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Calculus

Subject: Mathematics

Course Description:

Walk in the footsteps of Newton and Leibnitz!

An interactive text and graphing software combine with the exciting on-line course delivery to make Calculus an adventure. This course includes a study of limits, continuity, differentiation, and integration of algebraic, trigonometric and transcendental functions, and the applications of derivatives and integrals.

Major Topics:

---Semester 1---

Module 0: Preparation for Calculus Suggested Pace: 2 weeks

Topics

  • Understanding the properties of real numbers and the number line
  • Using the Cartesian coordinate system to graph functions
  • Comparing relative magnitudes of functions – contrasting exponential, logarithmic and polynomial growth

Content

  • Orientation to course
  • Real numbers and the real number line
  • Cartesian plane
  • Graphs and models
  • Linear models and rates of change
  • Functions and their graphs

Major Assignments and Assessments

  • Problem Sets
  • Entry Quiz
  • Oral Review: Discussion about using Calculator zoom features to examine a graph in a good viewing window and calculator operations to find the zeros of a graph and the point of intersection of two graphs
  • Quiz – Functions, Graphs, and Rates of Change

Module 1: Limits and Continuity Suggested Pace: 2 weeks

Topics

  • Intuitive understanding of limit process
  • Calculating limits using algebraic methods
  • Estimating limits using tables of data
  • Estimating limits using graphs
  • Understanding asymptotes graphically
  • Describing asymptotic behavior in terms of limits involving infinity
  • Intuitive understanding of continuity
  • Understanding continuity in terms of limits
  • Understanding graphs of continuous or non-continuous functions geometrically

Content

  • Preview of calculus
  • Finding limits graphically and numerically
  • Evaluating limits analytically
  • Continuity and one-sided limits
  • Infinite limits

Major Assignments and Assessments

  • Problems sets
  • Quiz – Calculating Limits
  • Oral Review: Discussion about using the Calculator to experiment and produce a table of values to examine a function and estimate a limit as x approaches a point and as x grows without bound. Discussion about the limitation of a graphing calculator to show discontinuities in functions and the value of using a calculator to support conclusions found analytically.
  • Elluminate Session: Discussion about conditions of continuity.
  • Test – Limits and Continuity

Module 2: Differentiation Suggested Pace: 5 weeks

Topics

  • Derivative defined as the limit of the difference quotient
  • Graphic, numeric and analytic interpretations of the derivative
  • Knowledge of derivatives of power and trigonometric functions
  • Basic rules for the derivatives of sums, products, and quotients of functions
  • Derivative interpreted as instantaneous rate of change
  • Continuity and differentiability
  • Slope of curve at a point
  • Tangent line to a curve at a point
  • Local linear approximation
  • Instantaneous rate of change as the limit of average rate of change
  • Approximate rate of change from graphs and tables of values
  • Chain rule and implicit differentiation
  • Equations involving derivatives and problems using their verbal descriptions
  • Modeling rates of change and solving related rates problems

Content

  • The derivative and the tangent line problem
  • Basic differentiation rules and rates of change
  • The product and quotient rules
  • The chain rule
  • Implicit differentiation
  • Related rates

Major Assignments and Assessments

  • Problem sets
  • Quiz – Definition and computation of derivatives
  • Oral Review: Discussion about using a calculator to find the value of a derivative at a point, and how to graph the derived function using a calculator. Discussion about the limitations of the calculator to find the numerical derivative (for example, f ‘(0) for f (x) = |x|).
  • Test – Differentiation

Module 3: Applications of Differentiation Suggested Pace: 6 weeks

Topics

  • Corresponding characteristics of graphs of f and f’
  • Relationship between the increasing and decreasing behavior of f and the sign of f’
  • Corresponding characteristics of graphs of f, f’, and f’’
  • Relationship between the concavity of f and the sign of f’
  • Points of inflection as places where concavity changes
  • Mean Value Theorem and geometric consequences
  • Analysis of curves including monotonicity and concavity
  • Optimization – absolute and relative extrema
  • Equations involving derivatives and problems using their verbal descriptions

Content

  • Extrema on an interval
  • Rolle’s Theorem and the Mean Value Theorem
  • Increasing and decreasing functions
  • Concavity and the second derivative test
  • Limits at infinity
  • Curve sketching
  • Optimization
  • Differentials

Major Assignments and Assessments

  • Problem sets
  • Quiz – Extrema and Concavity
  • Oral Review: Discussion about using the calculator to find the critical values of a function by examining the graph of the function and the graph of the function’s derivative.
  • Test – Applications of Derivatives
  • Semester Exam

---Semester 2---

Module 4: Integration Suggested Pace: 4 weeks

Topics

  • Definite integral as a limit of Riemann sums
  • Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:
  • Basic properties of definite integrals
  • Use of the Fundamental Theorem of Calculus to evaluate definite integrals
  • Use of the Fundamental Theorem of Calculus to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined
  • Find antiderivatives including the use of substitution
  • Finding specific antiderivatives using initial conditions, including applications to motion along a line
  • Use of Riemann sums and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically and by tables of values

