits Topic 2: Vertical asymptotes
	Lesson 16: Limits at Infinity	Topic 1: End-behavior of functions & horizontal asymptotes Topic 2: Limit laws for infinite limits Topic 3: Oblique asymptotes
	Lesson 17: Limits of Special Trigonometric Functions	Topic 1: Special limits involving the sine function Topic 2: Special limits involving the cosine function Topic 3: Special limits involving the tangent function
	Lesson 18: Continuity	Topic 1: Continuity at a point Topic 2: Continuity on a closed interval Topic 3: Continuity on an open interval Topic 4: Intermediate Value Theorem
Unit 3: Derivatives	Lesson 19: Definition of the Derivative	Topic 1: The derivative as the slope of a tangent Topic 2: The derivative as the rate of change Topic 3: The derivative as a function
	Lesson 20: Differentiation Rules	Topic 1: Constant rule, constant multiple rule Topic 2: Power rule Topic 3: Sum rule, difference rule Topic 4: Product rule Topic 5: Quotient rule Topic 6: Trigonometric rules
	Lesson 21: The Chain Rule	Topic 1: Chain rule definition Topic 2: The chain rule with other rules Topic 3: Higher-order derivatives
	Lesson 22: Derivatives of Exponential Functions	Topic 1: Exponential rule Topic 2: Base-a exponentials rule
	Lesson 23: Derivative of Logarithmic Functions	Topic 1: Natural logarithmic rule Topic 2: General logarithmic rule
	Lesson 24: Derivatives of Inverse Functions	Topic 1: Inverse trig rule Topic 2: Derivatives of Inverses
	Lesson 25: Differentiability and Continuity	Topic 1: Differentiability implies continuity Topic 2: Non-differentiable functions
	Lesson 26: Implicit Differentiation	Topic 1: Derivatives of implicitly defined functions
	Lesson 27: Logarithmic Differentiation	Topic 1: derivatives of complicated expressions
	Lesson 28: Parametric Derivatives	Topic 1: Parametric form of the derivative Topic 2: Parametric form of the second derivative
	Lesson 29: Differentiation with Polar Curves	Topic 1: Tangents to polar slopes
	Lesson 30: Differentiation of Vector-Valued Functions	Topic 1: Limits & continuity Topic 2: Derivatives of vector-valued functions Topic 3: Differentiation rules & examples
Unit 4: Application of the Derivative	Lesson 31: Tangent and Normal Lines	Topic 1: Tangent lines Topic 2: Normal lines
	Lesson 32: Position, Velocity, and Acceleration (PVA)	Topic 1: Position & velocity Topic 2: Acceleration
	Lesson 33: Related Rates	Topic 1: Defining the problem Topic 2: Example problems
	Lesson 34: Relative Extrema and the First Derivative Test	Topic 1: Relative extrema & critical numbers Topic 2: Increasing, decreasing functions & the first derivative
	Lesson 35: Concavity and the Second Derivative Test	Topic 1: Concavity Topic 2: The second derivative test
	Lesson 36: Absolute Extrema and Optimization	Topic 1: Extreme value theorem Topic 2: Sample problem
	Lesson 37: Rolle's Rule and the Mean Value Theorem	Topic 1: Rolle’s rule Topic 2: Mean value theorem
	Lesson 38: Differentials	Topic 1: Linear approximation Topic 2: Differentials
	Lesson 39: L'Hospital's Rule	Topic 1: Indeterminate forms & L’Hospital’s rule Topic 2: Proof of L’Hospital’s rule Topic 3: Other indeterminate forms
Unit 5: Functions and Graphs	Lesson 40: Differential Equations and Slope Fields	Solve simple differential equations and initial value problems. Generate a slope field for a differential equation.
	Lesson 41: Antiderivatives	Define the antiderivative and the indefinite integral. Explore basic antiderivative rules. Investigate rules for trigonometric antiderivatives.
	Lesson 42: The Chain Rule for Antiderivative	Use simple substitutions to find antiderivatives. Find antiderivatives of trigonometric integrals
	Lesson 43: Antiderivatives of Exponentials	Find antiderivatives for exponential functions.
	Lesson 44: Antiderivatives of Logarithm	Find antiderivatives for logarithmic functions.
	Lesson 45: Antiderivatives of Inverse Trigonometric Functions	Use inverse trigonometric functions to evaluate integrals.
	Lesson 46: Integration by Part	Define the integration by parts formula. Use integration by parts to evaluate integrals.
	Lesson 47: Integration by Partial Fractions	Review partial fraction decomposition of rational functions. Use partial fractions to integrate rational functions.
	Lesson 48: Trigonometric Substitutions	Use right triangle trigonometry to create substitutions for integrals. Recognize and integrate functions using trigonometric substitutions.
	Lesson 49: The Definite Integral	Define a Riemann sum. Define a definite integral. Find the area between two curves on the coordinate plane. Explore techniques for approximating definite integrals.
	Lesson 50: Fundamental Theorem of Calculus	Investigate properties of the definite integral. Define the Fundamental Theorem of Calculus. Explore integral defined functions. Find the average value of a function on an interval. Define the Mean Value Theorem for Integration.
	Lesson 51: Improper Integrals	Explore definite integrals with infinite limits. Explore definite integrals with discontinuous functions. Define convergence of an integral.
	Review and Final Exam	Review and Final Exam