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Advanced Placement Calculus BC, Part 2 (MTH520B)

Advanced Placement Calculus BC, Part 2 (MTH520B)

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Quick Overview

This course is the equivalent of an introductory college-level calculus course. Calculus helps scientists, engineers, and financial analysts understand the complex relationships behind real-world phenomena. Students learn to evaluate the soundness of proposed solutions and apply mathematical reasoning to real-world models. This course includes all the subjects in the AB course and several additional topics; thus, the pace is significantly more rigorous. This is the second semester of MTH520.

Teacher-Led Course (one-time payment)   $475.00

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Course Overview

This course is the equivalent of an introductory college-level calculus course. In this course, students study functions, limits, derivatives, integrals, and infinite series. Calculus helps scientists, engineers, and financial analysts understand the complex relationships behind real-world phenomena. Students learn to evaluate the soundness of proposed solutions and apply mathematical reasoning to real-world models. Students also learn to understand change geometrically and visually (by studying graphs of curves), analytically (by studying and working with mathematical formulas), numerically (by seeing patterns in sets of numbers), and verbally. Students prepare for the AP Exam and further studies in science, engineering, and mathematics. This is the second semester of MTH520.


Course Outline

SEMESTER ONE

Unit 1: The Basics

Students prepare to study calculus by reviewing some basic pre-calculus concepts from algebra and trigonometry. They learn what calculus is, why it was invented, and what it is used for.

  • Pre-Calculus Review
  • Introduction to Calculus
  • Using a Graphing Calculator
  • Combining Functions
  • Composite and Inverse Functions
  • Graphical Symmetry
  • Patterns in Graphs

Unit 2: Limits and Continuity

This unit addresses Topic I: Functions, Graphs, and Limits of the College Board's Calculus BC topic outline. Students learn two important concepts that underlie all of calculus: limits and continuity. Limits help students understand differentiation (the slope of a curve) and integration (the area inside a curved shape). Continuity is an important property of functions.

  • Finding Limits Analytically
  • Asymptotes as Limits
  • Relative Magnitudes for Limits
  • When Limits Do and Don't Exist
  • Continuity
  • Intermediate and Extreme Value Theorems

Unit 3: The Derivative

This unit addresses Topic II: Derivatives of the College Board's Calculus BC topic outline. Students learn how to calculate a derivative, the slope of a curve at a specific point. They learn techniques for finding derivatives of algebraic functions (such as y = x2) and trigonometric functions (such as y = sin x). Students also interpret the derivative as a rate of change and move fluidly between multiple representations including graphs, tables, and equations.

  • Slope and Change
  • Derivative at a Point
  • The Derivative
  • The Power Rule
  • Sums, Differences, Products, and Quotients
  • Graphs of Functions and Derivatives
  • Continuity and Differentiability
  • Rolles and Mean Value Theorems
  • Higher-Order Derivatives
  • Concavity
  • Chain Rule
  • Implicit Differentiation

Unit 4: Rates of Change

This unit focuses on Second Derivatives and Applications of Derivatives within Topic II: Derivatives of the College Board's Calculus BC topic outline. Students learn how to use calculus to model and analyze changing aspects of our world. In addition to the AB topics in this unit, BC students analyze polar and vector-valued functions.

  • Extrema
  • Optimization
  • Tangent and Normal Lines
  • Tangents to Polar Curves
  • Tangent Line Approximation
  • Rates and Derivatives
  • Related Rates
  • Rectilinear Motion
  • Motion with Vector Functions

Unit 5: The Integral, Part 1

This unit focuses on Topic III: Integrals in the College Board's Calculus BC topic outline. Students learn numerical approximations to definite integrals, interpretations and properties of definite integrals, the Fundamental Theorem of Calculus, and techniques of anti-differentiation. They learn how to find areas of curved shapes.

  • Riemann Sums
  • Area Approximations
  • The Definite Integral
  • Properties of Integrals
  • Graphing Calculator Integration
  • Applications of Accumulated Change
  • Antiderivatives
  • Composite Functions

SEMESTER TWO

Unit 1: The Integral, Part 2

This unit focuses on Topic III: Integrals in the College Board's Calculus BC topic outline. Students learn the Fundamental Theorem of Calculus, and techniques of anti-differentiation. They learn how to find areas of curved shapes.