Content

  • Antiderivatives and Indefinite Integration
  • Area
  • Riemann sums and definite integrals
  • The Fundamental Theorem of Calculus
  • Integration by substitution
  • Numerical integration
  • Application of definite integrals including area, volume, position/velocity/acceleration and accumulation functions
  • The Integral as a function

Major Assignments and Assessments

  • Problem Sets
  • Quiz – Integration and Area
  • Quiz – The Fundamental Theorem of Calculus
  • Oral Review: Discussion about using the calculator to estimate the value of a definite integral and to support solutions derived analytically.
  • Test – Integration

Module 5: Transcendental Functions Suggested Pace: 3 weeks

Topics

  • Use of implicit differentiation in finding the derivative of the inverse of a function
  • Geometric interpretation of differential equations via slope fields
  • Relationship between slope fields and solution curves for differential equations
  • Knowledge of derivatives of exponential, logarithmic, and inverse trigonometric functions
  • Basic properties of definite integrals
  • Use of the Fundamental Theorem of Calculus to evaluate definite integrals
  • Find antiderivatives including the use of substitution
  • Application of integrals

Content

  • The natural logarithmic function and differentiation
  • The natural logarithmic function and integration
  • Inverse functions including the relationship between the derivative of a function and its inverse at a point
  • Exponential functions
  • Bases other than e and applications
  • Inverse trigonometric functions and differentiation
  • Inverse trigonometric functions and integration

Major Assignments and Assessments

  • Problem Sets
  • Quiz – – Natural Logarithmic Functions and  Exponential Functions
  • Oral Review: Examine the limitations of the graphing calculator in graphing Natural Log functions. Students are required to verbally express the concepts related to the derivatives and integrals of exponential, logarithmic, and inverse trigonometric functions.
  • Test: Transcendental Functions

Module 6: Differential Equations and Slope Fields: Suggested Pace: 3 weeks

Topics

  • Differential equations: growth and decay
  • Differential equations: separation of variables
  • Slope fields

Content

  • Solving separable differential equations and using them in modeling
  • Geometric interpretation of differential equations via slope fields
  • Relationship between slope fields and solution curves for differential equations

Major Assignments and Assessments

  • Problem Sets
  • Oral Review: Examine the limitations of the graphing calculator in graphing Natural Log functions. Students are required to verbally express the concepts related to the derivatives and integrals of exponential, logarithmic, and inverse trigonometric functions.
  • Test: Differential Equations and Slope Fields

Module 7: Applications of Integration Suggested Pace: 3 weeks

Topics

  • Application of integrals – area and volume

Content

  • Area of a region between two curves
  • Volume

Major Assignments and Assessments

  • Problem Sets
  • Oral Review – Discuss setup on a graphing calculator to find volumes for functions that cannot be integrated by hand. Students are required to be able to explain how the calculator is used to assist with the integration portion of solving a volume problem.
    Students have opportunity to demonstrate their solutions to other members of the class as well as the teacher using the whiteboard, application sharing of MathType and Graphmatica solutions, and the audio feature during this session.
  • Test – Applications of Integration

Module 8: Integration Techniques and L'Hopital's Rule Suggested Pace: 2 weeks

Topics

  • Techniques of Integration
  • Techniques for using Differentiation to find Limits

Content

  • Basic rules of integration
  • Indeterminate forms and L’Hopital’s Rule

Major Assignments and Assessments

  • Problem Sets
  • Oral Review – Students must verbally demonstrate the ability to use a calculator generated table to show limiting values of functions and comparative rates of growth of functions.
  • Test – Integration Techniques

This course is designed to be highly teacher facilitated. Instructors give specific and timely feedback for each of the 100 plus lessons. Students are required to complete one-on-one oral examinations with their teacher for each module, discussions with other students where they practice communicating their mathematical concepts verbally, and have opportunity to schedule whiteboard sessions with the teacher. Teachers conduct synchronous Elluminate (whiteboard, application sharing, audio and video sharing) sessions that require students to practice critical thinking and analysis with free response type questions and present their work verbally to the teacher and other students during those sessions.

Each assignment gives students practice in communicating their mathematics in written form using sentences as well as symbols. The oral reviews done in each module and the synchronous Elluminate sessions give students opportunities to express their understanding of mathematics verbally.

Most lessons include practice using the calculator. The students are informed in the first lesson of the type of calculator needed and the four operations required to be used.. Students are required throughout the course to give evidence of calculator proficiency. The required calculator operations and techniques for using the calculator appropriately are discussed in the oral reviews of each module and in the synchronous Elluminate sessions as well as in scheduled one-on-one monthly phone conferences held between the teacher and each student.

Participation Requirements:

Besides engaging students in challenging curriculum, INVS guides students to reflect on their learning and to evaluate their progress through a variety of assessments. Assessments can be in the form of self-checks, practice lessons, multiple choice questions, projects, oral assessments and discussions. Instructors evaluate progress and provide interventions through the variety of assessments built into a course, as well as through contact with the student in other venues.

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Course Details

  • Course Code:
  • Course Credits: 2.0

Prerequisits:
Algebra I, Geometry, Algebra II, Pre-Calculus or Trigonometry/Analytical Geometry.

Estimated Completion:
2 Semesters