www.ctm-academy.cz
datum: 02.12.2024

Kód kurzu:

FLVS_APCBC

Název kurzu:

AP Calculus BC Online

Délka kurzu:

2 semesters

Rok školní docházky / Grade:

10 - 13

Partner:

FLVS - Florida Virtual School Global

Detailní popis

AP Calculus BC includes all topics in AP Calculus AB, as well as additional topics, such as differential and integral calculus (including parametric, polar, and vector functions) and series. It is equivalent to at least one year of calculus at most colleges and universities. AP Calculus BC is an extension of AP Calculus AB, and each course is challenging and demanding and requires a similar depth of understanding of topics.

AP Calculus AB and AP Calculus BC focus on students’ understanding of calculus concepts and provide experience with methods and applications. Through the use of big ideas of calculus (e.g., modeling change, approximation and limits, and analysis of functions), each course becomes a cohesive whole, rather than a collection of unrelated topics. Both courses require students to use definitions and theorems to build arguments and justify conclusions. The courses feature a multi-representational approach to calculus, with concepts, results, and problems expressed graphically, numerically, analytically, and verbally. Exploring connections among these representations builds understanding of how calculus applies limits to develop important ideas, definitions, formulas, and theorems. A sustained emphasis on clear communication of methods, reasoning, justifications, and conclusions is essential. Teachers and students should regularly use technology to reinforce relationships among functions, to confirm written work, to implement experimentation, and to
assist in interpreting results.

AP Calculus BC is designed to be the equivalent to both first and second semester college calculus courses. AP Calculus BC applies the content and skills learned in AP Calculus AB to parametrically defined curves, polar curves, and vector-valued functions; develops additional integration techniques and applications; and introduces the topics of sequences and series.

 RECOMMENDED PREREQUISITES
Before studying calculus, all students should complete the equivalent of four years of secondary mathematics designed for college-bound students: courses that should prepare them with a strong foundation in reasoning with algebraic symbols and working with algebraic structures. Prospective calculus students should take courses in which they study algebra, geometry, trigonometry, analytic geometry, and elementary functions. These functions include linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric and piecewise-defined functions. In particular, before studying calculus, students must be familiar with the properties of functions, the composition of functions, the algebra of functions, and the graphs of functions. Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and descriptors such as increasing and decreasing). Students should also know how the sine and cosine functions are defined from the unit circle and know the values of the trigonometric functions at the numbers 0, π/6, π/4, π/3, π/2, and their multiples. Students who take AP Calculus BC should have basic familiarity with sequences and series, as well as some exposure to parametric and polar equations.

Study materials (e-books, Discovery Education, etc. ...) for FLVS Global courses are INCLUDED in the price of the course.

Calculus je důležitým a nezbytným nástrojem nejen pro matematiky. Bez porozumění funkcím a jejich vlastnostem, diferenciálnímu a integrálnímu počtu se jen těžko studuje také fyzika, některé části chemie, ekonomie, ale také třeba biologie. Calculus je tak skutečně základním stavebním kamenem pro vybudování dalších, nejen matematických disciplín. V našem kurzu provedeme studenta světem funkcí a jejich vlastností, limitami, derivacemi i jejich aplikacemi. V kurzu jsou obsažena témata, která se běžně nevyučují na našich středních školách. Student se tak seznámí s tématy diferenciální rovnice, nekonečné řady a jejich konvergence, nevlastní integrál nebo polární souřadnice. Kurz AP Calculus BC je přípravou na AP Exam. Studenti se seznámí s konceptem AP exam a nacvičí si typové úlohy. Naši absolventi jsou na tuto náročnou zkoušku výborně připraveni.

RNDr. Ing. Jana Kalová, PhD., instruktorka CTM Online kurzů


Calculus is an important and essential tool not just for mathematicians. Without understanding of functions and their properties, derivatives and integrals, it is very difficult to conduct further studies in Physics, certain parts of Chemistry, Economy or even Biology. Calculus is therefore truly the foundation for developing other, not only mathematical disciplines. In this course we will guide the student through the world of functions and their properties, limits, derivatives and their applications. The course also contains topics, which are not commonly parts of the Czech high school curriculum. Students will therefore encounter topics such as differential equations, infinite series and their convergence, improper integrals or polar co-ordinates. The AP Calculus BC course is also a preparation for an AP exam. Students will be introduced to the AP exam concept and they will practice relevant types of questions. Our alumni will become thoroughly prepared for this challenging examination.