  • The Fundamental Theorems of Calculus
  • Definite Integrals of Composite Functions
  • Analyzing Functions and Integrals

Unit 2: Applications of the Integral

This unit focuses on Topic III: Integrals in the College Board's Calculus BC topic outline. Students learn to use integrals and antiderivatives to solve problems. In addition to the AB topics, BC students learn to calculate arc length for a smooth curve.

  • Area Between Curves
  • More Areas and Averages
  • Volumes of Revolution
  • Cross Sections
  • Arc Length
  • More Rectilinear Motion
  • Other Applications of the Definite Integral

Unit 3: Inverse and Transcendental Functions

This unit focuses on Topic II: Derivatives and Topic III: Integrals in the College Board's Calculus BC topic outline. Students learn to calculate and use derivatives, antiderivatives, and integrals of exponential functions (such as y = 3x where the input variable is an exponent), logarithmic functions (the inverses of exponential functions), and trigonometric functions (such as y = secant x). In addition to the AB topics, BC students learn how to use L'Hopital's Rule and the methods of partial fractions and integration by parts. Also, students learn how to find improper integrals, and derivatives and integrals of parametric functions.

  • Derivatives of Inverses
  • Inverse Trigonometric Functions
  • Logarithmic and Exponential Review
  • Transcendentals and 1/x
  • Derivatives of Logarithms and Exponentials
  • L'Hopital's Rule
  • Analysis of Transcendental Curves
  • Integrating Transcendental Functions
  • Partial Fractions
  • Integration by Parts
  • Improper Integrals
  • Applications of Transcendental Integrals
  • Derivatives of Parametric Functions
  • Integrating Parametric and Polar Functions

Unit 4: Separable Differential Equations and Slope Fields

This unit focuses on Topic II: Derivatives of the College Board's Calculus BC topic outline, specifically, on Equations Involving Derivatives. Students investigate differential equations and solve the equations using a technique called "separating the variables." In addition to the topics covered in AB, BC students also learn to use Eulers method to estimate the solution of differential equations and use logistic equations to model growth.

  • Slope Fields
  • Differential Equations as Models
  • Euler's Method
  • Exponential Growth and Decay
  • Logistic Growth
  • More Applications of Differential Equations

Unit 5: Sequences and Series

This unit focuses on Topic IV: Polynomial Approximations and Series of the College Board's Calculus BC topic outline, specifically, on Series of Constants and Taylor Series.

  • Sequences
  • Series
  • Convergence Tests
  • Radius of Convergence
  • Functions Defined by Power Series
  • Taylor and Maclaurin Series
  • Taylor's Theorem and Lagrange Error

Unit 6: AP Exam Review and Final Exam

Students review what they have learned and become more familiar with AP-type questions in preparation for the AP Exam. Students are also provided with access to previously released AP Exams for practice.

  • Exam Strategies
  • Review of Topics
  • Practice Exams

Unit 7: Calculus Project

Teachers may choose to assign a final project.

  • Project Days

Additional Information

Course Length 4 Months
Prerequisites N/A
Course Materials No
Course Start Date

Courses Taught by a K12 Teacher

Courses with a teacher have designated start dates throughout Fall, Spring, and Summer. Full-year courses last 10 months and semester courses last 4 months. Courses are taught by teachers in K12 International Academy. For details on start dates, click here.

Teacher Assisted Yes, this course is taught by a K12 International Academy teacher. If you are looking for a teacher-supported option with additional flexibility and year-round start dates, click here to learn about the Keystone School, another K12 online private schooling option.
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To use this course, you'll need a computer with an Internet connection.  Some courses require additional free software programs, which you can download from the Internet.

Hardware and Browsers (Minimum Recommendations)

Windows OS

  • CPU: 1.8 GHz or faster processor (or equivalent)

  • RAM: 1GB of RAM

  • Browser: Microsoft Internet Explorer 9.0 or higher, Mozilla Firefox 10.0 versions or higher, Chrome 17.0 or higher

  • At this time our users are encouraged not to upgrade to Windows 10 or Edge (the new browser)

 

Mac OS

  • CPU: PowerPC G4 1 GHz or faster processor; Intel Core Duo 1.83 GHz or faster processor

  • RAM: 1GB of RAM

  • Browser: Firefox 10.0 versions or higher, Chrome 17.0 or higher (Safari is not supported!)

Internet Connections

It is highly recommended that a broadband connection be used instead of dial up.