**AP Calculus AB Course and Exam Description**

Kód kurzu:

**FLVS_APCAB**

Název kurzu:

**AP Calculus AB Online**

Délka kurzu:

2 semesters

Rok školní docházky / Grade:

10 - 13

Partner:

FLVS - Florida Virtual School Global

### Detailní popis

**AP Calculus AB** includes techniques and applications of the derivative, the definite integral, and the Fundamental Theorem of Calculus. It is equivalent to at least a semester of calculus at most colleges and universities.

This online course offers a combination of assessment and instruction in an online environment containing but not limited to the areas of functions, functions and limits, differential calculus, and integral calculus. The course applies differential calculus to finding the slope of a curve, solving problems with related rates, calculating motion properties of moving particles, etc. It then applies integral calculus to finding the areas of irregular regions in a plane, volumes of rotation by various methods, and other scientific applications.

The purpose of this course is to provide students with a deep understanding of the concepts of calculus in order to prepare them for the AP exam and for further college and university calculus courses.

**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 AB** is designed to be the equivalent of a first semester college calculus course devoted to topics in differential and integral calculus.

**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.

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 také témata, která se běžně nevyučují na našich středních školách, např. diferenciální rovnice. Kurz AP Calculus AB 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, such as differential equations. The AP Calculus AB 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**

**Semester 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

**Segment Two**

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

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 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
- Finding Solutions Using Separation of Variables
- Exponential 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

### Sylabus kurzu

### AP Calculus AB Course and Exam Description

### Materiály

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

Description | Number | Type | Source |
---|---|---|---|

Graphing Calculator | Tool | NOT included in the price of the course | |

Study Forge | Software | included in the price of the course | |

### Cena

cena kurzu: 19 300,- Kč / 811,- EUR

### Zkušenosti studentů

First of all, I would like to thank for the opportunity to take this course. I have learned things that I would never learn in my school. It helped me a lot throughout this year and made it possible to take my AP exam without any trouble. I will definitely remember and revise occasionally what I learned in this course. Overall, I am very happy with this course and I will recommend it to anyone who is thinking about taking AP exams.

**Dominik H., 2021**

This course introduced me to calculus, even thou I might not be able to pass the final test completely, I at least grasped the overall knowledge of calculus and I am thankful for that. This course has been a ride for me and I make out of the things I learned.

**Samuel S., 2021**

The course gave me a great insight of the topics, that were discussed. The most important source of information were for me the videos, which had a great quality. It took me a lot of time, but I definitely enjoyed it and I think I now have really strong foundation in math on university level.

**Kryštof P., 2021**