RNDr. Ing. Jana Kalová, PhD., CTM Online instructor

 

Struktura kurzu

Study Scope and Sequence

Segment One

Module 01 - Limits and Continuity

  • Using Limits to Analyze Instantaneous Change
  • Estimating Limit Values from Graphs and Tables
  • Determining Limits Using Algebraic Properties and Manipulation
  • Selecting Procedures for Determining Limits
  • Squeeze Theorem and Representations of Limits
  • Determining Continuity and Exploring Discontinuity
  • Connecting Limits, Infinity, and Asymptotes
  • The Intermediate Value Theorem (IVT)

Module 02 - Differentiation: Definition and Fundamental Properties

  • Average and Instantaneous Rates of Change and the Derivative Definition
  • Determining Differentiability and Estimating Derivatives
  • Derivative Rules: Constant, Sum, Difference, Constant Multiple, and Power
  • The Product Rule and the Quotient Rule
  • Derivatives of Trigonometric Functions
  • Derivatives of Exponential and Logarithmic Functions

Module 03 - Differentiation: Composite, Implicit, and Inverse Functions

  • The Chain Rule
  • Implicit Differentiation
  • Differentiating Inverse Functions
  • Differentiating Inverse Trigonometric Functions
  • Selecting Procedures for Calculating Derivatives
  • Calculating Higher-Order Derivatives

Module 04 - Contextual Applications of Differentiation

  • Interpreting and Applying the Derivative in Motion
  • Rates of Change in Applied Contexts Other Than Motion
  • Related Rates
  • Approximating Values of a Function Using Local Linearity and Linearization
  • L\'Hospital\'s Rule

Module 05 - Analytical Applications of Differentiation

  • Mean Value and Extreme Value Theorems
  • Determining Function Behavior and the First Derivative Test
  • Using the Candidates Test to Determine Absolute Extrema
  • Determining Concavity of Functions and the Second Derivative Test
  • Connecting Graphs of Functions and Their Derivatives
  • Optimization Problems
  • Exploring Behaviors of Implicit Relations

Segment Two

Module 06 - Integration and Accumulation of Change

  • Exploring Accumulations of Change
  • Riemann Sums and the Definite Integral
  • Accumulation Functions Involving Area and the Fundamental Theorem of Calculus
  • Applying Properties of Definite Integrals
  • Finding Antiderivatives and Indefinite Integrals
  • Integrating Using Substitution
  • Integrating Using Integration by Parts
  • Integrating Using Linear Partial Fractions
  • Evaluating Improper Integrals
  • Integrating Functions Using Long Division and Completing the Square
  • Selecting Techniques for Antidifferentiation

Module 07 - Differential Equations

  • Solutions of Differential Equations
  • Sketching and Reasoning Using Slope Fields
  • Approximating Solutions Using Euler\'s Method
  • Finding Solutions Using Separation of Variables
  • Exponential Models with Differential Equations
  • Logistic Models with Differential Equations

Module 08 - Applications of Integration

  • Average Value and Connecting Position, Velocity, and Acceleration Using Integrals
  • Using Accumulation Functions and Definite Integrals in Applied Contexts
  • Finding the Area Between Curves
  • Finding the Area Between Curves That Intersect at More Than Two Points
  • Volumes with Discs
  • Volumes with Washers
  • Volumes with Cross Sections
  • The Arc Length of a Smooth, Planar Curve and Distance Traveled

Module 09 - Parametric, Polar, and Vector-Valued Equations

  • Differentiating Parametric Equations and Finding Arc Length
  • Differentiating and Integrating Vector-Valued Functions
  • Solving Motion Problems Using Parametric and Vector-Valued Functions
  • Defining Polar Coordinates and Differentiating in Polar Form
  • Finding Area Bounded by Polar Curves

Module 10 - Infinite Sequences and Series

  • Convergent and Divergent Infinite Series and Geometric Series
  • Integral Test for Convergence, Harmonic Series, and p-Series
  • Comparison Tests for Convergence
  • Additional Tests to Determine Convergence
  • Alternating Series and Their Error Bound
  • Taylor Polynomial Approximations of Functions and Evaluating Error
  • Radius and Interval of Convergence of Power Series
  • Finding Taylor or Maclaurin Series for a Function
  • Representing Functions as Power Series

 

 

 

 

Sylabus kurzu

Materiály

 

 

K vašemu kurzu potřebujete tyto učebnice a studijní pomůcky (pokud ke kurzu potřebujete laboratorní sadu, zkuste si nejdříve dohodnout možnost využívání školní laboratoře):

DescriptionNumberTypeSource
Graphing CalculatorToolNOT included in the price of the course
Study ForgeSoftwareincluded in the price of the course

Cena

cena kurzu: 19 300,- Kč / 811,- EUR

Zkušenosti studentů

Díky CTM Online programu jsem měl možnost během středoškolského studia absolvovat online kurzy matematiky zaštítěné CTY při univerzitě Johnse Hopkinse v USA. Ve druháku jsem začal kurzem Honors Pre-Calculus, kde jsem se naučil myslet matematicky a hledat za každým vzorcem jeho původ a pravý význam. Ve třeťáku jsem pak pokračoval kurzem AP Calculus BC, který mi dal velmi solidní základy diferenciálního a integrálního počtu, na kterých jsem mohl s přehledem stavět při studiu na University of Aberdeen, na níž jsem ve studiu matematiky pokračoval. Během studia těchto kurzů jsem začal mít matematiku skutečně rád. Profesor Edward Burger, který připravoval videa do těchto kurzů, je jedním z nejlepších učitelů matematiky v USA. A snad právě díky němu jsem začal mít matematiku opravdu rád a posléze ji i začal studovat na vysoké škole. Ukázal mi, že je to svět zábavy, svět hledání pravdy, svět, kde všechno dává smysl.


Velice důležitou součástí této mé zkušenosti byla paní profesorka Jana Kalová. Byla mojí instruktorkou – pomáhala mi pochopit, co jsem nepobíral, pomáhala mi pravidelně na kurzech pracovat a často mi i psala různé zajímavosti navíc, díky čemuž jsem poznal, že skvělí, milí a nápomocní matematici jsou všude po světě. Během svého středoškolského studia jsem měl přístup k mnoha zdrojům vzdělání - absolvoval jsem různé stáže, psal SOČku, učil se pečlivě do některých předmětů - a všechny byly přínosné, ale CTM Online kurzy rozhodně nejpřínosnější! Doporučil bych je každému a kdykoli!

---

Thanks to the CTM Online programme, I had the opportunity to study online Maths courses, made by the CTY Johns Hopkins University, while I was still at the Bishop Grammar School in Žďár nad Sázavou. I started with the Honors Pre-Calculus course and then continued by AP Caculus BC. These courses taught me what is the mathematical thinking and how to think about maths problems. It also helped me to develop a wide foundation on which I was building during my studies at the University of Aberdeen, where are continued my studies of mathematics later on. In AP Calculus BC, the concepts of integration and differentiation were amazingly explained by professor Edward Burger, one of the best teachers of mathematics in the USA. But it was not just professor Burger who showed my what a joy doing maths is.

Professor Jana Kalova, my supervisor in this programme, was helping me to understand what I was not able to grasp yet, to help me keep pace with the course tempo and to motivate me when I felt it is too much work for me. During my high school studies, I was using many sources to learn maths – I was taking part in internships, reading books, watching interesting videos on YouTube – but nothing was as beneficial as doing these CTM Online courses. I am eternally grateful to everyone who gave me this opportunity and I would recommend this courses to anyone anytime.

Daniel Mužátko, student CTM, nyní student University of Aberdeen